##
**Fuzzy mathematical models in engineering and management science.**
*(English)*
Zbl 0683.90024

Amsterdam etc.: North-Holland. xxiii, 338 p. $ 100.00; Dfl. 190.00 (1988).

The first part of the book (comprising about half of its contents) deals with theoretical aspects of the fuzzy set theory, especially fuzzy numbers: Chapter 2 presents basic notions of the fuzzy set theory. - In chapter 3 fuzzy numbers are treated, with emphasis on the so-called triangular and trapezoidal fuzzy numbers. In the following chapters the authors confine themselves in fact to using only fuzzy numbers of these types. - In chapter 4 a proposal for ordering fuzzy numbers is presented. It consists in ordering fuzzy numbers according to a certain indicator connected with the number, called “an ordinary representative” of the fuzzy number. If some fuzzy numbers are indistinguishable in relation to this indicator, it is suggested to use the mode criterion and the divergence one as further criteria for ordering. - In chapter 5 several measures of inaccuracy of a fuzzy number are suggested. - The next chapter is concerned with the problem of an approximation of the results of the following operations of one or two arguments carried out on triangular fuzzy numbers by triangular fuzzy numbers: A(\(\cdot)B\), \(A^{-1}\), A(:)B, ln(A), exp(A), \(A^ n\). - Chapter 7 deals with the problem of solving equations of the form \(A(+)X=C\) and \(A(\cdot)X=C\), in which A and C are triangular fuzzy numbers. - The subsequent chapter contains a discussion about t-norms and t-conorms. - In chapter 9 fuzzy numbers on the interval [0,1] and generalized \(\wedge\) and \(\vee\) operations on these numbers are treated. - Chapter 10 is concerned among other things with the equations \(A(\vee)X=C\) and \(A(\wedge)X=C\), in which A and C stand for triangular fuzzy numbers on the interval [0,1].

The second part of the book discusses various known mathematical models and methods connected with them which are used in the engineering and management science. They are: the so-called zero-base budgeting method, the Delphi method for forecasting, the decounting method, smoothing (filtering) of data, some problems related to reliability modeling and evaluation, the critical path method, some optimization models connected to the investment problem, a model and solution method for the transportation problem.

In each of the considered cases the authors apply an identical approach. For a known classical model and/or method they propose to replace numerical data and operations carried out on them by fuzzy numbers and generalized fuzzy operations on these numbers, respectively. They illustrate the effect of such a substitution each time by carrying out the calculations for simple examples of the models. In this way they put emphasis unfortunately only on the numerical side of the problems, without giving much attention to the question of an interpretation of the fuzzy models and their usefulness in practice. The fact that the book is overweighted with examples of calculations constitutes one of its main defects (this applies to both parts of the book). The calculations, presented in a detailed way, carried out on simple examples (the reader is not capable of following all these calculations - and, as a matter of fact, this would be of no use to him), will occupy more than half of the book! For the reasons mentioned above the title of the book seems to be exaggerated and does not correspond very exactly to the contents of the book.

The book contains a lot of errors and mistakes which may cause some difficulties, especially to a reader who is not familiar with the elements of fuzzy set theory. I will mention only two of these errors as examples. The first one: The \(\alpha\)-cut of any fuzzy number A for \(\alpha =0\) is equal to the whole space of real numbers, i.e. \(A_ 0=R\). The authors assume in the whole book, however, that \(A_ 0\) equals the closure of the support of A (i.e. the closure of the set of x’s for which \(\mu (x)>0)\). The second error: an arbitrary fuzzy set, and not only a convex one, possesses the property of nesting \((\alpha '<\alpha)\Rightarrow (A_{\alpha}\subseteq A_{\alpha '})!\)

In conclusion I would like to give some attention to the approach to solving the transportation problem with fuzzy delivery unit costs as proposed by the authors. The authors should have provided a justification for the fact that the stepping stone method really leads to the best solution in the sense of the earlier presented relation of order of fuzzy numbers. This proof would have been very easy. It follows almost immediately from the following properties of the notion “ordinary representative of the fuzzy number”:

(1) \(o.r.f.n.(A(+)B)=o.r.f.n.(A)=o.r.f.n.(B)\) and

(2) \(o.r.f.n.(kA)+k*o.r.f.n.(A)\) for \(k>0.\)

However, the same properties imply that, in order to obtain the desired solution, it suffices to use the stepping stone method with ordinary numbers, replacing in the transportation problem model the fuzzy costs with their ordinary representatives. If the solution is a unique one, it is the best one. If more solutions have been obtained, one of them should be selected by applying successively the mode criterion and the divergence criterion for total transportation costs. In view of the above comments the remark (3) presented on page 267 becomes irrelevant and the method of solving the transportation problem with fuzzy costs can be simplified considerably.

The second part of the book discusses various known mathematical models and methods connected with them which are used in the engineering and management science. They are: the so-called zero-base budgeting method, the Delphi method for forecasting, the decounting method, smoothing (filtering) of data, some problems related to reliability modeling and evaluation, the critical path method, some optimization models connected to the investment problem, a model and solution method for the transportation problem.

In each of the considered cases the authors apply an identical approach. For a known classical model and/or method they propose to replace numerical data and operations carried out on them by fuzzy numbers and generalized fuzzy operations on these numbers, respectively. They illustrate the effect of such a substitution each time by carrying out the calculations for simple examples of the models. In this way they put emphasis unfortunately only on the numerical side of the problems, without giving much attention to the question of an interpretation of the fuzzy models and their usefulness in practice. The fact that the book is overweighted with examples of calculations constitutes one of its main defects (this applies to both parts of the book). The calculations, presented in a detailed way, carried out on simple examples (the reader is not capable of following all these calculations - and, as a matter of fact, this would be of no use to him), will occupy more than half of the book! For the reasons mentioned above the title of the book seems to be exaggerated and does not correspond very exactly to the contents of the book.

The book contains a lot of errors and mistakes which may cause some difficulties, especially to a reader who is not familiar with the elements of fuzzy set theory. I will mention only two of these errors as examples. The first one: The \(\alpha\)-cut of any fuzzy number A for \(\alpha =0\) is equal to the whole space of real numbers, i.e. \(A_ 0=R\). The authors assume in the whole book, however, that \(A_ 0\) equals the closure of the support of A (i.e. the closure of the set of x’s for which \(\mu (x)>0)\). The second error: an arbitrary fuzzy set, and not only a convex one, possesses the property of nesting \((\alpha '<\alpha)\Rightarrow (A_{\alpha}\subseteq A_{\alpha '})!\)

In conclusion I would like to give some attention to the approach to solving the transportation problem with fuzzy delivery unit costs as proposed by the authors. The authors should have provided a justification for the fact that the stepping stone method really leads to the best solution in the sense of the earlier presented relation of order of fuzzy numbers. This proof would have been very easy. It follows almost immediately from the following properties of the notion “ordinary representative of the fuzzy number”:

(1) \(o.r.f.n.(A(+)B)=o.r.f.n.(A)=o.r.f.n.(B)\) and

(2) \(o.r.f.n.(kA)+k*o.r.f.n.(A)\) for \(k>0.\)

However, the same properties imply that, in order to obtain the desired solution, it suffices to use the stepping stone method with ordinary numbers, replacing in the transportation problem model the fuzzy costs with their ordinary representatives. If the solution is a unique one, it is the best one. If more solutions have been obtained, one of them should be selected by applying successively the mode criterion and the divergence criterion for total transportation costs. In view of the above comments the remark (3) presented on page 267 becomes irrelevant and the method of solving the transportation problem with fuzzy costs can be simplified considerably.

Reviewer: S.Chanas

### MSC:

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90B25 | Reliability, availability, maintenance, inspection in operations research |

90B50 | Management decision making, including multiple objectives |

03E72 | Theory of fuzzy sets, etc. |

90C08 | Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) |

93E11 | Filtering in stochastic control theory |