An inverse theory of sequential decision processes, including the standard control process and allocation process, is developed. A finite-stage deterministic invertible (main) dynamic program (DP) whose reward functions depend not only on action but also on state is formulated as a sequential decision process. The main DP is transformed into an equivalent inverse DP by an algebraic inversion. The main DP maximizes a generalized total reward, while the inverse DP minimizes a generalized total state. An inverse theorem is established. It characterizes optimal solutions (optimal reward functions and optimal policy) of inverse DP by those of main DP through inverse and composition. The main DP includes a linear equation and quadratic criterion (main) control process on the half-line and a typical multi-stage (main) allocation process. Therefore, the inverse DP generates an inverse control process and an inverse allocation process, respectively. Not solving the recursive equation directly but applying the inverse theorem, optimal solutions of both inverse processes are easily calculated by use of those of the corresponding main processes.

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Manufactured goods on order, for example, general-purpose induction motors for specific uses, can be assembled in a flow shop by installation of a preassembling system, which mainly consists of automatic warehouses and marshalling shops. In a marshalling shop, a human picker takes parts out of a parts-oriented pallet and puts them in a kit pallet according to assembling instructions. The preassembling system contains four working elements; stacker cranes, shuttle cars, human pickers and a kit-pallet line, operations of which are needed to optimize by curtailment of total working hour in dealing many given order sheets. It becomes clear by system analysis that the number of operating times of a stacker crane in a day can be reduced by using repeatedly parts-pallets which have been already outputted on a waiting table by previous instructions. Reduction of this number of times substantially results in that of the total working hour, which becomes possible mainly by improvement of the dealing sequence of order sheets. This sequencing problem has the following two features: there are many (about one hundred) order sheets to be sequenced, and the performance of each decision is strongly dependent on the previous decisions. This paper introduces a finite state system to express the problem, and then proposes a dynamic suboptimal sequencing algorithm which is a dynamic decision process in the finite state system , Selecting one hundred typical order sheets for a day from six-month actual data, we solve the sequencing problem, and calculate the number of outputting times of stacker cranes and the total working hour by digital simulations. Results show that both of them decrease remarkably to about one third and two thirds, respectively.

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In this paper we deal with a one machine scheduling problem to minimize the mean weighted flow-trme subject to the constraint that the job tardiness is not greater than a specified value. An algorithm to obtain a pair. wise local optimum schedule for this problem has been. presented by Smith [4]. We give a necessary condition under which a pairwise local optimum schedule should not coincide with the global optimum one, and give an improved schedule for this case. On the basis of the analysis, an efficient algorithm is developed to obtain global optimum schedules for more cases than Smith's algorithm. Computational experiment is performed to show the quality of the solution, the computational time, and core memory size required for the algorithm in comparison with the previous three algorithms: Smith's, Burns', and DP.

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An algorithm for solving a weighted minimax real-valued flow problem in a polynomial time is given. The weighted minimax flow is that which minimizes the maximum value of arc-flow multiplied by arc-weight among all flows of maximum flow value. We use the capacity modification technique and the method of solving a ratio minimization problem for our problem.

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When weights of arcs in a graph are normal variates, we seek a spanning tree maximizing the probability that the sum of weights of arcs in the spanning tree is not greater than a given constant. An O(e^2n) algorithm for it is given.

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A unifying framework, named tree programming, for solution algorithms of combinatorial optimization problems, such as branch-and-bound algorithms, dynamic programming, backtrack programming, additive implicit enumeration and cutting plane methods, is proposed. Constituents of tree programming are a selection rule, a branching rule, an upper bounding function, an elimination rule and a terminating condition. The essential difference from conventional models of such algorithms is in the abstract definition of elimination rules. The validity of tree programming, finiteness and correctness, is examined. It is shown that finiteness is mainly dominated by the selection rule and the elimination rule, and that correctness by the terminating condition. As examples of tree programming, algorithms cited above are reformulated along the proposed framework.

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