This paper is concerned with solving a knapsack problem subject to the constraint that total number of variables used in the solution is bounded above by a given number. The approach taken in [6,7] for solving the (equalty-constraint) knapsack problem is extended to this case. Like the case of the equality-constraint knapsack problem [6,7], the feasibility of the problem is not trivial and examined first. A necessary and sufficient condition for the feasibility is obtained through a special structure of the problem. All dual feasible bases of the linear programming relaxation problem of the problem are obtained explicitly so that the cone spanned by all right hand side vectors for which the problem is feasible is decomposed into the cones generated by the columns of the dual feasible bases. Associated with each dual feasible basis is a group relaxation problem of the problem. The group minimization problem is written down explicitly and studied. Unlike the case of the single constraint integer linear programming problems [6,7], there may still remain an infinite number of right hand side vectors for which the problem is not solved optimally through its group minimization problem. To resolve this difficulty (the infinity of the number of unsolved right hand side vector), a new relaxation problem is introduced in which the nonnega-tivity of only one of the two variables corresponding to a dual feasible basis is relaxed. It is determined for what right hand side vectors the original problem is solved or is not solved by solving the new relaxation problem thus introduced. By solving all the new relaxation problems, one finds that only a finite number of problems remain unsolved. After solving the problem for all right hand side vectors, a tree structure is exploited to represent all the optimal solutions to the problem for which its right hand side vectors are changed all over a dual feasible basis.

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The Laguerre transform, introduced by Keilson and Nunn (1979), Keilson, Nunn and Sumita (1981) and further studied by Sumita (1981), provides an algorithmic framework for the computer evaluation of repeated combinations of continuum operations such as convolution, integration, differentiation and multiplication by polynomials. The procedure enables one to numerically evaluate many distribution results of interest, which have been available only formally behind the 'Laplacian curtain'. Since the initial development, the formalism has been extended to incorporate matrix and bivariate functions and finite signed measures. The purpose of this paper is to summarize theoretical results on the Laguerre transform obtained up to date. In a sequel to this paper, a summary is given focusing on algorithmic aspects. The two summary papers will enable the reader to use the Laguerre transform with ease.

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In this note we shall describe the results obtained by applying the decomposition principle established in [2], [3] to poly-linking systems. In order to state those results in a logically self-consistent way, we shall introduce a new notion of 'minors' of generalized polymatroids. Firstly, it is observed that the subsystems defined from a poly-linking system through our decomposition method are not in general poly-linking systems any more. This difficulty can be overcome by considering a poly-linking system as a special case of generalized polymatroids. In fact, it can be shown by an easy calculation that a poly-linking system is equivalent to a special case of generalized polymatroids. And when a poly-linking system is considered as a generalized polymatroid, its resultant subsystems through our decomposition method are seen to be the 'minors' of this generalized polymatroid. The notion of a 'minor' of a generalized polymatroid is first introduced in this paper. Hence from the point of view of our decomposition principle, the notion of 'poly-linking system' is not self-consistent, and it should be treated as a special case of generalized polymatroids. Secondly, as a direct consequence of the results in [2], [3], a direct-sum decomposition of the optimal solutions of the intersection problems on poly-linking systems is induced. Lastly, we shall investigate the parametrized type of the intersection problem on poly-linking systems where the rank functions are multiplied by positive real parameters.

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There are many studies which consider an optimal tour on a given graph including the well-known Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP). Most of these studies, however, assume that "all" nodes on a given graph should be visited exactly once or at least once. In this paper, we relax this assumption and consider the following problem: "Given items with known values located at nodes of network, one wants to collect items so that their total value is maximized, under the assumption that a tour starting from the "center" node and returning to the center node is completed within a predetermined time limit." Because of the added (time) constraint, one may not visit all nodes. The addition of this simple-looking constraint makes the problem difficult as it introduces an added dimention of selecting nodes to visit. After a brief introduction, Section 2 presents two formulations of the problem, a native formulation and an improved one based on the introduction of self-loops to the graph corresponding to the problem. The latter formulation allows us to utilize solution strategies developed for the standard TSP. A branch and bound solution strategies together with a solution method of a relaxation problem is described in Section 3. A relaxation problem is generally an assignment problem with an added constraint for which an efficient branch and bound procedure is proposed. In particular, a recommended strategy for the selection of the branching variable is identified, and also an efficient procedure to transmit information from a branching problem to its sub-problems is proposed. Section 4 presents results of computational time requirements and basic characteristics of the proposed algorithm are clarified.

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A stochastic production planning problem with a finite number of planning periods is analyzed where cumulative demands up to each period are independent random variables with continuous probability distributions. In the problem, backlogging is permitted and production is restricted by its capacity. Dynamic but linear costs of inventory holding and backlogging, and of production with setup charge are considered. A branch-and-bound algorithm is developed to find an optimal plan within finite searching steps, and its computational effectiveness is evaluated.

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We present a network simplex method for the maximum balanced flow problem, i.e., the problem of finding a maximum flow in a source-to-sink network such that each arc-flow value does not exceed a fixed propor-tion of the total flow value from the source to the sink. We generalize the notion of strong feasibility in the network simplex method for the maximum flow problem to give a finiteness proof for the new algorithm. We also consider the maximum balanced integral flow problem, and show that the problem can be solved in polynomial-time for a special case when the balancing rate function is constant.

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This note gives a characterization of the universal pair of bases of a pair of polymatroids as the nearest pair of bases with respect to a class of pseudo-distances including the Kullback-Leibler divergence. N. Megiddo considered the lexico-optimal flow problem in a multiterminal network N = (V, A, c; S^+, S^-) (V: vertex set, A: arc set, c: capacity, S^+: supply (source) vertices, S^-: demand (sink) vertices), which is to find a maximal flow such that the supply flow (s^+ (v) | v ∈ S^+)[resp., the demand flow (s^- (v) | v ∈ S^-)] is as proportional as possible to a given weight vector. This problem is treated by S. Fujishige as a special case of the lexico-optimal base problem for a single polymatroid. This paper considers the problem of finding a maximal flow such that the supply flow s^+ and the demand flow s^- are as "near" as possible (where a one-to-one correspondence between S^+ and S^- is assumed to be given), and generalizes it to the problem of finding a "nearest" pair of bases of a pair of polymatroids It is shown that the "nearest" pair coincides with the universal pair if either of the following criteria is adopted. (1) The f-divergence (a generalization of the Kullback-Leibler divergence) between the bases should be minimized; (2) The vector consisting of the ratios of the corresponding components of the bases should be lexicographically maximized.

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