单机最优交货期决策及工件排序

本文考虑n个独立工件在一台机器上加工的排序问题,每个工件J_i的交货期设置为d_i=kP_i~α(α≥1),目标是寻找工件最优加工时间乘子及工件最优排序S(?),使工件完工时间与交货期的最大偏差最小。给出寻找最优加工时间乘子k(?)及工件最优排序S(?)的方法。     

A Note on the Common Due-Date Assignment Problem

This note considers the n-job, one-machine scheduling problem with common due-dates. Previous research has shown that the optimal value of the common due-date coincides with one of the job completion times for a given job sequence. In this note, we propose a linear programming (L.P.) formulation of the problem and obtain the same optimal result via considering the L.P. dual problem.     

A Generalized Model of Optimal Due-Date Assignment by Linear Programming

A linear-programming model to find the optimal ‘CON due-date’ is considered for n independent jobs to be processed on a single machine. The term ‘CON due-date’ stands for constant-allowance due-date, where each job receives exactly the same due-date. The measure of performance considered is a more generalized version of similar problems studied earlier. Duality theory is used to obtain an optimal solution. Some earlier studies are shown to be special cases of the model studied in this paper. Numerical examples are presented for better understanding.     

Optimal Due-Date Determination and Sequencing of <Emphasis Type="Italic">n</Emphasis> Jobs on a Single Machine

Given a set of n jobs with deterministic processing times and the same ready times, the problem is to find the optimal processing-time multiple k* for the T.W.K. due-date assignment method, and the optimal sequence σ* to minimize the total amount of missed due-dates. It is found that k* is a constant for a given job set and σ* should be in S.P.T. sequence. After the theoretical treatment, a numerical example is given for discussion. The optimal results can readily be extended to situations in which the processing times are random variables with known means and having the same coefficient of variation. From a practical point of view, the main merit of this paper is that it demonstrates how, under certain production environments in which completion times of the jobs can be anticipated, to determine the optimal due-dates and obtain the optimal sequence.     

Common Best Proximity Points: Global Optimal Solutions

Let S:A→B and T:A→B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx=x and Tx=x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common solution, it is attempted to find an element x that is closest to both Sx and Tx in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx=x and Tx=x when there is no common solution. It is remarked that the functions x→d(x,Sx) and x→d(x,Tx) gauge the error involved for an approximate solution of the equations Sx=x and Tx=x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x→d(x,Sx) and x→d(x,Tx) by stipulating a common approximate solution of the equations Sx=x and Tx=x to fulfill the condition that d(x,Sx)=d(x,Tx)=d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yielding common optimal approximate solutions of certain fixed point equations. Besides establishing the existence of common best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

A Common Regularization for Three Reservoir Optimal Control Problems

Four optimal control problems of reservoir release are investigated. The first problem is to minimize the peak release in order to prevent flood and to reduce the flood height. The second problem is to maximize the lowest release in order to ensure irrigation, water supply, shipping and environment downstream. The third problem is to minimize the flooding duration in order to reduce damage to goods, possessions, plants, levees, etc. It is shown that these three problems may possess infinitely many different optimal solutions, but they all have a common optimal solution, which is the unique optimal solution of the fourth problem. Since this unique optimal solution depends continuously on the input data, the fourth problem is well-posed and it can be considered as a common regularization of the three ill-posed problems.     

On Optimal Random Nets

The possibility to approximate bounded linear mappings between Banach spaces depends on the degree of compactness. One way to measure this degree of compactness is the scale of entropy numbers, cf. (B. Carl and I. Stephani: 1990. "Entropy, Compactness and the Approximation of Operators," Cambridge Univ. Press, Cambridge). In the usual (worst-case) setting of numerical analysis this scale has been studied for a long time. Recent research is concerned with the study of the so-called average-case and randomized (Monte Carlo) settings. We propose the respective counterparts of the entropy numbers in these settings and obtain their behavior for Sobolev embeddings. It turns out that, at least in this situation, randomly chosen nets may not improve the approximability of operators in the Monte Carlo setting. However, we can use the results to improve previous estimates for average Kolmogorov numbers, as obtained in (P. Mathé, 1991, J. Complexity7, 261-281).     

On Optimal Lattice Decompositions

A lattice decomposition lemma by H. Rosenthal is given an optimal refinement, both in the real and in the complex valued setting.     

On Infinitesimally Optimal Paths

The concept of infinitesimal optimality is specified for discrete dynamic extremal problems.     

Optimal Pointer Algorithms for Finding Nearest Common Ancestors in Dynamic Trees

We consider the problem of finding the nearest common ancestor of two given nodes x and y (denoted by nca(x, y)) in a dynamic forest of rooted trees. Interspersed with nca-queries are online commands link(x, y), where x, but not necessarily y, is a tree root. The effect of a command link(x, y) is to combine the trees containing x and y, by making y the parent of x. This problem was originally proposed by A. V. Aho, J. E. Hopcroft, and J. D. Ullman (SIAM J. Comput.5, No. 1 (1976), 115–132). We present a pointer machine algorithm, which performs n link and m nca operations in time O(n + m log log n), matching a lower bound by D. Harel and R. E. Tarjan (SIAM J. Comput.13, No. 2 (1984), 338–355). The previous best bound on a pointer machine was O((n + m)log n), due to D. D. Sleator and R. E. Tarjan (J. Comput. System Sci.26, No. 3 (1983), 362–391).     

On Setting Optimal Sales Commissions

This paper treats the problem of determining the commission rates a company should choose to establish for each of n products sold by a salesman. It has been shown that a sales commission plan based on paying equal commission rates on the gross margins of the salesman's product lines is optimal under the assumption that sales generated as a function of time spent on a product is a deterministic increasing function of time. This paper explores a similar problem with relaxation of the deterministic assumption. It considers a stochastic sales function in order to show differences in preference functions between a company and salesman in determining optimal commissions.     

On the Optimal Sizing Problem

The paper studies the problem of determining the number and dimensions of sizes of apparel so as to maximize profits. It develops a simple one-variable bisection search algorithm that gives the optimal solution. An example is solved interactively using a Macintosh LC and Math CAD, a mathematical software package.     

ON THE OPTIMAL PARKING PROBLEM

In this paper,the concept of dual predictable projection is used,for the optimal parkingproblem.A strictly rigorous,simpler treatment is introduced and the optimal stopping rule is alsogiven explicitly.     

On Optimal 4-Dimensional Metrics

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics. Supported in part by NSF grant DMS-0604735.     

On the Common Substring Alignment Problem

The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings S₁, S₂ … S_c and a target string T, Y is a common substring of all strings S_i, that is, S_i = B_iYF_i. The goal is to compute the similarity of all strings S_i with T, without computing the part of Y again and again. Using the classical dynamic programming tables, each appearance of Y in a source string would require the computation of all the values in a dynamic programming table of size O(nℓ) where ℓ is the size of Y. Here we describe an algorithm which is composed of an encoding stage and an alignment stage. During the first stage, a data structure is constructed which encodes the comparison of Y with T. Then, during the alignment stage, for each comparison of a source S_i with T, the pre-compiled data structure is used to speed up the part of Y. We show how to reduce the O(nℓ) alignment work, for each appearance of the common substring Y in a source string, to O(n)-at the cost of O(nℓ) encoding work, which is executed only once.     

On the Logic of Common Belief

We investigate an axiomatization of the notion of common belief (knowledge) that makes use of no rules of inference (apart from Modus Ponens and Necessitation) and highlight the property of the set of accessibility relations that characterizes each axiom. Mathematics Subject Classification: 03B45, 68T25.     

On the Existence of Optimal Solutions to an Optimal Control Problem

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.Communicated by L. D. Berkovitz