Green manufacturing is used to describe an environmentally friendly manufacturing approach, which explicitly considers the impact of production on the environment and resources. Therefore, the production scheduling of solving energy conscious is in line with the focus of green manufacturing. In this paper, we consider the scheduling problems with rejection in the green manufacturing industry. The objective is to minimize the makespan of the accepted jobs plus the total rejection penalty of the rejected jobs, subject to the constraint that the total machine cost of the processed jobs is not more than a given threshold. We present pseudo-polynomial time algorithms and 2-approximation algorithms for the single-machine and the parallel-machine problems, respectively.

In this paper, we consider the two-agent scheduling with time-dependent and position-dependent processing time in a two-machine flow-shop system. The objective is to minimize the makespan of agent A under the constraint that the objective of agent B cannot exceed a given upper bound. In the time-dependent processing time model, the constraint is the makespan of agent B. We first show that the problem is NP-hard in the ordinary sense. Then we prove that there exists an optimal schedule which is a permutation schedule, and there exists an optimal schedule in which the adapted version of Johnson's rule do not hold for neither agent A nor agent B. Furthermore, we show that three special cases are polynomially solvable. In the position-dependent processing time model, if the constraint is the makespan of agent B, we show that the problem is NP-hard in the ordinary sense. If the constraint is the total completion times of agent B and jobs have different release dates, the problem is strongly NP-hard.

In this paper, we consider the uniform -machine scheduling problems under the green manufacturing system, in which the newer machine has smaller energy consumption cost. The objective are to minimize the makespan and the maximum lateness under the constrain that the total energy cost has a given upper bound Ub. Considering that these problems are intractable, we assume that the uniform machines have two speeds. For the problems of minimizing the makspan, we first present a polynomial -time algorithm for the twomachine preemptive scheduling problem and then generalize to the m -machine preemptive scheduling problem. Next, we present a 2 -approximation algorithm for the m -machine non -preemptive scheduling problem. Furthermore, for the m -machine non -preemptive scheduling problem of minimizing the maximum lateness, we propose and analyze an improved EDD algorithm.

This paper focuses on job scheduling with step learning and job rejection. The step learning model aims to reduce the processing time for jobs starting after a specific learning date. Our objective is to minimize the sum of the maximum completion time of accepted jobs and the total rejection penalty of rejected jobs. We examine special cases of processing times for both single-machine and parallel-machine scenarios. For the former, we design a pseudo-polynomial time algorithm, a 2-approximation algorithm and a fully polynomial-time approximation scheme (FPTAS) based on data rounding. For the latter, we present a fully polynomial-time approximation scheme achieved by trimming the state space. Additionally, for the general case of the single-machine problem, we propose a pseudo-polynomial time algorithm.

We consider the single-machine scheduling with coupled task and rejection, in which each coupled task is either accepted and processed on a single machine or rejected with a certain rejection penalty. Each accepted coupled task Ja is made of two tasks with processing times aa and ba, respectively, and a fixed exactly time interval La between the two tasks is required. The objective is to minimize the makespan of the accepted coupled task and the total penalty of the rejected coupled task. We present 3-approximation algorithm for the cases of aa = La = p and ba = La = p, respectively, and prove that the problems with common ba or common aa is polynomially solvable. Furthermore, when aa = ba = p, we provide a 2-approximation algorithm for La > p and show that the problem with for La & LE; p can be solved in polynomial time.

The exploration of submodular optimization problems on the integer lattice offers a more precise approach to handling the dynamic interactions among repetitive elements in practical applications. In today’s data-driven world, the importance of efficient and reliable privacy-preserving algorithms has become paramount for safeguarding sensitive information. In this paper, we delve into the DR-submodular and lattice submodular maximization problems subject to cardinality constraints on the integer lattice, respectively. For DR-submodular functions, we devise a differential privacy algorithm that attains a (1-1/e-ρ ) -approximation guarantee with additive error O(rσln |N|/ϵ ) for any ρ >0 , where N is the number of groundset, ϵ is the privacy budget, r is the cardinality constraint, and σ is the sensitivity of a function. Our algorithm preserves O(ϵ r^2) -differential privacy. Meanwhile, for lattice submodular functions, we present a differential privacy algorithm that achieves a (1-1/e-O(ρ )) -approximation guarantee with additive error O(rσln |N|/ϵ ) . We evaluate their effectiveness using instances of the combinatorial public projects problem and the budget allocation problem within the bipartite influence model.

In this paper, we consider the parallel-machine scheduling with step-deteriorating jobs. The actual processing time of each job deteriorates as a step function if its starting time is beyond a given deteriorating date. We focus on the case of the common job deteriorating date. For the minimization problem of total completion time, we first show that the problem is NP-hard in the strong sense. Then we propose one property of any optimal schedule. Furthermore, we prove that two special cases of common normal processing time or common penalty are polynomially solvable. For the minimization problem of total weighted completion time, we analyze the NP-hardness and present a polynomial time optimal algorithm for the case of common normal processing time and common penalty.

考虑了共享制造环境下的同类机排序问题.在共享制造环境中,每个工件Jj 都有一个可以加工的机器集Mj,Jj 可以被分别给Mj 的某一台机器加工,也可以一定服务成本分配给其他剩余机器进行加工.该文的目标是最小化工件的最大完工时间加总服务成本.对于机器台数是固定常数情况,该文对经典加工模型和简单退化加工模型分别提出了基于程序划分的全多项式时间近似方案.对总服务成本不超过给定上界的限制下最小化最大完工时间问题,给出了其整数规划模型.

By taking transmission losses and directed communication into considered, a more practical scenario of economic dispatch problem is studied in the paper. It is formulated as a non-convex optimization problem. Using convex relaxation, the non-convex optimization problem is transformed into convex optimization problem, and the conditions to ensure that they have the same solution are given. Then, a consensus-based distributed algorithm with time-varying feedback gains is presented, which can be used to the general directed communication networks. Especially, the convergence and optimality of the proposed algorithm are proved by using multi-parameter eigenvalue perturbation theory and graph theory. Finally, simulation results validate the theoretical results and illustrate the effectiveness and advantages of the proposed algorithm.

In this paper, we consider the multitasking scheduling with alternate odd-period and even-period. For the minimization of makespan on one single machine, we present a 2-approximation algorithm for the general case and a 4/3 -approximation algorithm for a special case when jobs have identical release dates. And we prove that the problem is strongly NP-hard when jobs have different release dates. For the minimization of makespan on identical parallel machines, we present a (5/2-1/m) -approximation algorithm for the general case and a pseudo-polynomial time algorithm when the number of machines is constant. Furthermore, we prove that the single-machine scheduling of minimizing the lateness is strongly NP-hard.

We consider the problem of local privacy where actions are subsets of a ground multiset and expectation rewards are modeled by a [Formula: see text]decomposable monotone submodular function. For the DR-submodular maximization problem under a polymatroid constraint, Soma and Yoshida [26] provide a continuous greedy algorithm for no-privacy setting. In this paper, we obtain the first differentially private algorithm for DR-submodular maximization subject to a polymatroid constraint. Our algorithm achieves a [Formula: see text]approximation with a little loss and runs in [Formula: see text][Formula: see text] times where [Formula: see text] is the rank of the base polymatroid and [Formula: see text] is the size of ground set. Along the way, we analyze the utility and privacy of our algorithm. A concrete experiment to simulate the privacy Uber pickups location problem is provided, and our algorithm performs well within the agreed range.

We consider the time-dependent scheduling with proportional and delivery times on a single machine. Three models of the processing times are addressed here, they are proportional deterioration, proportional-linear shortening and proportional-linear increasing. The objective is to minimize the time by which all jobs are delivered. For the first model, we prove that the problem is polynomial solvable when jobs have identical release dates. When jobs arrive dynamically, we first give the proof of the NP-hardness and present a two-approximation algorithm. Then we propose a fully polynomial time approximation scheme for the case where the number of distinct release dates is a constant by applying the "rounding-the-input-data " technique. For the second and third models, when jobs have identical release dates, we prove that they are polynomial solvable, when jobs have different release dates, we present two-approximation algorithms for each of them.

In this paper, we consider the single-machine scheduling with step-deteriorating jobs and rejection. Each job is either rejected by paying a rejection penalty, or accepted and processed on the single machine, and the actual processing time of each accepted job is a step function of its starting time and the common deteriorating date. The objective is to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs. For the case of common deteriorating penalty, we first show that the problem is NP-hard in the ordinary sense. Then we present two pseudo-polynomial algorithms and a 2-approximation algorithm. Furthermore, we propose a fully polynomial time approximation scheme. For the case of common normal processing time, we present two pseudo-polynomial time algorithms, a 2-approximation algorithm and a fully polynomial time approximation scheme.

In this paper, we aim to establish a concrete representation, as a family of sets, for every algebraic L-domain. We generalize the notion of a topped algebraic intersection structure to a locally algebraic intersection structure. Just as topped algebraic intersection structures are concrete representations of algebraic lattices, locally algebraic intersection structures are concrete representations of algebraic L-domains. This result extends the classic Stone's representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it will be seen that many well-known representations of algebraic L-domains can be analyzed with the framework of locally algebraic intersection structures.

研究了具有累积效应的两台同类机排序问题,目标是极小化机器总载重.半积函数在组合优化通常用于算法设计与分析.对该文中涉及的问题,用该函数设计了一个γ-完全多项式近似方案,并进行了算法分析.

In this paper, we focus on a semi-online scheduling problem with rejection on identical parallel machines, where "semi-online" means that the ratio of the longest processing time among all jobs to the shortest one is no more than gamma with gamma >= 1. In particular, in this setting, there are a coupling of independent jobs arriving online over time with the flexibility of rejection, which implies that each job will be either accepted and scheduled on one of identical machines or rejected at the cost of penalty cost. Our objective is minimizing the total completion time of the accepted jobs plus the total penalty cost of the rejected jobs. For this problem, we design a deterministic polynomial time semi-online algorithm entitled as alpha Delayed Shortest Processing Time with Rejection (ADSPTR). In competitive analysis, by adopting the approach "Improved Instance Reduction", we obtain the competitive ratio of ADSPTR is at most 1 + root 1+gamma (gamma-1)-1/ gamma . (c) 2020 Elsevier Inc. All rights reserved.

Many machine learning problems, such as medical data summarization and social welfare maximization, can be modeled as the problems of maximizing monotone submodular functions. Differentially private submodular functions under cardinality constraints are first proposed and studied to solve the Combinatorial Public Projects (CPP) problem, in order to protect personal data privacy while processing sensitive data. However, the research of these functions for privacy protection has received little attention so far. In this paper, we propose to study the differentially private submodular maximization problem over the integer lattice. Our main contributions are to present differentially private approximation algorithms for both DR-submodular and integer submodular function maximization problems under cardinality constraints and analyze the sensitivity of our algorithms.

This paper investigates a classic online scheduling problem with learning effect on a single machine. Specifically, a number of independent jobs that arrive online over time will be processed on a single machine and learning effect implies that the real processing time of job [Formula: see text] is a non-increasing function of its position [Formula: see text], i.e., [Formula: see text], where [Formula: see text] is the basic processing time of job [Formula: see text] and [Formula: see text] is the learning index. Our goal is to minimize the total completion time of all jobs. For the problem, we develop a deterministic polynomial time online algorithm called Delayed Shortest Basic Processing Time (DSBPT) and state that it is an online algorithm with a competitive ratio of 2, which matches the lower bound of the online scheduling problem we focus on.

研究具有入树约束的两台同类机排序问题,工件具有不同的到达时间且具有单位的加工时间,目标为极小化最大完工时间.对于该NP-难问题,首先设计了一个分支定界算法并证明了算法的最优性.然后通过一个具体算例,详细验证了算法的执行过程.