For the intrinsic complexity of multi-level programming problems, metaheuristic algorithms, such as genetic algorithm (GA), particle swarm optimization (PSO) and tabu search, have been used to solve the problems.

Artificial neural networks or, simply, neural networks (NNs) have been widely developed in solving mathematical programming (MP) or optimization problems.

Auction mechanism has been applied to distributed resource scheduling problems in recent years. The demand for distributed resource scheduling is more evident in manufacturing systems with complex supply structure and where customer service is prioritized (Kutanoglu and Wu 1999).

Fuzzy set theory is a powerful tool for representing vague phenomena and for linguistic representation in modern computers. The basic concept has been applied in many different areas. For the convenience of later discussions, some of the fuzzy concepts are summarized in this chapter. However, we shall not discuss the most basic aspects and assume the reader has some familiarity with fuzzy set theory. For more details, the reader can consult the many excellent books on this subject.

Traditional MLP concerns decentralized planning problems with multiple DMs in a multi-level or hierarchical organization where decisions interact with one another (Wen and Hsu in J Oper Res Soc 42:125–133 1991). Motivated by the concept of dynamic programming (DP) and the operation of fuzzy dynamic programming (Bellman in Dynamic programming. Princeton University Press, Princeton, New Jersey 1957; Kacprzyk in Multistage decision-making under fuzziness theory and applications. Verlag TUV Rheinland, Koln 1983), this chapter extends the decentralized planning problems to a temporal fashion where the hierarchical decisions can be made stage by stage.

Although the various approaches proposed for fuzzy multi-level programming in the previous chapter are very powerful, it is only a start in this fruitful direction. Further developments are needed to consider other aspects and thus to make the actual solution process more practical in the sense of solving more realistic problems with reduced computation. Two of the most important aspects which need to be considered are the problem of aggregation of the fuzzy representation and the representation of systems whose description is vague and not well defined. The first aspect is concerned with the basic problem of fuzzy combination, where, in the previous chapter, only the non-compensatory min is used. The second aspect is concerned with the handling of vague systems. Notice that the problem representing the system at each division or each level in the previous chapter is crisp, not fuzzy. The fuzziness comes from the mutual interaction or mutual competition of the different decision-makers in each level or each division. In this chapter, fuzzy systems represented by fuzzy parameters or fuzzy coefficients, not just fuzzy objectives and fuzzy decisions, will be considered.

I was fortunate to have the opportunity to contact with Professor Zadeh and fuzziness in the early 1970’s. The frequent contacts, encouragements as well as the directions personally from Professor Zadeh became a continuous inspiration to my work. It is difficult to summarize and to re-call all the contacts, developments and meetings on fuzziness through the years but, at this remarkable point of time, I would like to congratulate the successful developments and applications on fuzziness during these past nearly 50 years and celebrate on Professor Zadeh’s 90th anniversary.

This study applies the Analytic Network Process (ANP) to forecast the sales volume of printers in Taiwan for adjusting the recycling and treatment fee as an incentive for recycling industries. When historical data are lacking and when a broad spectrum of social impact is involved, the ANP, with the capacity to manage dependence and feedback among the factors, can serve as a tool to forecast outcomes by using expert judgment. The priorities derived from numerical judgment are similar to probabilities. They are obtained from the limit supermatrix of the ANP that represents forecasts for the next period. The result of back testing has shown that the ANP's percentage error is small compared with those of some naïve statistical techniques. Sensitivity analysis is also made to ensure robustness of the model. Finally, the characteristic strengths of the Analytic Hierarchy Process (AHP) and ANP in forecasting are discussed to simplify their use in future applications.

A production and inventory control problem is solved by a modified computational dynamic programming procedure. Specifically a two dimensional problem described by differential system equations with an integral objective function is solved by the proposed scheme. The numerical results are examined for different number of stages. The advertising rate, inventory, sales, and profit are illustrated through tables and diagrams.

In this paper, the construction and approximation problem of a single input and single output (SISO) fuzzy system with normal implication and center-of-gravity defuzzifier is discussed. First, the method of a non-singleton fuzzifier for the input variable and the concept of an adaptive universe for the output fuzzy set are proposed. Then, by using this method and this concept of an adaptive universe, SISO fuzzy systems based on center-of-gravity defuzzifier and normal implications such as the Kleene–Dienes implication or the Lukasiewicz implication are constructed. The constructed fuzzy systems have the general form S¯(x)=Ai∗(x)f(xi)+Ai+1∗(x)f(xi+1), with Ai∗(x)+Ai+1∗(x)=1, and, furthermore, they are universal approximators. The sufficient conditions for the proposed fuzzy systems to be universal approximators are also obtained. To illustrate the universal property, an example is also given.

In this paper, it is investigated how to sequence jobs with fuzzy processing times and predict their due dates on a single machine such that the total weighted possibilistic mean value of the weighted earliness–tardiness costs is minimized. First, an optimal polynomial time algorithm is put forward for the scheduling problem when there are no precedence constraints among jobs. Moreover, it is shown that if general precedence constraints are involved, the problem is NP-hard. Then, four reduction rules are proposed to simplify the constraints without changing the optimal schedule. Based on these rules, an optimal polynomial time algorithm is proposed when the precedence constraint is a tree or a collection of trees. Finally, a numerical experiment is given.

In this paper, we consider minimizing multiple linear objective functions under a max-t-norm fuzzy relational equation constraint. Since the feasible domain of a max–Archimedean t-norm relational equation constraint is generally nonconvex, traditional mathematical programming techniques may have difficulty in yielding efficient solutions for such problems. In this paper, we apply the two-phase approach, utilizing the min operator and the average operator to aggregate those objectives, to yield an efficient solution. A numerical example is provided to illustrate the procedure.

Cellular manufacturing is a useful way to improve overall manufacturing performance. Group technology is used to increase the productivity for manufacturing high quality products and improving the flexibility of manufacturing systems. Cell formation is an important step in group technology. It is used in designing good cellular manufacturing systems. The key step in designing any cellular manufacturing system is the identification of part families and machine groups for the creation of cells that uses the similarities between parts in relation to the machines in their manufacture. There are two basic procedures for cell formation in group technology. One is part-family formation and the other is machine–cell formation. In this paper, we apply a fuzzy relational data clustering algorithm to form part families and machine groups. A real data study shows that the proposed approach performs well based on the grouping efficiency proposed by Chandrasekharan and Rajagopalan.

We study two deferent concepts of semicontinuity of fuzzy mappings by establishing characterizations of these fuzzy mappings. Relationships between semicontinuity and continuity of fuzzy mappings are explored. Some basic properties of these fuzzy mappings are presented and proved.

In current PC computing environment, the fuzzy clustering method based on perturbation (FCMBP) is failed when dealing with similar matrices whose orders are higher than tens. The reason is that the traversal process adopted in FCMBP is exponential complexity. This paper treated the process of finding fuzzy equivalent matrices with smallest error from an optimization point of view and proposed an improved FCMBP fuzzy clustering method based on evolutionary programming. The method seeks the optimal fuzzy equivalent matrix which is nearest to the given fuzzy similar matrix by evolving a population of candidate solutions over a number of generations. A new population is formed from an existing population through the use of a mutation operator. Better solutions survive into next generation and finally the globally optimal fuzzy equivalent matrix could be obtained or approximately obtained. Compared with FCMBP, the improved method has the following advantages: (1) Traversal searching is avoided by introducing an evolutionary programming based optimization technique. (2) For low-order matrices, the method has much better efficiency in finding the globally optimal fuzzy equivalent matrix. (3) Matrices with hundreds of orders could be managed. The method could quickly get a more accurate solution than that obtained by the transitive closure method and higher precision requirement could be achieved by further iterations. And the method is adaptable for matrices of higher order. (4) The method is robust and not sensitive to parameters.

Although there have been many researches on cluster analysis considering feature (or variable) weights, little effort has been made regarding sample weights in clustering. In practice, not every sample in a data set has the same importance in cluster analysis. Therefore, it is interesting to obtain the proper sample weights for clustering a data set. In this paper, we consider a probability distribution over a data set to represent its sample weights. We then apply the maximum entropy principle to automatically compute these sample weights for clustering. Such method can generate the sample-weighted versions of most clustering algorithms, such as k-means, fuzzy c-means (FCM) and expectation & maximization (EM), etc. The proposed sample-weighted clustering algorithms will be robust for data sets with noise and outliers. Furthermore, we also analyze the convergence properties of the proposed algorithms. This study also uses some numerical data and real data sets for demonstration and comparison. Experimental results and comparisons actually demonstrate that the proposed sample-weighted clustering algorithms are effective and robust clustering methods.

In this paper, a weight selection procedure in the W-k-means algorithm is proposed based on the statistical variation viewpoint. This approach can solve the W-k-means algorithm’s problem that the clustering quality is greatly affected by the initial value of weight. After the statistics of data, the weights of data are designed to provide more information for the character of W-k-means algorithm so as to improve the precision. Furthermore, the corresponding computational complexity is analyzed as well. We compare the clustering results of the W-k-means algorithm with the different initialization methods. Results from color image segmentation illustrate that the proposed procedure produces better segmentation than the random initialization according to Liu and Yang’s (1994) evaluation function.

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.