Many problems from inventory, production and capacity planning, and from network design, exhibit scale economies and can be formulated in terms of finding minimum-additive-concave-cost nonnegative network flows. We reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic-programming method, called the send-and-split method, to solve it. The main work of the method entails repeatedly solving set-splitting and minimum-cost-chain problems. In uncapacitated networks with n nodes, a arcs, and d + 1 demand nodes, i.e., nodes with nonzero exogenous demand, the algorithm requires up to n2−13d + s2d operations (additions and comparisons) where s = n log2n + 3a is the number of operations required to solve a minimum-cost-chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph. If also the network is k-planar, i.e., the graph is planar with all demand nodes lying on the boundary of k faces, the method requires at most n2−kd3k + sd2k operations. The algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones. These results unify, significantly generalize (e.g., to cyclic problems), and sometimes improve upon (e.g., for tandem facilities) known polynomial-time dynamic-programming algorithms for Wagner and Whitin's (Wagner, H. M., Whitin, T. M. 1958. Dynamic version of the economic lot size model. Management Sci. 5 89–96.) dynamic economic-order-quantity problem, Zangwill's (Zangwill, W. I. 1969. A backlogging model and a multi-echelon model of a dynamic economic lot size production system—A network approach. Management Sci. 15 506–527.) generalization to tandem facilities, and Veinott's (Veinott, Jr., A. F. 1969. Minimum concave-cost solution of Leontief substitution models of multi-facility inventory systems. Oper. Res. 17 262–291.) increasing-capacity warehousing problem. The networks for the finite-period versions of these problems are each 1-planar. The method improves upon Zangwill's (Zangwill, W. I. 1968. Minimum concave cost flows in certain networks. Management Sci. 14 429–450.) related O(and) running-time dynamic-programming method for finding minimum-additive-concave-cost non-negative flows in circuitless single-source networks. We also implement the method to solve in polynomial time the (d + 1)-demand-node and k-planar versions of the minimum-cost forest and Steiner problems in graphs. The running time for the (d + 1)-demand-node Steiner problem in graphs is comparable to that of Dreyfus and Wagner's (Dreyfus, S. E., Wagner, R. A. 1971. The Steiner problem in graphs. Networks 1 195–207.) method.
Experience with the maximum entropy spectral analysis (MESA) method suggests that (1) it can produce inaccurate frequency estimates of short sample sinusoidal data, and (2) it sometimes produces calculated values for the filter coefficients that are unduly contaminated by rounding errors. Consequently, in this report we develop an algorithm for solving the underlying least‐squares linear prediction (LSLP) problem directly, without forcing a Toeplitz structure on the model. This approach leads to more accurate frequency determination for short sample harmonic processes, and our algorithm is computationally efficient and numerically stable. The algorithm can also be applied to two other versions of the linear prediction problem. A Fortran program is given in Part II.
We present a set of Fortran subroutines which implement the least‐squares algorithm described in Part I (Barrodale and Erickson, 1980, this issue) for solving three variants of the linear prediction problem. When combined with a suitable driver program, these subroutines can be used to calculate maximum entropy spectra without imposing a Toeplitz structure on the model.
The existence of an optimal stopping policy that is stationary and halting in a discrete-time-parameter finite-state finite-action branching Markov decision chain is here characterized by the finite termination of successive approximations. We call a policy stopping if the expected population size at time N converges to zero as N approaches infinity, and halting if the expected population at time N is zero for some N. We show that when the rewards are real valued, the Nth iterate of successive approximations is a fixed point of the optimal return operator for some N when initiated with the value of a stationary halting policy if and only if there exists a halting stationary optimal stopping policy. Similar results are shown for a Gauss-Seidel improvement of successive approximations.
In 1980 two different recursive algorithms were published, complete with Fortran programs, for autoregressive (AR) spectral estimation, based on least‐squares solutions for the AR parameters using forward and backward linear prediction. The first of these to appear, by Barrodale and Erickson (1980a, b) forms the normal equations for the Mth order AR parameters from the corresponding normal equations for the [Formula: see text] order parameters. However, for each value of M = 1, 2, …, MMAX, the normal equations are solved by Cholesky’s method ab initio, i.e., without reference to the solution of the previous normal equations of lower order. In contrast, the later algorithm by Marple (1980) calculates the Mth order AR parameters in a recursive manner from the [Formula: see text] order parameters.
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