Dynamic programming for proportionate flow-shop scheduling with due-date assignment

A note on study on proportionate flowshop scheduling with due-date assignment and position-dependent weights

Single machine scheduling with due date assignment to minimize the total weighted lead time penalty and late work

Given processing times of n jobs on a single machine with penalties for earliness and tardiness and penalties associated with assignment of due-dates, the objective is to select optimal due-dates and optimal sequence. Scheduling procedure for the solution of this problem is presented along with proof of optimality and illustrative numerical examples.

This paper considers single machine scheduling and due date assignment problems in which the jobs need to be delivered to customers after processing. It is assumed that the delivery times are proportional to the length of the already processed jobs, and a job's processing time depends on its position in a sequence. The objective functions include total earliness, the weighted number of tardy jobs and the cost of due date assignment. We analyze the problems with two different due date assignment methods and conclude that the problems are polynomial time solvable. We provide a dynamic programming algorithm with O(n3) times for the problems.

An open-shop schedule has no restrictions on the processing order of the jobs. The problem of minimizing the total earliness and tardiness penalties in an m-machine nonpreemptive open-shop is NP-hard. A network structure based on the spanning tree is constructed in this work to simplify the mathematical model of a nonpreemptive open-shop problem whose objective is to minimize total earliness and tardiness penalties. The idle time rule developed in this work serves as the decision criterion in testing the performance. The proposed heuristic is compared with SPT, EDD, SLACK, and MWSTR/EST methods using 7,200 cases which are constructed by different features. Expenmental results mdlcate that the proposed heuristic performs much better than the other four methods under the total tardiness and earliness penalties criterion.

We study the problem of two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times. Under the condition of the due-date assignment with common flow allowance (i.e. slack (SLK) due-date assignment), we provide a bi-criteria analysis where the first criterion is to minimise the scheduling criteria (i.e. the weighted sum of earliness, tardiness and flow allowance costs), and the second criterion is to minimise the resource consumption cost (i.e. the weighted sum of resource consumption cost). The objective is to determine the optimal job sequence, resource allocations and common (flow allowance) slack time that minimise the three different versions of the two criteria. We prove that these problems can be solved in polynomial time.

In this short note, one data driven model predictive control is studied to design the optimal control sequence. The idea of data driven means the actual output value in cost function for model predictive control is identi_ed through input-output observed data in case of unknown but bounded noise and martingale di_erence sequence. After substituting the identi_ed actual output in cost function, the total cost function in model predictive control is reformulated as the other standard form, so that dynamic programming can be applied directly. As dynamic programming is only used in optimization theory, so to extend its advantage in control theory, dynamic programming algorithm is proposed to construct the optimal control sequence. Furthermore, stability analysis for data drive model predictive control is also given based on dynamic programming strategy. Generally, the goal of this short note is to bridge the dynamic programming, system identi_cation and model predictive control. Finally, one simulation example is used to prove the e_ciency of our proposed theory

Recently, as the market share of electric vehicles (EVs) has increased, how to handle the increased electricity demand for EV charging in the power grid and how to use EV batteries from a grid-operating aspect have become more important. Also, from the perspective of individual EVs, Vehicle-to-Grid (V2G) technologies that reduce the cost for each vehicle’s charging in conjunction with the power grid are significant. In this paper, the V2G control problem at the individual vehicle level is studied using a Dynamic Programming (DP) algorithm that considers EVs’ charging efficiency. The DP algorithm is developed to generate an optimized charging/discharging power profile that minimizes electricity costs, while satisfying the constraints of the initial and final battery states of charge, for given a time-of-use electricity price. To show the effectiveness of the proposed algorithm, simulation is conducted for three different charging scenarios (unidirectional charging, bidirectional charging, and unidirectional charging with cost variations based on electricity usage), and the results showed that DP can achieve significant cost savings of about 30% compared to the normal charging method. Also, the result of DP is compared with that of Linear Programming, demonstrating that DP outperforms Linear Programming in cost savings for the V2G control problem.

Multiple Sequence Alignment (MSA) is one of the most fundamental problems in computational molecular biology. The running time of the best known scheme for finding an optimal alignment, based on dynamic programming, increases exponentially with the number of input sequences. Hence, many heuristics were suggested for the problem. We consider a version of the MSA problem where the goal is to find an optimal alignment in which matches are restricted to positions in predefined matching segments. We present several techniques for making the dynamic programming algorithm more efficient, while still finding an optimal solution under these restrictions. We prove that it suffices to find an optimal alignment of the predefined sequence segments, rather than single letters, thereby reducing the input size and thus improving the running time. We also identify "shortcuts" that expedite the dynamic programming scheme. Empirical study shows that, taken together, these observations lead to an improved running time over the basic dynamic programming algorithm by 4 to 12 orders of magnitude, while still obtaining an optimal solution. Under the additional assumption that matches between segments are transitive, we further improve the running time for finding the optimal solution by restricting the search space of the dynamic programming algorithm.

In this paper, a dynamic programming (DP) algorithm is proposed to compute the maximum a posteriori probability (MAP) estimate of the random location of multiple sources by passive sensor array. Based on the principle of optimality, the multivariate MAP decision problem can be transformed into a multistage one‐dimensional maximization problem. By a recursive computation and backward optimum searching technique, the MAP solution can be obtained. The computational complexity is much less than that of the direct MAP searching method, especially when the number of sensors and/or the number of sources is large. In addition, the DP computation technique is equally applicable both in the case of correlated sources and the case of uncorrelated sources.

In this work, we study dynamic programming (DP) algorithms for partially observable Markov decision processes with jointly continuous and discrete state-spaces. We consider a class of stochastic systems which have coupled discrete and continuous systems, where only the continuous state is observable. Such a family of systems includes many realworld systems, for example, Markovian jump linear systems and physical systems interacting with humans. A finite history of observations is used as a new information state, and the convergence of the corresponding DP algorithms is proved. In particular, we prove that the DP iterations converge to a certain bounded set around an optimal solution. Although deterministic DP algorithms are studied in this paper, it is expected that this fundamental work lays foundations for advanced studies on reinforcement learning algorithms under the same family of systems.

Routing problems are a class of combinatorial problems with many practical applications. Recently, end-to-end deep learning methods have been proposed to learn approximate solution heuristics for such problems. In contrast, classical dynamic programming (DP) algorithms guarantee optimal solutions, but scale badly with the problem size. We propose Deep Policy Dynamic Programming (DPDP), which aims to combine the strengths of learned neural heuristics with those of DP algorithms. DPDP prioritizes and restricts the DP state space using a policy derived from a deep neural network, which is trained to predict edges from example solutions. We evaluate our framework on the travelling salesman problem (TSP), the vehicle routing problem (VRP) and TSP with time windows (TSPTW) and show that the neural policy improves the performance of (restricted) DP algorithms, making them competitive to strong alternatives such as LKH, while also outperforming most other ‘neural approaches’ for solving TSPs, VRPs and TSPTWs with 100 nodes.KeywordsDynamic ProgrammingDeep LearningVehicle Routing

The powertrain model of the series-parallel plug-in hybrid electric vehicles (PHEVs) is more complicated, compared with series PHEVs and parallel PHEVs. Using the traditional dynamic programming (DP) algorithm or Pontryagin minimum principle (PMP) algorithm to solve the global-optimization-based energy management strategies of the series-parallel PHEVs is not ideal, as the solution time is too long or even impossible to solve. Chief engineers of hybrid system urgently require a handy tool to quickly solve global-optimization-based energy management strategies. Therefore, this paper proposed to use the Radau pseudospectral knotting method (RPKM) to solve the global-optimization-based energy management strategy of the series-parallel PHEVs to improve computational efficiency. Simulation results showed that compared with the DP algorithm, the global-optimization-based energy management strategy based on the RPKM improves the computational efficiency by 1806 times with a relative error of only 0.12%. On this basis, a bi-level nested component-sizing method combining the genetic algorithm and RPKM was developed. By applying the global-optimization-based energy management strategy based on RPKM to the actual development, the feasibility and superiority of RPKM applied to the global-optimization-based energy management strategy of the series-parallel PHEVs were further verified.

This paper considers a joint order acceptance and scheduling problem under a general scenario. A manufacturer receives multiple orders with a given revenue, processing time, release date, due date, deadline, and earliness and tardiness penalties. The manufacturer can be seen as a single-machine system. Due to limited capacity, the manufacturer cannot process every order and needs to determine the optimal set of accepted orders and corresponding production schedule such that the total profit is maximized. The manufacturer can extend its capacity with overtime by paying an additional cost. A time-indexed formulation is presented to model the problem. Two exact algorithms are proposed. The first algorithm, denoted by DPIA-GR, is a dynamic programming (DP)-based algorithm that starts by solving a relaxed version of the original model and successively recovers the relaxed constraint until an optimal solution to the original problem is achieved. The second algorithm, denoted by DPIA-LRGR, improves DPIA-GR by incorporating Lagrangian relaxation (LR). The subgradient method is employed to find the optimal Lagrangian multipliers. The relaxed model in DPIA-GR and the LR model in DPIA-LRGR can be represented using a weighted di-graph. Both algorithms are equivalent to finding the longest path in the graph and applying a graph reduction strategy to prevent unnecessary computational time and memory usage. A genetic algorithm (GA) is also proposed to solve large-scale versions of the problem. Numerical experiments show that both DPIA-GR and DPIA-LRGR solve the problem efficiently and outperform CPLEX and GA, but DPIA-LRGR offers better performance.

This study mainly analyses the fuel consumption of plug-in hybrid vehicles during operation. A new control method for automobiles based on energy management strategy and dynamic programming algorithm is proposed. The new method plans and analyses the minimum electricity consumption, and then uses dynamic programming algorithms to analyse this parameter. The research results indicated that the vehicle state was constantly changing with the variation of SOC value during driving. The energy mobilization of the vehicle was more obvious after adding dynamic programming strategy. The efficiency of the vehicle was relatively high in driving state 1, with a minimum value of 70%, which was about 20% higher than in driving state 4. The average fuel consumption in driving state 2 was 1.8L higher than in other driving states. The overall efficiency of automobiles after incorporating dynamic programming was improved, with a shorter time to reach the lowest efficiency point compared with not incorporating dynamic programming algorithms. The highest efficiency value was 7.86% higher than that of not incorporating dynamic programming models. The new control method can reduce energy consumption and improve the energy management and control effect. The study provides a better research direction for energy management and control of hybrid electric vehicles in the future.