This paper considers a class of general dynamic programs which satisfies the monotonicity and contraction assumption, and in which the sets of cost functions and policies are closed under the monotone contraction operators. This class of dynamic programs includes, piecewise linear, affine dynamic programs, partially observable Markov decision processes, and many sequential decision processes under uncertainty such as machine maintenance control models and search problems with incomplete information. An algorithm based on generalized policy improvement has the property that it only generates cost functions and policies belonging to distinguished subsets of cost functions and policies, respectively.

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For a distributive lattice D ⫋ 2^E and a submodular function f on D with φ ∈ D and f(φ) = 0, the pair (D,f) is called a submodular system and, when E ∈ D, the polyhedron given by B(f) = {x | x ∈ R^E, ∀X ∈De : x(X) ≦ f(X), x(E) =f(E)} is called the base polyhedron associated with (D, f)・ We examine the structure of the base polyhedron B(f) and give a characterization of all the faces of B(f). Faces of B(f) are made correspond one-to-one to certain sublattices of D, so that the collection D of all such sublattices of D is anti-isomorphic with the collection F of all the nonempty faces of B(f). Here, D and Fare considered as posets relative to set inclusion. The incidence relation among faces, dimensions of faces, and extreme points and extreme rays of faces are given based on the structure of the sublattices in D. These include as special cases recent results on (1) a poset structure of a polymatroid extreme point and connected components by Bixby, Cunningham and Topkis, (2) extreme rays of a cone determined by a distributive lattice by Tomizawa, and (3) adjacency for polymatroid extreme points by Topkis. Moreover, given a sublattice D_1 of D_2 on which f is modular, F(D_1) = {x | x ∈ B(f), ∀X ∈D_1 : x(X) = f(X) } is a nonempty face of B(f) and there uniquely exists a sublattice D_2 in D which corresponds to the face F(D_1). We show a theorem which characterizes the relationship between D_1 and D_2. D_2 is considered as a closure of D_1 and this closure operation is closely related to the concept of maximal skeleton recently considered by Nakamura and Iri. Algorithmic aspects of these characterizations are also discussed.

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In this paper, the stationary waiting time distributions Fq (x) and F(x) are explicitly formulated for the GI/E_k/m queue under the first-come first-served discipline. The transition probability matrix and the imbedded probabilities play the important roles in this study. Some numerical results are presented for various systems as E_l/E_k/m, U_l/E_k/m, D/E_k/m, etc. Further the properties of F(x) are considered.

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This paper is concerned with the convex cost multicommodity flow problem. First, it is shown that the Lagrangian dual of the problem can be formulated as a convex nonsmooth optimization problem. Then, an algorithm is presented for solving a problem which is obtained by slightly modifying the dual problem so as to be. amenable to shortest chain algorithms. Since the proposed algorithm mainly consists of successively solving shortest chain problems and linearly constrained subproblems whose size is independent of the underlying network structure, we may expect that it can solve fairly large problems. Convergence of the algorithm is proved and several remarks on its implementation are given. Finally, limited computational experience with the proposed algorithm is reported.

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This paper describes a formulation of Fuzzy Linear Programming (FLP) Problem with fuzzy coefficients by the extension principle. An order relation among fuzzy sets is defined by fuzzy max which is defined through the extension principle. Mathematically speaking, it means that a__~ > b__~ <=> a^^-_α >__- b^^-_α and a__-_α >__- b__-_α The constraints and the object are both fuzzified by fuzzy linear function. Two FLP problems are considered as follows: (i) Problem (A) is to decide the non-fuzzy solution x that maximizes y__~=c__~x^t subject to A__~x^t<__- b__~^t and (ii) Problem (B) is to decide the fuzzy solution x__~ that maximizes y__~=cx__~^t subject to Ax__~^t <__- b__~^t This fuzzy solution means the possibility distribution of solution in the problem(B) . Two concepts of optimality are used as maximizing the fuzzy objective function in a whole sense of fuzzy set and minimizing its fuzziness. Since the FLP problem (A) takes the. possibility distribution of coefficients into consideration, its solution is robust to the uncertainty of model, compared with the solution in conventional LP problem. The FLP problem (B) provides us with the possibility of solution which reflects the fuzziness of parameters. This formulation can be used as a model of top level decision problem in a fuzzy environment. This approach seems tractable and applicable to the real world decision problem where human estimation is influential. Fuzzy sets are restricted to a class of trianguler membership functions. Owing to this simplification, the FLP problem can be turned into a conventional LP problem with twice numbers of constraints in the FLP problem. Numerical examples are discribed to explain our FLP problems.

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