Restricted Dynamic Programming Research Articles

Bilevel interactive optimisation for rebatching scheduling problem with selectivity banks in high variety flow line production

Mass customization enables the integration of traditional flow line production with product platforms to accommodate abundant product-process varieties. These platform-based flow lines explore common process routes while highlighting rebatching scheduling with selectivity banks (RBS) to handle large process varieties across production stages at minimum setup cost. Given the inherent coupling between decision making in job diverging and retrieval quality, an interactive optimization approach is necessary for the RBS problem. This study proposes a bilevel interactive optimization (BIO) model for RBS to accommodate high variety flow line production. The model addresses the conflicting goals of lane occupancy cost, process setup cost, and job divergence and retrieval efficiency. Regarding job divergence at the leader-level, a vehicle routeing problem with precedence constraints is formulated and solved by a constructed genetic algorithm (GA). Concerning job retrieval at the follower-level and the ongoing characteristic of selectivity banks, a dispatching problem with various batch size preference and dynamic time window is established and dealt with a restricted dynamic programming (RDP) algorithm after balancing search efficiency and accuracy. Thus, to solve the BIO, a hybrid GA-RDP is developed and implemented. A practical application to an automotive painting shop illustrates the operational benefits of the BIO model for the RBS problem.

A risk-constrained time-dependent cash-in-transit routing problem in multigraph under uncertainty

The Cash-in-Transit (CIT) deals with the transportation of banknotes, coins, and other valuable items. Due to the high-value density of these products, incorporating security strategies in the carrier operations is crucial. This paper proposes new CIT models involving deterministic and stochastic time-varying traffic congestion. Since risk exposure of a vehicle is proportional to the time-dependent travel time, a new formula is introduced to measure the risk of traveling. Moreover, this study covers one of the important weaknesses of previous CIT routing models by investigating the problem in multigraph networks. Multigraph representation maintains a set of non-dominated parallel arcs, which are differentiated by two attributes including travel time and robbery risk. Considering maximum allowable time duration together with a risk threshold yields to design a more balanced routing scheme. Multi-attribute parallel arcs in a stochastic time-dependent network bring high computational challenges. Herein, we introduce efficient algorithms including a novel flexible restricted Dynamic Programming and a self-adaptive caching Genetic Algorithm. The proposed algorithms are tested on both a real case study in Isfahan metropolis and generated instances. Ultimately, sensitivity analyses are conducted to assess the importance of the use of multigraph networks in the CIT and to provide significant managerial insights for administrators and practitioners.

Improving dynamic programming for travelling salesman with precedence constraints: parallel Morin–Marsten bounding

The precedence constrained traveling salesman (TSP-PC), also known as sequential ordering problem (SOP), consists of finding an optimal tour that satisfies the namesake constraints. Mixed integer-linear programming works well with the ‘lightly constrained’ TSP-PCs, close to asymmetric TSP, as well as the with the ‘heavily constrained’ (Gouveia, Ruthmair, 2015). Dynamic programming (DP) works well with the heavily constrained (Salii, 2019). However, judging by the open TSPLIB SOP instances, the worst for any method are the ‘medium’. We implement a parallel Morin–Marsten branch-and-bound scheme for DP (DPBB). We show how the lower bound heuristic parameterizes DPBB's worst-case complexity and DPBB ‘inherits’ the abstract travel cost aggregation feature of the DP, permitting its direct use with both the conventional and bottleneck TSP-PC. The scheme was tested on TSPLIB instances, with best known upper bounds (TSP-PC), or those found by restricted DP (Bottleneck TSP-PC), and lower bounds from a greedy-type heuristic. Our OPENMP-based parallel implementation achieved 20-fold speedup for larger instances. We close the long-standing kro124p.4.sop (conventional TSP-PC) and both kro124p.4.sop and ry48p.2.sop (Bottleneck TSP-PC).

Performance Comparison of Two Recent Heuristics for Green Time Dependent Vehicle Routing Problem

This article concerns a green Time Dependent Capacitated Vehicle Routing Problem (TDCVRP) which is confronted in urban distribution planning. The problem is formulated as a Markovian Decision Process and a dynamic programming (DP) approach has been used for solving the problem. The article presents a performance comparison of two recent heuristics for the green TDCVRP that explicitly accounts for time dependent vehicle speeds and fuel consumption (emissions). These heuristics are the classical Restricted Dynamic Programming (RDP) algorithm, and the Simulation Based RDP that consists of weighted random sampling, RDP heuristic and simulation. The numerical experiments show that the Simulation Based Restricted Dynamic Programming heuristic can provide promising results within relatively short computational times compared to the classical Restricted Dynamic Programming for the green TDCVRP.

Efficient solution algorithms for a time-critical reliable transportation problem in multigraph networks with FIFO property

Relief distribution in urban environments is one of the major activities in emergency logistics management. The effective and time-saving dispatching process in affected areas is pivotal in rescue operations. In this study, we formulate a reliable time-dependent vehicle routing problem with time windows in a multigraph based network. In such networks, there exist parallel arcs with multiple attributes between nodes. The purpose of the provided model is to minimize delays in delivering prioritized items in disaster response operations. It also controls the minimum reliability of each route. Controlling the reliability in relief distribution gives this assurance that emergency packages on vehicles can reach their destinations safely and in a timely manner. In order to solve the problem, a novel restricted dynamic programming is applied to the problem through the giant-tour representation. The proposed algorithm can reach the optimal solution when utilized in an unrestricted way. In addition, a modified caching genetic algorithm and a three-phase optimization method based on the tabu search heuristic are provided to deal with larger instances in reasonable computation times. Finally, a real transportation case is presented to illustrate the potential applicability of the model in urban environments. The results accentuate the efficiency of the proposed methods and show the significance of multigraph to accelerate the distribution operations for reliable emergency logistics planning.

A Simulation Based Restricted Dynamic Programming approach for the Green Time Dependent Vehicle Routing Problem

This paper addresses a Green Time Dependent Capacitated Vehicle Routing Problem that accounts for transportation emissions. The problem has been formulated and solved using Dynamic Programming approach. The applicability of Dynamic Programming in large sized problems is, however, limited due to exponential memory and computation time requirements. Therefore, we propose a generic heuristic approach, Simulation Based Restricted Dynamic Programming, based on weighted random sampling, the classical Restricted Dynamic Programming heuristic and simulation for the model to solve large sized instances. These decision support tools can be used to aid logistics decision-making processes in urban distribution planning. The added values of the proposed model and the heuristic have been shown based on a real life urban distribution planning problem between a pharmaceutical warehouse and a set of pharmacies, and ten relatively larger instances. The results of the numerical experiments show that the Simulation Based Restricted Dynamic Programming heuristic can provide promising results within relatively short computation times compared to the classical Restricted Dynamic Programming for the Green Time Dependent Capacitated Vehicle Routing Problem. The Simulation Based Restricted Dynamic Programming algorithm yields 2.3% lower costs within 93.1% shorter computation times on average, compared to the classical Restricted Dynamic Programming. Moreover, the analyses on the effect of traffic congestion in our base case reveal that 2.3% benefit on total emissions and 0.9% benefit on total routing cost could be obtained if vehicles start delivery after heavy congested period is passed.

Hybrid segmentation of left ventricle in cardiac MRI using gaussian-mixture model and region restricted dynamic programming

Segmentation of the left ventricle from cardiac magnetic resonance images (MRI) is very important to quantitatively analyze global and regional cardiac function. The aim of this study is to develop a novel and robust algorithm which can improve the accuracy of automatic left ventricle segmentation on short-axis cardiac MRI. The database used in this study consists of three data sets obtained from the Sunnybrook Health Sciences Centre. Each data set contains 15 cases (4 ischemic heart failures, 4 non-ischemic heart failures, 4 left ventricle (LV) hypertrophies and 3 normal cases). Three key techniques are developed in this segmentation algorithm: (1) ray scanning approach is designed for segmentation of images with left ventricular outflow tract (LVOT), (2) a region restricted technique is employed for epicardial contour extraction, and (3) an edge map with non-maxima gradient suppression approach is put forward to improve the dynamic programming to derive the epicardial boundary. The validation experiments were performed on a pool of data sets of 45 cases. For both endo- and epi-cardial contours of our results, percentage of good contours is about 91%, the average perpendicular distance is about 2mm. The overlapping dice metric is about 0.92. The regression and determination coefficient between the experts and our proposed method on the ejection fraction (EF) is 1.01 and 0.9375, respectively; they are 0.9 and 0.8245 for LV mass. The proposed segmentation method shows the better performance and is very promising in improving the accuracy of computer-aided diagnosis systems in cardiovascular diseases.

Automatic Left Ventricle Segmentation in Cardiac MRI Using Topological Stable-State Thresholding and Region Restricted Dynamic Programming

Segmentation of the left ventricle (LV) is very important in the assessment of cardiac functional parameters. The aim of this study is to develop a novel and robust algorithm which can improve the accuracy of automatic LV segmentation on short-axis cardiac magnetic resonance images (MRI). The database used in this study consists of 45 cases obtained from the Sunnybrook Health Sciences Centre. The 45 cases contain 12 ischemic heart failures, 12 non-ischemic heart failures, 12 LV hypertrophies, and 9 normal cases. Three key techniques are developed in this segmentation algorithm: 1) topological stable-state thresholding method is proposed to refine the endocardial contour, 2) an edge map with non-maxima gradient suppression approach, and 3) a region-restricted technique that is proposed to improve the dynamic programming to derive the epicardial boundary. The validation experiments were performed on a pool of data sets of 45 cases. For both endo- and epicardial contours of our results, percentage of good contours is about 91%, the average perpendicular distance is about 2 mm, and the overlapping dice metric is about 0.91. The regression and determination coefficient for the experts and our proposed method on the ejection fraction is 1.05 and 0.9048, respectively; they are 0.98 and 0.8221 for LV mass. An automatic method using topological stable-state thresholding and region restricted dynamic programming has been proposed to segment left ventricle in short-axis cardiac MRI. Evaluation results indicate that the proposed segmentation method can improve the accuracy and robust of left ventricle segmentation. The proposed segmentation approach shows the better performance and has great potential in improving the accuracy of computer-aided diagnosis systems in cardiovascular diseases.

Vehicle routing under time-dependent travel times: The impact of congestion avoidance

Daily traffic congestion forms a major problem for businesses such as logistic service providers and distribution firms. It causes late arrivals at customers and additional costs for hiring the truck drivers. Such costs caused by traffic congestion can be reduced by taking into account and avoiding predictable traffic congestion within vehicle route plans. In the literature, various strategies are proposed to avoid traffic congestion, such as selecting alternative routes, changing the customer visit sequences, and changing the vehicle-customer assignments. We investigate the impact of these and other strategies in off-line vehicle routing on the performance of vehicle route plans in reality. For this purpose, we develop a set of vehicle routing problem instances on real road networks, and a speed model that reflects the key elements of peak hour traffic congestion. The instances are solved for different levels of congestion avoidance using a modified Dijkstra algorithm and a restricted dynamic programming heuristic. Computational experiments show that 99% of late arrivals at customers can be eliminated if traffic congestion is accounted for off-line. On top of that, about 87% of the extra duty time caused by traffic congestion can be eliminated by clever congestion avoidance strategies.

Restricted dynamic programming: A flexible framework for solving realistic VRPs

Most successful solution methods for solving large vehicle routing and scheduling problems are based on local search. These approaches are designed and optimized for specific types of vehicle routing problems (VRPs). VRPs appearing in practice typically accommodate restrictions that are not accommodated in classical VRP models, such as time-dependent travel times and driving hours regulations. We present a new construction framework for solving VRPs that can handle a wide range of different types of VRPs. In addition, this framework accommodates various restrictions that are not considered in classical vehicle routing models, but that regularly appear in practice. Within this framework, restricted dynamic programming is applied to the VRP through the giant-tour representation. This algorithm is a construction heuristic which for many types of restrictions and objective functions leads to an optimal algorithm when applied in an unrestricted way. We demonstrate the flexibility of the framework for various restrictions appearing in practice. The computational experiments demonstrate that the framework competes with state of the art local search methods when more realistic constraints are considered than in classical VRPs. Therefore, this new framework for solving VRPs is a promising approach for practical applications.

A Dynamic Programming Heuristic for the Vehicle Routing Problem with Time Windows and European Community Social Legislation

In practice, apart from the problem of vehicle routing, schedulers also face the problem of finding feasible driver schedules complying with complex restrictions on drivers' driving and working hours. To address this complex interdependent problem of vehicle routing and break scheduling, we propose a restricted dynamic programming heuristic for the vehicle routing problem with time windows and the full European social legislation on drivers' driving and working hours. The problem we consider includes all rules in this legislation, whereas in the literature only a basic set of rules has been addressed. In addition to this basic set of rules, the legislation contains a set of modifications that allow for more flexibility. To include the legislation in the restricted dynamic programming heuristic, we propose a break scheduling heuristic. Computational results show that our method finds solutions to benchmark instances—which only consider the basic set of rules—with 18% fewer vehicles and 5% less travel distance than state-of-the-art approaches. Moreover, our results are obtained with significantly less computational effort. Furthermore, the results show that including a set of rules on drivers' working hours—which has been generally ignored in the literature—has a significant impact on the resulting vehicle schedules: 3.9% more vehicle routes and 1.0% more travel distances are needed. Finally, using the modified rules of the legislation leads to an additional reduction of 4% in the number of vehicles and of 1.5% in travel distances. Therefore, the modified rules should be exploited in practice.

Reliable Pitch Marking of Affective Speech at Peaks or Valleys Using Restricted Dynamic Programming

The affective communication channel plays a key role in multimodal human-computer interaction. In this context, the generation of realistic talking-heads expressing emotions both in appearance and speech is of great interest. The synthetic speech of talking-heads is generally obtained from a text-to-speech (TTS) synthesizer. One of the dominant techniques for achieving high-quality synthetic speech is unit-selection TTS (US-TTS) synthesis. Affective US-TTS systems are driven by affective annotated speech databases. Since affective speech involves higher acoustic variability than neutral speech, achieving trustworthy speech labeling is a more challenging task. To that effect, this paper introduces a methodology for achieving reliable pitch marking on affective speech. The proposal adjusts the pitch marks at the signal peaks or valleys after applying a three-stage restricted dynamic programming algorithm. The methodology can be applied as a post-processing of any pitch determination and pitch marking algorithm (with any local criterion for locating pitch marks), or their merging. The experiments show that the proposed methodology significantly improves the results of the input state-of-the-art markers on affective speech.

A dynamic programming heuristic for vehicle routing with time-dependent travel times and required breaks

For the intensively studied vehicle routing problem (VRP), two real-life restrictions have received only minor attention in the VRP-literature: traffic congestion and driving hours regulations. Traffic congestion causes late arrivals at customers and long travel times resulting in large transport costs. To account for traffic congestion, time-dependent travel times should be considered when constructing vehicle routes. Next, driving hours regulations, which restrict the available driving and working times for truck drivers, must be respected. Since violations are severely fined, also driving hours regulations should be considered when constructing vehicle routes, even more in combination with congestion problems. The objective of this paper is to develop a solution method for the VRP with time windows (VRPTW), time-dependent travel times, and driving hours regulations. The major difficulty of this VRPTW extension is to optimize each vehicle’s departure times to minimize the duty time of each driver. Having compact duty times leads to cost savings. However, obtaining compact duty times is much harder when time-dependent travel times and driving hours regulations are considered. We propose a restricted dynamic programming (DP) heuristic for constructing the vehicle routes, and an efficient heuristic for optimizing the vehicle’s departure times for each (partial) vehicle route, such that the complete solution algorithm runs in polynomial time. Computational experiments demonstrate the trade-off between travel distance minimization and duty time minimization, and illustrate the cost savings of extending the depot opening hours such that traveling before the morning peak and after the evening peak becomes possible.

Restricted dynamic programming based neighborhoods for the hop-constrained minimum spanning tree problem

In this paper we develop, study and test new neighborhood structures for the Hop-constrained Minimum Spanning Tree Problem (HMSTP). These neighborhoods are defined by restricted versions of a new dynamic programming formulation for the problem and provide a systematic way of searching neighborhood structures based on node-level exchanges. We have also developed several local search methods that are based on the new neighborhoods. Computational experiments for a set of benchmark instances with up to 80 nodes show that the more elaborate methods produce in a quite fast way, heuristic solutions that are, for all cases, within 2% of the optimum.

KnotSeeker: Heuristic pseudoknot detection in long RNA sequences

Pseudoknots are folded structures in RNA molecules that perform essential functions as part of cellular transcription machinery and regulatory processes. The prediction of these structures in RNA molecules has important implications in antiviral drug design. It has been shown that the prediction of pseudoknots is an NP-complete problem. Practical structure prediction algorithms based on free energy minimization employ a restricted problem class and dynamic programming. However, these algorithms are computationally very expensive, and their accuracy deteriorates if the input sequence containing the pseudoknot is too long. Heuristic methods can be more efficient, but do not guarantee an optimal solution in regards to the minimum free energy model. We present KnotSeeker, a new heuristic algorithm for the detection of pseudoknots in RNA sequences as a preliminary step for structure prediction. Our method uses a hybrid sequence matching and free energy minimization approach to perform a screening of the primary sequence. We select short sequence fragments as possible candidates that may contain pseudoknots and verify them by using an existing dynamic programming algorithm and a minimum weight independent set calculation. KnotSeeker is significantly more accurate in detecting pseudoknots compared to other common methods as reported in the literature. It is very efficient and therefore a practical tool, especially for long sequences. The algorithm has been implemented in Python and it also uses C/C++ code from several other known techniques. The code is available from http://www.csse.uwa.edu.au/~datta/pseudoknot.

Near-optimal data allocation over multiple broadcast channels

Data broadcast is known as a scalable way to transmit database items to large client populations through a wireless channel. However, with a large set of items, the expected delay of receiving desired data increases due to the sequential nature of the broadcast channel. One possible solution is to increase the number of available channels and allocate items evenly over these channels, then cyclically broadcast data in each channel. In view of data access skew (the access frequencies of data are usually different), some channels may be reserved exclusively for those few frequently requested items, while the infrequently accessed bulk of data are allocated on other channels. In this paper, an O(N log K) restricted dynamic programming (RDP) algorithm is proposed to partition N items over K channels assuming data access is skewed with the object of minimizing the average expected delay (aed) of clients. To speed up the DP process, for any partition, we predict a possible location which may be very close to the optimal cut by using a low bound as the actual aed for the remaining items. Thus, the number of comparisons in DP can be restricted to a small interval around the predicted cut point. To further reduce the costs in DP, the hierarchical property of optimal solution is adopted. Simulation results show that, the hit rate obtained by RDP is higher than 90% and it also outperforms the existing algorithm 200%. We extend the work of RDP and develop an O(N log N log K) PKR algorithm; simulation results show that the solution is optimal.

An O(N log K) Restricted Dynamic Programming Algorithm for Data Allocation over Multiple Channels

Data broadcast has become a promising approach to achieving information dissemination in wireless environments due to the limited channel bandwidth and the power constraints of portable devices. In this paper, a restricted dynamic programming approach which generates broadcast programs is proposed to partition data items over multiple channels near optimally. In our approach, a function to predict the optimal average expected delay, in terms of the number of channels, the summation of the access frequencies of data items, and the ratio of the data items is developed by employing curve fitting. Applying this function, we can find a cut point that may be very close to the optimal cut;' Thus, the search space in dynamic programming can be restricted to the interval around a determined cut point. Therefore, our approach only takes O(N log K) time, where N is number of data items and K is the number of broadcast channels. Simulation results show that the solution obtained by our proposed algorithm is near-optimal.

Multiresolution banded refinement to accelerate surface reconstruction from polygons

We propose a method for constructing a tiling between a pair of planar polygons. Our technique uses multiresolution: tilings of lower resolution polygons are used to construct a tiling for the full resolution polygons. The tilings are constructed using banded refinement, by restricted dynamic programming, in roughly linear time and space. By contrast, the optimal dynamic programming method requires quadratic time and space. In our empirical study of surface reconstruction of brain contours our algorithm exhibited significant speedup over the optimal dynamic program, yet nearly always found an optimal reconstruction. Our approach appears to be generalizable to other geometric problems solvable by dynamic programming, and flexible enough to be tuned for varying data set characteristics.

A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem

Dynamic programming (DP) algorithms for the traveling salesman problem (TSP) can easily incorporate time dependent travel times, time windows, and precedence relationships which present difficulties for algorithms based on linear or nonlinear programming formulations and for many TSP heuristics. However, exact DP algorithms for the TSP have exponential storage and computational time requirements and can solve only very small problems. We present a restricted DP heuristic (a generalization of the nearest neighbor heuristic) that can include all the above considerations but solves much larger problems. The heuristic cannot guarantee optimality because only a userspecified specified number of partial tours is retained at each stage. In this paper, the heuristic is implemented for the time dependent traveling salesman problem and is tested on a personal computer on randomly generated problems. The quality of the solution improves, on average, as more computational time is permitted.

On-line recognition of hand-written characters utilizing positional and stroke vector sequences

An on-line recognition method for hand-written characters utilizing stroke vector sequences and a positional vector sequence has been developed. The number of target characters is about 2000, and fairly good recognition scores have been attained. Our scheme uses the number of strokes as the primary parameter. We employ three types of recognition strategy depending on the number of strokes. The general stroke vector sequence method, devised to analyze the shape, can represent both skeleton and local characteristics by a small amount of information; and the restricted dynamic programming method is effective to determine the shape of a stroke. The similarity of two shapes and the complexity of a stroke have been introduced to reduce the dictionary size and the processing time, respectively.