A set of n items N {1,2,...i...,n} is given and each item is processed by machines M_1 and M_2 in this order. Each item i, iεN, goes through three operation, namely <a_i>, <b_i>, and <p_i>. The operation <p_i> can be assigned to either M_1 or M_2 , while operations <a_i> and <b_i> are assinged to M_1 and M_2 respectively. Item i is said to be of I-type job, i^I, when <p_i> is assigned to M_1 and of II-type job, i^<II> when assigned to M_2. This paper deals with a problem of finding an optimal schedule, i.e., determining the job type of each item and the processing sequence of all the items, which minimizes makespan. The assignment of <p_i>, iεN, is specified by a set λ, where λ={i|i^I, iεN}, and an optimal sequence corresponding to a given λ, which is denoted by S_λ, can be obtained by Johnson's condition. Thus an optimal schedule, S_<λ*>, exists among the 2^n optimal sequences corresponding to the 2^n possible S_λ's. One item k is chosen from among the II-type jobs, and its job type is reversed from k^<II> to k^I. For this new set of job types, an optimal sequence is obtained. To describe this process towards an optimal schedule, a network structure can be constructed encompassing all of the solutions which are obtained only by an operation of "one way change" of the job type from k^<II> to k^I. The summary of this paper is as follows: (1) It is shown that the makespan on an arbitrary path from S_φ to S_<λ*> is strictly monotone decreasing. (2) The lower-bound of the makespan is obtained for the set of all the schedules generated from an arbitrary S_λ. (3) An algorithm is developed to solve for an optimal solution for this problem.

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We propose a combinatorial framework for fixed point algorithms and constructive proofs of combinatorial lemmas in topology. The framework consists of two sets of pseudomanifolds and an operator relating them. They have lattice structures which are dual to each other. We show that the set of "joins" of pseudomanifolds related by the operator is a homogeneous and orientable pseudomanifold under several conditions. By exploiting this framework we generalize Sperner's lemma on convex polytope. Namely, let C be a convex polytope with m facets F_1,…,F_m, S be a finite triangulation of C and S^^- = {σ|σ is a face of some simplex of S}. Given a nondenerate vertex v of C and a labelling function l from the set of vertices of S to {1,…, m}, the set of indices of facets, there is an odd number of simplices σ of S^^- such that l(σ)∪{i|1≦i≦m, σ⊂F_i} strictly includes {i|1≦i≦m, v∈F_i} We also prove the generalization of Sperner's lemma by Fan and van der Laan-Talman-Van der Heyden's lemma as corollaries to the result.

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This paper addresses itself to the maximization of the yield of some chemical process subject to various operational as well as physical constraints. Some of the operational constraints on such factors as the level of flows and the number of flow change are discrete in nature and standard mathematical formulation leads to a medium scale mixed integer programming problem. A typical problem contains 150 integer variables in addition to 150 continuous variables and 200 constraints. A problem of this size may not possibly be solved by the general purpose mixed integer programming code in accordance with our basic requirement, i.e., in less than one minute on 1 MIPS computer. Thus we introduce a series of relaxation schemes by elaborating the special structure of the problem and reduce the original problem into a set of subproblems, all of which can be solved by standard methods. We tested this algorithm on a number of real scale problems and always obtained almost optimal solution within I minute, whose discrepancy from the true optimum was less than 1% relative to the value of the objective function.

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The problem to schedule n independent tasks nonpreemptively on m unrelated processors with the objective of minimizing the finishing time is considered. For the case of m=2, an approximate algorithm which has a worst-case performance ratio 1+ε and runs in time O (nlogn) is proposed. For general m, by restricting the number of task types to k, a polynomial time (in n, m) exact algorithm is presented.

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Variational inequalities have extensively been studied to formulate equilibrium problems which arise in many fields including economics and operations research. Also, various numerical methods such as projection method and diagonalization method have recently been developed for the solution of variational inequalities. This paper considers modified variational inequalities which allow the constraint set to be non-convex and therefore contain classical variational inequalities as a special case. First, a solution method is presented for unconstrained problems and conditions for global convergence are established. Then, for inequality constrained variational in-equalities, a solution method is proposed by modifying the multiplier methods for constrained optimization, and its convergence property is examined. When this method is applied to a dual formulation of the asymmetric traffic equilibrium problem, in which variables are travel costs and all constraints are inequalities, path flows can be obtained as the optimal Lagrange multipliers of the variational inequality problem. Finally, some numerical examples containing traffic equilibrium problems of medium size are solved to exhibit the effectiveness of the proposed methods.

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We consider a single-machine scheduling problem in which penalties occur for jobs which are completed either before or after their due date. The objective is to minimize the sum of the penalties for all jobs. A dominance relation based on pairwise precedence relation of adjacent jobs is proposed in this paper to improve the efficiency of the branch and bound algorithm which was proposed previously. A method is presented to obtain the pairwise precedence interval on which job i precedes its adjacent job j . The effectiveness of the proposed algorithm is demonstrated by numerical examples with various types of due dates. The results are summarized as follows. 1. The efficiency of the branch-and-bound algorithm can be improved by introducing the pairwise precedence relation. 2. For the problems in which the due dates of jobs distribute uniformly over the total processing time or over its latter half, the value of the objective function can be largely improved by sequencing jobs in the order of earliest due date (EDD sequence) than in the order of random sequence. In this type of problems the optimal or a good solution can be obtained in some reasonable time. When the due dates are concentrated or are distributed over the first half of total processing time, EDD sequence may not provide an effective initial sequence. 3. The computing time and total nodes generated by attaining a first optimal sequence are less than the half of them required by completing the computation.

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In this paper, a fuzzy linear programming problem is formulated under the assumption that a multiattribute value function is given, and its method of solution is proposed. The fuzzy linear programming problem in this paper is the unconstrained linear programming problem with several objective functions whose coefficients are fuzzy numbers. These fuzzy numbers are regarded as the possibility distribution of the coefficients. The measurable multiattribute value function is given in the objective function space. In the analogy of the principle of maximizing expected utility which is a decision procedure under the probability distribution, the principle of maximizing possible value and the principle of maximizing necessary value are proposed as the decision procedures under the possibility distribution. The possible value and the necessary value are represented as fuzzy integrals of the value function with respect to the possibility measure and the necessity measure respectively, whereas the expected utility is represented as Lebesgue's integral of the utility function with respect to probability measure. The fuzzy linear programming problem is formulated using each of these decision procedures. These problems are reduced to nonlinear programming problems. A method of solution using simplex method is proposed.

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