The branch-and-bound procedure is formulated in rather general terms and necessary conditions for the branching and bounding functions are precisely specified. Results include the standard properties for finite procedures, plus several convergence conditions for infinite procedures. Discrete programming which includes integer programming and combinatorial optimization problems, is discussed and Fibonacci search is presented as an example of a nonfinite branch-and-bound procedure employing an optimal convergence rule.

Each of n jobs must be run first on machine I and then on machine I. Running times for each job on each machine are given. Also specified are arbitrary time lags which prescribe that a job may not be started (completed) on machine II until at least a certain time has elapsed since starting (completing) the job on machine I. A rule is given for determining the sequence in which jobs are to be run on the machines—using the same sequence for both machines—in order to minimize the time between the start of production of the first job on machine I and the completion of production of the last job on machine II.

Any multistage process may be constructed by a sequence of serial and/or parallel compositions. With certain restrictions on the form of the return composition function, optimal processes may be constructed in an efficient manner from optimal subprocesses. A number of simple sufficient conditions on the return functions are given and several approximation procedures are outlined.

The dynamic programming recursive procedure has provided an efficient method for solving a variety of multi-stage decision problems in which the objective is measured by a real valued utility function. In this paper we propose that the real valued objective function be replaced by preference relations. Sufficient conditions are given on the structure of the preference relations to insure that the recursive dynamic programming procedure yields an optimal sequence of decisions. The solution method is well adapted to an interactive mode of implementation in which there is a dialogue between the decision maker and a source of information and analysis (e.g., a computer). The "computer" collects, analyzes, and presents information on a set of alternatives to the decision maker who then communicates to the "computer" his choice of the best alternatives in the set. The process is repeated, stage by stage, thus generating an optimal sequence of decisions. The approach should be particularly useful in dealing with multi-stage decision problems involving design and/or operation of facilities and multi-period public projects where a variety of desiderata must be considered (i.e., a simple cost or benefit function is inadequate).

A solution is presented for the problem of determining the sequence in which a set of manufacturing operations should be carried out in order to minimize in-process inventory costs while meeting a variety of technological constraints. Both strict precedence constraints and contiguity constraints are considered, the latter dealing with the requirement that certain sets of operations be performed contiguously (but without specification of any order).