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The chapter reviews and extends the theory of exact and superlative index numbers. Exact index numbers are empirical index number formula that are equal to an underlying theoretical index, provided that the consumer has preferences that can be represented by certain functional forms. These exact indexes can be used to measure changes in a consumer's cost of living or welfare. Two cases are considered: the case of homothetic preferences and the case of nonhomothetic preferences. In the homothetic case, exact index numbers are obtained for square root quadratic preferences, quadratic mean of order r preferences, and normalized quadratic preferences. In the nonhomothetic case, exact indexes are obtained for various translog preferences.

A simple adaption of Dijkstra's algorithm for finding a minimum weight path on a weighted directed graph serves to solve the optimal control problem posed by Arbib for tolerance automata with cost function. Cumulative cost automata and the application of network theory to cost automata are also discussed.

In this paper, we define and examine the concept of a fuzzy recognizer. If L(M) is the language recognized by an incomplete fuzzy recognizer M, we show that there is a completion M of M such that L(M) = L(M). We also show that if A is a recognizable set of words, then there is a complete accessible fuzzy recognizer M_A such that L(M_A) = A. We lay groundwork to determine rational decompositions of recognizable sets.

Part I, presented in the preceding issue of Kybernetes, concluded with the presentation of a theorem in the form of an algorithm to aid in the modeling of the inherited ability of an automaton to modify and extend its knowledge structures. Part II continues with the presentation of the proof of the theorem, and the theorem is applied to a case study to determine the states of an automaton when the states could not be observed during a period of performance. The application of the disciplines of the General Systems Theory is further indicated for a second case study of an industrial system of 10 interactive automata.

Studies the concept of the Cartesian composition of fuzzy finite state machines. Shows that fuzzy finite state machines and their Cartesian composition share many structural properties. Some of these properties are singly generated; retrievability, connectedness, strong connectedness, commutativity, perfection and state independence. Thus a fuzzy finite state machine which is a Cartesian composition of submachines can be studied in terms of smaller machines.

The restructuring of shipping and shipbuilding companies in the midst of rapidly shrinking global shipping demand has become a prominent issue in Korea. In shipping finance, loan syndication featuring many creditors surges as the preferred option. However, increasing the numbers of creditors in the syndicate results in two opposite effects. First is the beneficial effect from their enhanced monitoring power. On the other hand, there is the adverse effect resulting from increased difficulty in coordination when syndicate members increase, particularly in bankruptcy. Our aim of this paper is to analyze the role of finance in the shipping and shipbuilder markets, and determine the theoretical optimal number of creditors for the shipping finance syndicate based on Bolton and Scharfstein (1996). The two issues above result from moral hazard and non-verifiability: coordination among many creditors for collection of bonds in case of default, and the enhancement of monitoring private benefit exploitation by the ship-owner during default. Considering the two conflicting forces result from an increase in creditor membership, we draw conclusions on determining the optimal number of creditors by considering trade-offs between these two factors: More creditors are preferred when the monitoring effect dominates. Otherwise, less creditors are preferred.

The analysis of the wing/fuselage and fuselage/tail unit interaction forces is extended to cover the case when the attached component is more conveniently analysed by the Matrix Displacement Method. The flexibility matrix of the complete aircraft, supported on the wing/fuselage attachment points, follows from the results derived in this and previous sections and takes into account the elastic interaction between the various components. The dynamical matrix of the complete free aircraft is set up and for completeness the theory and properties of the normal modes of vibration are given. A final sub‐section discusses some points of detail in the mass distribution and the definition of the forces on the aircraft.

The rapprochement suggested by Arbib between continuity in control theory and tolerance in the theory of automata is discussed with reference to orbits, and in particular to the proof of Takens' version of Zeeman's tolerance stability conjecture. The corresponding result for tolerance automata is established.

Defines the order preservation and operator uniqueness assumption and the active assumption. Under these assumptions, presents the invariance of optimum solution in a multi‐objective fuzzy environment and gives illustrative examples. Presents applications, including in the heat transfer area, illustrated by numerical results and three dimensional figures. The results may have useful applications in fuzzy languages, fuzzy logics and other related areas.

This article considers the effect of freight discounts and disposals on the inventory policies of a profit‐maximising firm with price‐elastic demand and with price as a mark‐up on the acquisition cost. In addition, the cost of placing an order is in part dependent upon a freight‐cost structure, sensitive to the size of the order and upon the possibility of selling unwanted inventory at a lower price. A numerical example is used to illustrate the resulting inventory policies.