A target x is a point on the real line given by the value of a random variable X, which has some distribution function F. A searcher starts looking for x from some point on the line, using a continuous path. He makes for x with un upper bound on his speed till he finds it. The target being sought for might be in either direction from the starting point, so the searcher has in general to retrace his steps many times before he attains his goal. It is desired to search in an optimal manner so as to minimize the expected cost of the search. All previous papers treated this problem using the origin as the starting point of the search. They have been proved that one can minimize the expected cost if the underlying distribution satisfies certain conditions. In this paper, the problem will be treated in the "General Case", which means that the search may start from any point on the real line. Conditions under which we can minimize the expected cost in the general case will be given.

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The j-th best subset problem for the generalized least squares is formulated in which statistical criteria as well as non-statistical conditions are introduced. Non-statistical conditions are based on a knowledge of the scientific field to which research is related, natural logic and common sense, while statistical criteria are t-test, Durbin-Watson serial correlation test, absolute relative error test, turning point test and fitting test, depending on the covariance matrix of a disturbance term and type of data. Various technical methods are devised to make a computer solve the first (j=1) to the J-th (e.g., J=10) best subset problems in one computer-run, depending on whether or not a researcher has a new criterion or appropriate values for the parameters used to evaluate meaningful subsets before estimation. Then, the ultimately best subset among the best J subsets is regarded as a practical solution to the variable selection problem for the generalized least squares. The System OEPP can handle the proposed variable selection method.

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Here the validity of a no-derivative Complex Method for the optimization of constrained nonlinear programming (NLP) problems is discussed. This method, starting with N (N>__-n+1, where n is the dimension of the problem) feasible points, determines the optimum by a typical descent method. Though this method is capable of determining a nearly optimal feasible solution, its convergence in the general case is not guaranteed. However, this method has good convergence properties for unconstrained problems; hence transformation of the constrained NLP problem to a series of smooth unconstrained problems by the use of a penalty function and the application of the complex method to these functions is proposed for the determination of the optimum. A number of problems are solved by the complex method as well as by the use of a penalty function and the complex method, and the results are compared.

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The concepts of possibility distribution and measure have been introduced by L.A. Zadeh. The possibility distribution is regarded as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. The possibility measure of an event may be interpreted as the possibility degree of occurrence of the event under a possibility distribution. On the other hand, possibility, necessity, impossibility and contingency have been naturally extended to deal with fuzzy sets in the framework of Venn's diagram, using h-level sets by H. Prade. Zadeh's possibility theory is recovered in this approach. However, these modalities are considered as absolute indices. Namely, the fact that the possibility of an event is very large does not imply the reason that the possibility of opposite event is small. In this paper, the relative modalities are defined using Prade's modalities. The relative possibility is defined as the degree where the possibility of an event is larger than the possibility of opposite event. In the same way, the relative necessity, the relative impossibility and the relative contingency are defined. The relative necessity is not dual of the. relative possibility but equal to the relative possibility. Therefore, the dual relative possibility is defined. And relations between the relative modalities and the absolute modalities are investigated. As one of the applications of these modalities, the decision procedures based on the modal concept are considered. and divided into eight classes, so that fuzzy multi-objective linear programming problems are formulated using these procedures. These problems are reduced to nonlinear programs which are not necessarily convex programs. An approximating method of the global optimal solution using simplex method is proposed.

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The Shortest Path Problem is one of the classic and fundamental problems in mathematical programming. Many real-world problems are formulated as a shortest path problem and solved by a Dynamic Programming approach. Recently, there have been several studies on shortest path problems with one or more knapsack type side constraints. In a constrained shortest path problem, every edge of the network has not only a distance but also some weight such as time or cost. In this paper, we consider a new type of constrained shortest path problem in which the network has a weight imposed on its nodes. For example, given a water price and its demand at every node, find a sales route with maximal profit (= sales - cost) that does not exceed the total supply Q. We call this the Water Salesman Problem. It is a subproblem derived from an existing vehicle routing problem with supply constraints in a steel company. In this paper, we discuss two algorithms for solving a generalized water salesman problem, including the original one, where split deliveries of water are allowable. One is based on Dynamic Programming where every state consists of two factors, the current node and the remaining water supply. The other is an algorithm using the k-th shortest path method. The algorithms are tested out on randomly generated problems. The first algorithm can solve any water salesman problem with 10 to 80 nodes but takes relatively more time. On the other hand, the second one can solve almost all problems rapidly. However, some problems are forced to stop before obtaining the optimum solution because of memory capacity.

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