Pseudo-polynomial Time Dynamic Programming Algorithm Research Articles

Multitasking scheduling with job switching allowed between alternate periods

This paper explores the multitasking scheduling problem via alternate periods, which is common in manufacturing and service industries. The shift pattern comprises regular shifts interspersed with breaks, allowing job switching. We focus on two alternate periods consisting of odd periods and even periods. Each job is not only processed completely within odd periods or completely within even periods, but job switching is also allowed between alternate periods with additional overhead. For this problem, we propose two job switching models. The first model is alternate periods multitasking scheduling with switching time, where a fixed switching time is incurred whenever a job is switched from one period to its adjacent period. The second model is alternate periods multitasking scheduling with diverse speeds, where an urgent switching cost is produced if a job is processed through the alternate periods with different speeds. For the two models, we study three scheduling objectives including minimising the total completion time, minimising the maximum lateness, and minimising the number of tardy jobs, respectively. For each of the problems under consideration, we investigate the structural properties of the optimal schedule and develop the corresponding pseudo-polynomial time dynamic programming algorithm. Furthermore, we give examples to effectively illustrate the proposed algorithms.

Pulsed Power Load Coordination in Mission- and Time-critical Cyber-physical Systems

Many mission- and time-critical cyber-physical systems deploy an isolated power system for their power supply. Under extreme conditions, the power system must process critical missions by maximizing the Pulsed Power Load (PPL) utility while maintaining the normal loads in the cyber-physical system. Optimal operation requires careful coordination of PPL deployment and power supply processes. In this work, we formulate the coordination problem for maximizing PPL utility under available resources, capacity, and demand constraints. The coordination problem has two scenarios for different use cases, fixed and general normal loads. We develop an exact pseudo-polynomial time dynamic programming algorithm for each scenario with a proven guarantee to produce an optimal coordination schedule. The performance of the algorithms is also experimentally evaluated, and the results agree with our theoretical analysis, showing the practicality of the solutions.

Single-Machine Scheduling with a Deteriorating Maintenance Activity and DeJong’s Learning Effect

In this paper, we study the single-machine scheduling problems with DeJong’s learning effect and deteriorating maintenance activity, where DeJong’s learning effect is a special position-based learning effect and the duration of the deteriorating maintenance activity is a linear increasing function of its starting time. Our goal is to determine the job sequence of all jobs and the position of the maintenance activity to minimize some performance measures. When the performance measures are the makespan and the total completion time, we show that both of them can be solved in polynomial time. When the performance measure is the total weighted completion time, we develop a pseudo-polynomial time dynamic programming algorithm under a special case. When the performance measure is the maximum lateness, we show that the earliest due date first (EDD) order is a bad algorithm for the general case, and develop a polynomial time algorithm under a special case.

On the exact separation of cover inequalities of maximum-depth

We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly {mathcal {N}}{mathcal {P}}-hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

Iterated local search for single machine total weighted tardiness batch scheduling

This paper presents an iterated local search (ILS) algorithm for the single machine total weighted tardiness batch scheduling problem. To our knowledge, this is one of the first attempts to apply ILS to solve a batching scheduling problem. The proposed algorithm contains a local search procedure that explores five neighborhood structures, and we show how to efficiently implement them. Moreover, we compare the performance of our algorithm with dynamic programming-based implementations for the problem, including one from the literature and two other ones inspired in biased random-key genetic algorithms and ILS. We also demonstrate that finding the optimal batching for the problem given a fixed sequence of jobs is $$\mathcal {NP}$$ -hard, and provide an exact pseudo-polynomial time dynamic programming algorithm for solving such problem. Extensive computational experiments were conducted on newly proposed benchmark instances, and the results indicate that our algorithm yields highly competitive results when compared to other strategies. Finally, it was also observed that the methods that rely on dynamic programming tend to be time-consuming, even for small size instances.

Due date assignment and two-agent scheduling under multitasking environment

This paper addresses a two-agent scheduling problem with due date assignment under multitasking environment, in which the due dates of the jobs from the first agent are decision variables to be determined using the unrestricted (usually referred to as DIF) due date assignment method. Each agent requests the processing of its own set of jobs on a machine and wishes to minimize a certain scheduling criterion related to the completion times of its jobs only. Under multitasking, when a job (primary job) is processed, it is inevitably interrupted by other jobs (waiting jobs) that are available but unfinished, and the amount of time that each waiting job interrupting the primary job is a linear function of the remaining processing time of the waiting job. The overall objective is to determine the optimal primary job sequence along with the due dates of the jobs from the first agent as to minimize the weighted sum of the due date assignment cost and weighted number of late jobs from the first agent, while maintaining the total completion time of the jobs from the second agent not exceeding a given threshold. We show that the problem is $$\mathcal {NP}$$-hard, devise a pseudo-polynomial time dynamic programming algorithm, establishing that it is $$\mathcal {NP}$$-hard in the ordinary sense, and demonstrate that it admits a fully polynomial-time approximation scheme.

A single-item lot-sizing problem with a by-product and inventory capacities

The high pace of waste accumulation in landfills and the depletion of scarce natural resources lead us to seek pathways for converting unavoidable production outputs into useful and high added-value products. In this context, we formalize and propose a model for the single-item lot-sizing problem, which integrates the management of unavoidable production residues classified as by-products. During the production process of a main product, a by-product is generated, stored in a limited capacity and transported with a fixed transportation cost. This problem is investigated for two cases of the by-product inventory capacity: time-dependent and constant. We prove that the problem with inventory capacities is NP-Hard. To solve it optimally, we develop a pseudo-polynomial time dynamic programming algorithm. For the case with stationary inventory capacities, a polynomial time dynamic programming algorithm is proposed.

Automated partial atomic charge assignment for drug-like molecules: a fast knapsack approach

A key factor in computational drug design is the consistency and reliability with which intermolecular interactions between a wide variety of molecules can be described. Here we present a procedure to efficiently, reliably and automatically assign partial atomic charges to atoms based on known distributions. We formally introduce the molecular charge assignment problem, where the task is to select a charge from a set of candidate charges for every atom of a given query molecule. Charges are accompanied by a score that depends on their observed frequency in similar neighbourhoods (chemical environments) in a database of previously parameterised molecules. The aim is to assign the charges such that the total charge equals a known target charge within a margin of error while maximizing the sum of the charge scores. We show that the problem is a variant of the well-studied multiple-choice knapsack problem and thus weakly mathcal {NP}-complete. We propose solutions based on Integer Linear Programming and a pseudo-polynomial time Dynamic Programming algorithm. We demonstrate that the results obtained for novel molecules not included in the database are comparable to the ones obtained performing explicit charge calculations while decreasing the time to determine partial charges for a molecule from hours or even days to below a second. Our software is openly available.

Designing Green Communication Systems for Smart and Connected Communities via Dynamic Spectrum Access

Smart and connected communities (SCCs) are emerging as a novel paradigm that allows the community residents to be connected with surrounding environments through smart technologies. However, there remain important challenges to fully exploit the potential of SCCs in improving societal well-being and prosperity. In particular, there is a need for designing green communication systems that are also capable of providing high quality of service (QoS) to distribute and collect information to and from SCCs. However, simultaneously satisfying both of these criteria is difficult due to varying demands posed by heterogeneous sensing modalities, lack of dedicated infrastructure in rural/sub-urban areas, and certain sustainability constraints. While low-power short-range technologies often fail to achieve high QoS, using 3G or 4G technologies (LTE, LTE-A, GSM) for SCCs will eventually face spectrum scarcity and cross technology interference. In recent times, Dynamic spectrum access (DSA) has been proposed as a solution to overcome policy constraints and improve spectrum scarcity by spectrum sharing. In this article, we show that harnessing DSA in the context of SCCs can also achieve notable benefits in terms of energy efficiency and sustainability. Specifically, we propose a novel architecture for designing sustainable SCCs using a small-scale DSA-enabled overlay network that improves end-to-end energy efficiency of the network while guaranteeing QoS. We also propose a dynamic spectrum band selection approach that intelligently matches any message requirement to a suitable band type by exploiting distinct electro-magnetic characteristics of various bands. Since data generated in SCCs are typically valuable only when delivered within a certain hard (or soft ) deadline, we formulate a linear optimization problem for determining the most energy-efficient path that ensures a delivery time within the hard deadline. After proving that such a problem is NP-Hard, we propose an exact pseudo-polynomial time dynamic programming algorithm to solve it followed by a polynomial time greedy heuristic. Additionally, we formulate a non-linear optimization problem to find the optimal path when the message delivery time is defined as a soft deadline and extend our greedy heuristic to handle soft deadlines. Compared to the homogeneous band access approaches that opportunistically access free channels within a given spectrum band, our extensive simulation study shows that the proposed dynamic multi-band selection approach significantly improves the achievable energy efficiency while meeting various hard and soft deadlines.

Cost allocation in rescheduling with machine unavailable period

We study a rescheduling problem faced by multiple job owners sharing a single machine, where jobs need to be rescheduled when the machine becomes unavailable for a period of time. The disruption caused by any new schedule is restricted such that the difference between the completion times in the initial and the new schedules of any job is no more than a given threshold. A natural way to reschedule is to process the jobs in the initial sequence, each as early as possible. This defines a feasible schedule over which cost saving can potentially be achieved by optimal rescheduling as long as the cost saving can be fairly shared by job owners. We define a cooperative game for job owners accordingly, to share the cost saving. Given that the optimization problem is computationally intractable, we find several optimal properties and develop an optimal pseudopolynomial time dynamic programming algorithm for rescheduling. We provide a simple closed form core allocation of the total cost saving for all the jobs, and also provide the Shapley value of the game in a computable form. Then we computationally evaluate the extra cost caused by machine unavailability, the cost saving from optimization relative to the naturally constructed schedule, the likelihood for the Shapley value to be a core allocation, and how the Shapley value allocates cost saving among job owners. Managerial insights are derived from the computational studies. This work contributes to the literature by explicitly incorporating two classic scheduling topics: sequencing game and rescheduling.

Complexity results for an integrated single machine scheduling and outbound delivery problem with fixed sequence

In this paper, we consider an integrated production and outbound delivery scheduling problem. In particular, we address the situation in which the scheduling sequence and the delivery sequence are the same and predefined. A set of jobs are processed on a single machine, and finished jobs are delivered to the customers by a single capacitated vehicle. Each job has a processing time, and transportation times between customers are taken into account. Because the sequence is given, the problem consists in forming batches of jobs and our objective is to minimize the sum of the delivery times or general functions of the delivery times. The NP-hardness of the general problem is established, and a pseudopolynomial time dynamic programming algorithm is given. Some particular cases are treated, for which NP-hardness proofs and polynomial time algorithms are given. Finally, a fixed-parameter tractability result is given.

Complexity results and algorithms for an integrated single machine scheduling and outbound delivery problem with fixed sequence

In this paper, we consider an integrated production and outbound delivery scheduling problem. In particular, we address the situation in which the scheduling sequence and the delivery sequence are the same and predefined. A set of jobs are processed on a single machine and finished jobs are delivered to the customers by a single capacitated vehicle. Each job has a processing time and transportation times between customers are taken into account. Since the sequence is given, the problem consists to form batches of jobs and our objective is to minimize the sum of the delivery times or general functions of the delivery times. The NP-hardness of the general problem is established and a pseudopolynomial time dynamic programming algorithm is given. Some particular cases are treated, for which NP-hardness proofs and polynomial time algorithms are given. Finally, a fixed-parameter tractability result is given.

NP-hard and polynomial cases for the single-item lot sizing problem with batch ordering under capacity reservation contract

In this paper, we study the single-item lot sizing problem under a capacity reservation contract. A manufacturer is replenished by an external supplier with batch deliveries and a certain capacity is reserved at the supplier level with an advantageous cost. In addition to the classical ordering and inventory holding costs, for each batch ordered under the reserved capacity a fixed cost per batch is incurred; and for batches exceeding this capacity a higher fixed cost per batch is paid, typically through the purchase from the spot market. We identify various NP-hard cases, propose a pseudo-polynomial time dynamic programming algorithm under arbitrary parameters, show that the problem admits an FPTAS and give polynomial time algorithms for special cases. We finally state a list of open problems for further research.

An FPTAS for the parallel two-stage flowshop problem

We consider the NP-hard m-parallel two-stage flowshop problem, abbreviated as the (m,2)-PFS problem, where we need to schedule n jobs to m parallel identical two-stage flowshops in order to minimize the makespan, i.e. the maximum completion time of all the jobs on the m flowshops. The (m,2)-PFS problem can be decomposed into two subproblems: to assign the n jobs to the m parallel flowshops, and for each flowshop to schedule the jobs assigned to the flowshop. We first present a pseudo-polynomial time dynamic programming algorithm to solve the (m,2)-PFS problem optimally, for any fixed m, based on an earlier idea for solving the (2,2)-PFS problem. Using the dynamic programming algorithm as a subroutine, we design a fully polynomial-time approximation scheme (FPTAS) for the (m,2)-PFS problem.

The single processor total weighted completion time scheduling problem with the sum-of-processing-time based learning model

In this paper, we analyse the single processor total weighted completion time scheduling problem with the learning effect, where the processing time of each job is a non-increasing function dependent on the sum of the normal processing times of preceding jobs. We prove that the considered problem is at least NP-hard. Moreover, a pseudopolynomial time dynamic programming algorithm that optimally solves the problem with a step learning function (curve) is constructed. Furthermore, fast approximation algorithms for the general version of the problem, where job processing times are described by arbitrary functions dependent on the sum of the normal job processing times, are provided. Their efficiency is verified numerically and for Weighted Shortest Processing Times algorithm a worst case analysis is also performed.

Reverse time-restricted shortest paths: Application to air traffic management

In this paper, we consider a particular class of network flow problems that seeks a shortest path, if it exists, between a source node s and a destination node d in a connected digraph, such that we arrive at node d at a specified time τ while leaving node s no earlier than a lower-bounding time LB, and where the availability of each network link is time-dependent in the sense that it can be traversed only during specified intervals of time. We refer to this problem as the reverse time-restricted shortest path problem (RTSP), and it arises, for example, in the context of generating flight plans within air traffic management approaches under severe convective weather conditions. We show that this problem is NP-hard in general, but is polynomially solvable under a special regularity condition. A pseudo-polynomial time dynamic programming algorithm is developed to solve Problem RTSP, along with an effective heap implementation strategy. Computational results using real flight generation test cases as well as random simulated problems are presented.

Makespan minimization in single-machine scheduling with step-deterioration of processing times

This paper considers a single-machine scheduling problem of minimizing the maximum completion time for a set of independent jobs. The processing time of a job is a non-linear step function of its starting time and due date. The problem is already known to be 𝒩𝒫-hard in the literature. In this paper, we first show this problem to be 𝒩𝒫-hard in the ordinary sense by proposing a pseudo-polynomial time dynamic programming algorithm. Then, we develop two dominance rules and a lower bound to design a branch-and-bound algorithm for deriving optimal solutions. Numerical results indicate that the proposed properties can effectively reduce the time required for exploring the solution space.

A dynamic programming algorithm for scheduling jobs in a two-machine open shop with an availability constraint

This paper studies a two-machine open shop scheduling problem with an availability constraint, ie we assume that a machine is not always available and that the processing of the interrupted job can be resumed when the machine becomes available again. We consider the makespan minimization as criterion. This problem is NP-hard. We develop a pseudo-polynomial time dynamic programming algorithm to solve the problem optimally when the machine is not available at time s>0. Then, we propose a mixed integer linear programming formulation, that allows to solve instances with up to 500 jobs optimally in less than 5 min with CPLEX solver. Finally, we show that any heuristic algorithm has a worst-case error bound of 1.

Fabrication scheduling on a single machine with due date constraints

There is a fabrication machine available for processing a set of jobs. Each job is associated with a due date and consists of two parts, one is common among all products and the other is unique to itself. The unique components are processed individually and the common parts are grouped into batches for processing. A constant setup time is incurred when each batch is formed. The completion time of a job is defined as the time when both of its unique and common components are completed. In this paper, we consider two different objectives. The first problem seeks to minimize the maximum tardiness, and the second problem is to minimize the number of tardy jobs. To minimize the maximum tardiness, we propose a dynamic programming algorithm that optimally solves the problem in polynomial time. Next, we show NP-hardness proof and design a pseudo-polynomial time dynamic programming algorithm for the problem of minimizing the number of tardy jobs.

Minimizing the weighted number of tardy jobs and maximum tardiness in relocation problem with due date constraints

The relocation problem addressed in this paper is to determine a reconstruction sequence for a set of old buildings, under a limited budget, such that there is adequate temporary space to house the residents decanted during rehabilitation. It can be regarded as a resource-constrained scheduling problem where there is a set of jobs to be processed on a single machine. Each job demands a number of resources for processing and returns probably a different number of resources on its completion. Given a number of initial resources, the problem seeks to determine if there is a feasible sequence for the successful processing of all the jobs. Two generalizations of the relocation problem in the context of single machine scheduling with due date constraints are studied in this paper. The first problem is to minimize the weighted number of tardy jobs under a common due date. We show that it is NP-hard even when all the jobs have the same tardy weight and the same resource requirement. A dynamic programming algorithm with pseudo-polynomial computational time is proposed for the general case. In the second problem, the objective is to minimize the maximum tardiness when each job is associated with an individual due date. We prove that it is strongly NP-hard. We also propose a pseudo-polynomial time dynamic programming algorithm for the case where the number of possible due dates is predetermined.