The forest production greatly relies on nature and variations of the natural conditions related to the production are not small. Then, because of the long production time span, it is necessary to concider the forest management planning as a planning under uncertainty. The purpose of this paper is to present a forest management planning by coordinating the uncertainty and the efficiency of forest production. The theory of stochastic programming is applied. The forest management planning consists of two phases: One is long term and the other is short term. This paper is concerned with the long term planning such as from 60 to 80 years. This planning is how to derive the present forest condition to the objective forest condition and this is one of the most basic requirements in forest management planning. In this paper, the objective forest stand condition is the normal age-class arrangement. A planning method applied here is to minimize the variance of forest yield volume under the condition of keeping its expectation arbitrarily fixed, and the other planning method adopts one of the von Neumann-Morgenstern utility functions. They are formulated as a quadratic programming problem. The methods are applied to a real forest management planning and the unique optimal plan is provided for this case. Our models will be significant in the planning at the divisional forest-office in the national forest.

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Recently the importance of physical distribution system is emphasized, and Various research about it are studied. When comparing alternative physical distribution sytems on viewpoint of profitability, the distance of driving of trucks is one of the factors that mostly complicate the problem. When the total distance of driving required for carrying given goods is changed, the number of truck, driver and helper, etc., is changed too. It greatly influences the total cost of physical distribution system. On the other hand, because the total distance of driving is changeable by the other factors, such as the demand of goods, the period of supply, the size of truck, the way of dividing the delivery route of trucks, etc., it is rather difficult to infer it in advance theoretically. If it is able to infer the distance of driving simply with allowable error before adopt a new physical distribution system into practice, it is very convenient for examining the optimal physical distribution system. The purpose of this paper is to derive some approximate formula from which we could infer the distance of driving in advance, before adopted alternative physical distribution systems into practice. At first, the consideration of the shortest route on some typical models, such as radial type, join-2-points type, circle type, straight-line type, etc., instead of the delivery route of real world network has helped us to find out the common characteristrics about the distance of driving. Secondly, some approximate formula of the distance of driving on the real world network has been studied out by means of 'Go' Board model. Based on the experimental results with maps and actual data from one soft-drinking company, the degree of error of the approximate formula has been examined.

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In studying complex problems, in developing plans, in managing organizations, in working with systems and in various other kinds of human endeavor, it is often desirable and sometimes essential to synthesize hierachies. The process of arranging elements in a hierachy is usually dealt with intuitively. On the other hand, the concept of fuzzy sets gives an important mathematical clue for an approach to studies of such systems with no sharp boundaries. On the basis of fuzzy sets theory, we propose a method for structuring hierachy for the several problems mentioned above, that is Fuzzy Structural Modeling (FSM) method. An important requirement for structural modeling of complex systems is that the necessary data is acquired and organized into a form which a structural model can be developed. A fuzzy reachability matrix is one such form. The main purpose of this method is to describe and illustrate a formal procedure for constructing the graphic presentation of the hierachical arrangement, given the necessary information concerning the relation of each element to each other element. The procedure permits the automatic development of the graphic structure that portrays the hierachy. The entries in the matrix are taken to values on the interval [0,1] by virtue of a fuzzy binary relation. According to the matrix, a fuzzy digraph describes a contextual relation among the elements of the system and can be transformed into an interpretive structural model of the system with respect to the relation. A significant step in the development of the proposed fuzzy structural modeling consists of the relaxation of transitivity, irreflexivity and asymmetry with respect to the construction of subordination matrix. The relaxation extends the flexibility and applicability for the structural modeling of systems. In order to show how the proposed method works, a few examples have been illustrated and the structure for the system has been successfully identified.

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Optimization problems for non-linear systems often involve those cases which preclude the assumption of uni-modality of the objective functions. The methodology has been so far developed with regards to optimizing of multi-variable multi-modal objective functions. It has, however, failed in guaranteeing the global maxima or minima of solutions. This study is an extension of the search method previously developed by the author for single-variable multi-modal functions to multi-variable multi-modal objective functions. The algorithm presented in this paper searches the global optima:tinder the Lipschitz condition. It provides the approximation of the global optima and the upper bound of the global maxima. Through numerical tests, it is shown that the algorithm efficiently produces satisfactory results.

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This paper presents a method for determining economic lot size when the demand rate varies M times over finite time horizon. No shortages are to be permitted and lots may be added to inventory at arbitrary times during planning horizon.

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