Many practical scheduling problems involve processing several batches of related jobs on common facilities where a setup time is incurred whenever there is a switch from processing a job in one batch to a job in another batch. We extend various scheduling models to include batch setup times. The models include the one-machine maximum lateness, total weighted completion time, and number of late jobs problems. In all these cases, a dynamic programming approach results in an algorithm that is polynomially bounded in the number of jobs, but is exponential in the number of batches. We also study the parallel machine model with preemption and show that the maximum completion time, maximum lateness, total weighted completion time, and number of late jobs problems are NP-hard, even for the case of two identical parallel machines, and sequence independent setup times.

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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.

Abstract. We study some combinatorial and algorithmic problems associated with an arbitrary motion of input points in space. The motivation for such an investigation comes from two different sources: computer modeling and sensitivity analysis. In modeling, the dynamics enters the picture since geometric objects often model physical entities whose positions can change over time. In sensitivity analysis, the motion of the input points might represent uncertainties in the precise location of objects.The main results of the paper deal with state transitions in the minimum spanning tree when one or more of the input points move arbitrarily in space. In particular, questions of the following form are addressed: (i) How many different minimum spanning trees can arise if one point moves while the others remain fixed? (ii) When does the minimum spanning tree change its topology if all points are allowed to move arbitrarily?

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