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A pseudo-polynomial time algorithm for solving the resource dependent assignment problem

DISCRETE APPLIED MATHEMATICS 2015年 182卷 115-121页

作者： Shabtay, Dvir Steiner, George Yedidsion, Liron Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel McMaster Univ DeGroote Sch Business Operat Management Area Hamilton ON Canada Technion Israel Inst Technol Fac Ind Engn & Management IL-32000 Haifa Israel

In this paper the resource dependent assignment problem (RDAP) is considered. In the RDAP the cost of assigning agent j to task i is a multiplication of task i's cost parameter by a cost function of agent j and the cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for the RDAP problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second one is the total weighted resource consumption. Yedidsion et al. showed that the bicriteria variations of the problem are all NP-hard for any given set of task costs. However, whether these problems are strongly or ordinarily NP-hard remained an open question. In this paper we close this gap by providing pseudo-polynomial time algorithms for solving these problems. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Assignment problem pseudo-polynomial time algorithm Resource allocation Bicriteria optimization

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A pseudo-polynomial time algorithm for solving the resource dependent assignment problem

A pseudo-polynomial time algorithm for solving the resource ...

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8th International Colloquium on Graphs and Optimization (GO)

关键词： Assignment problem pseudo-polynomial time algorithm Resource allocation Bicriteria optimization

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Scheduling with step learning and job rejection

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OPERATIONAL RESEARCH 2025年第1期25卷 1-18页

作者： Song, Jiaxin Miao, Cuixia Kong, Fanyu Qufu Normal Univ Sch Math Sci Qufu 273165 Shandong Peoples R China Nanjing Univ Informat Sci & Technol Sch Management Sci & Engn Nanjing 210044 Jiangsu Peoples R China Qufu Normal Univ Inst Operat Res Rizhao 276826 Shandong Peoples R China

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.

关键词： Scheduling Step learning Rejection penalty pseudo-polynomial time algorithm Fully polynomial-time approximation scheme Approximation algorithm

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Proportionate flow-shop scheduling with rejection

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2016年第5期67卷 752-769页

作者： Shabtay, Dvir Oron, Daniel Ben Gurion Univ Negev IL-84105 Beer Sheva Israel Univ Sydney Sydney NSW 2006 Australia

In many heavily loaded manufacturing systems, managers routinely make use of outsourcing options in order to maintain reasonable Quality of Service for customers. Thus, there is a strong need to provide tools for managers to economically coordinate sourcing and scheduling decisions. Our main aim is to provide such tools for an important set of flow-shop scheduling problems where rejection (outsourcing) is allowed and processing times are machine independent. Our scheduling problems are essentially bicriteria problems, which combine a scheduling objective and the total outsourcing cost. We study several problems which differ according to the scheduling criterion considered. Moreover, each problem is divided into four different variations depending on the way the two criteria are dealt with. For example, in one variation the two criteria are aggregated into a single objective function;in two other variations the aim consists of minimizing one criterion subject to ensuring that the value of the other criterion will not exceed a predefined threshold. From a theoretical point of view, a computational complexity classification is provided for all variations of the problems under consideration. Moreover, optimization algorithms have been constructed to solve all problem variations, and approximation schemes have been developed for solving hard variations. Those schemes enable managers to solve large instances of hard variations while controlling the maximal gap between the obtained solution and the (unknown) optimal solution.

关键词： scheduling with rejection proportionate flow-shop pseudo-polynomial time algorithm FPTAS

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The just-in-time scheduling problem in a flow-shop scheduling system

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2012年第3期216卷 521-532页

作者： Shabtay, Dvir Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel

We study the problem of maximizing the weighted number of just-in-time (JIT) jobs in a flow-shop scheduling system under four different scenarios. The first scenario is where the flow-shop includes only two machines and all the jobs have the same gain for being completed JIT. For this scenario, we provide an O(n(3)) time optimization algorithm which is faster than the best known algorithm in the literature. The second scenario is where the job processing times are machine-independent. For this scenario, the scheduling system is commonly referred to as a proportionate flow-shop. We show that in this case, the problem of maximizing the weighted number of JIT jobs is NP-hard in the ordinary sense for any arbitrary number of machines. Moreover, we provide a fully polynomial time approximation scheme (FPTAS) for its solution and a polynomial time algorithm to solve the special case for which all the jobs have the same gain for being completed JIT. The third scenario is where a set of identical jobs is to be produced for different customers. For this scenario, we provide an O(n(3)) time optimization algorithm which is independent of the number of machines. We also show that the time complexity can be reduced to O(n log n) if all the jobs have the same gain for being completed JIT. In the last scenario, we study the JIT scheduling problem on m machines with a no-wait restriction and provide an O(mn(2)) time optimization algorithm. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Just-in-time scheduling Flow-shop Complexity pseudo-polynomial time algorithm No-wait restriction

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Single-machine scheduling with trade-off between number of tardy jobs and compression cost

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JOURNAL OF SCHEDULING 2007年第4-5期10卷 303-310页

作者： He, Yong Wei, Qi Cheng, T. C. E. Hong Kong Polytech Univ Dept Logist Kowloon Hong Kong Peoples R China Zhejiang Univ Inst Technol Ningbo 315100 Peoples R China Zhejiang Univ State Key Lab CAD & CG Hangzhou 310027 Peoples R China Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China

We consider a single-machine scheduling problem, in which the job processing times are controllable or compressible. The performance criteria are the compression cost and the number of tardy jobs. For the problem, where no tardy jobs are allowed and the objective is to minimize the total compression cost, we present a strongly polynomial time algorithm. For the problem to construct the trade-off curve between the number of tardy jobs and the maximum compression cost, we present a polynomial time algorithm. Furthermore, we extend the problem to the case of discrete controllable processing times, where the processing time of a job can only take one of several given discrete values. We show that even some special cases of the discrete controllable version with the objective of minimizing the total compression cost are NP-hard, but the general case is solvable in pseudo-polynomial time. Moreover, we present a strongly polynomial time algorithm to construct the trade-off curve between the number of tardy jobs and the maximum compression cost for the discrete controllable version.

关键词： single-machine scheduling problem pseudo-polynomial time algorithm strongly polynomial time algorithm controllable processing times

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Max-min weight balanced connected partition

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第4期57卷 1263-1275页

作者： Wang, Lele Zhang, Zhao Wu, Di Wu, Weili Fan, Lidan Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Xinjiang Peoples R China Univ Texas Dallas Dept Comp Sci Richardson TX 75080 USA

For a connected graph and a positive integral vertex weight function , a max-min weight balanced connected -partition of , denoted as , is a partition of into disjoint vertex subsets such that each (the subgraph of induced by ) is connected, and is maximum. Such a problem has a lot of applications in image processing and clustering, and was proved to be NP-hard. In this paper, we study on a special class of graphs: trapezoid graphs whose maximum degree is bounded by a constant. A pseudo-polynomial time algorithm is given, based on which an FPTAS is obtained for . A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of in terms of the total weight of the graph. In providing such a lower bound, a byproduct of this paper is that any 4-connected trapezoid graph on at least seven vertices has a 4-contractible edge, which may have a value in its own right.

关键词： Balanced connected partition pseudo-polynomial time algorithm FPTAS

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Faster Minimization of Tardy Processing time on a Single Machine

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algorithmICA 2022年第5期84卷 1341-1356页

作者： Bringmann, Karl Fischer, Nick Hermelin, Danny Shabtay, Dvir Wellnitz, Philip Saarland Univ Saarland Informat Campus SIC Saarbrucken Germany Saarland Informat Campus SIC Max Planck Inst Informat Saarbrucken Germany Ben Gurion Univ Negev Dept Ind Engn & Management Beer Sheva Israel

This paper is concerned with the 1 parallel to Sigma p(j)U(j) problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version in O (P . n) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for problem, each improving on Lawler and Moore's algorithm in a different scenario. Our first algorithm runs in (O) over tilde (P-7/4) time, and outperforms Lawler and Moore's algorithm in instances where n = (omega) over tilde (P-3/4). Our second algorithm runs in (O) over tilde (min {P . D-#, P+D}) time, where D-# is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore's algorithm when n = (omega) over tilde (D-#) or n = (omega) over tilde (D/P). Further, it extends the known (O) over tilde ( P ) algorithm for the single due date special case of 1 parallel to Sigma p(j) U-j in a natural way. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new "skewed" version of (max. min)-Convolution which is interesting in its own right.

关键词： Single machine scheduling Tardy processing time pseudo-polynomial time algorithm Fast polynomial multiplication (max, min)-Convolution

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Two-agent parallel-machine scheduling with rejection

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THEORETICAL COMPUTER SCIENCE 2017年 703卷 66-75页

作者： Li, Dawei Lu, Xiwen East China Univ Sci & Technol Sch Sci Shanghai 200237 Peoples R China

We study the two-agent scheduling with rejection on two parallel machines. There are two competing agents A and B with job families J(A) and J(B), respectively. A job in J(A) or J(B) is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on one of the two parallel machines. The objective is to minimize the sum of the given objective function f(A) of the accepted A-jobs and the total rejection penalty of the rejected A-jobs subject to an upper bound on the sum of the given objective function f(B) of the accepted B-jobs and the total rejection penalty of the rejected B-jobs, where f(A) and f(B) are non-decreasing functions on the completion time of the accepted A-jobs and accepted B-jobs, respectively. We consider four scheduling problems associated with different combinations of the two agents' objective functions, f(A) = Sigma C-j(A) and f(B) is an element of {C-max(B), L-max(B), Sigma C-j(B), Sigma W(j)(B)Uj(B)}. When (f(A), f (B) ) = (Sigma C-j(A),C-max(B)), we provide two pseudo-polynomial time algorithms and a fully polynomial-time approximation scheme (FPTAS). For the other problems, we give a pseudo-polynomial time algorithm, respectively. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Agent scheduling Parallel machines Rejection pseudo-polynomial time algorithm FPTAS

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Two-machine flow-shop scheduling with rejection

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COMPUTERS & OPERATIONS RESEARCH 2012年第5期39卷 1087-1096页

作者： Shabtay, Dvir Gasper, Nufar Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the RP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1+epsilon)R and a makespan value of at most (1+epsilon)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Flow-shop scheduling Scheduling with rejection Bicriteria optimization NP-hard Approximation algorithm pseudo-polynomial time algorithm FPTAS

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