Heuristic Decomposition Research Articles

A survey of 3D Space Path-Planning Methods and Algorithms

Due to their agility, cost-effectiveness, and high maneuverability, Unmanned Aerial Vehicles (UAVs) have attracted considerable attention from researchers and investors alike. Path planning is one of the practical subsets of motion planning for UAVs. It prevents collisions and ensures complete coverage of an area. This study provides a structured review of applicable algorithms and coverage path planning solutions in Three-Dimensional (3D) space, presenting state-of-the-art technologies related to heuristic decomposition approaches for UAVs and the forefront challenges. Additionally, it introduces a comprehensive and novel classification of practical methods and representational techniques for path-planning algorithms. This depends on environmental characteristics and optimal parameters in the real world. The first category presents a classification of semi-accurate decomposition approaches as the most practical decomposition method, along with the data structure of these practices, categorized by phases. The second category illustrates path-planning processes based on symbolic techniques in 3D space. Additionally, it provides a critical analysis of crucial influential approaches based on their importance in path quality and researchers’ attention, highlighting their limitations and research gaps. Furthermore, it will provide the most pertinent recommendations for future work for researchers. The studies demonstrate an apparent inclination among experimenters toward using the semi-accurate cellular decomposition approach to improve 3D path planning.

Decomposition heuristics for multiobjective problems. The Food bank network redesign case

Large MO-MILP problems are often very hard to solve by exact methods. In this study we consider the Food bank network redesign (FBNR) problem to introduce three decompose-and-fix heuristics that successfully solve large real-life based instances of the problem. The FBNR problem is modeled as a multi-period, multi-product supply chain redesign problem that accounts for economic, environmental and social objectives in three distinct functions. Each new heuristic decomposes the FBNR problem into two MO-MILP problems, which are sequentially solved following the lexicographic concept. Decomposition observe the nature and terms of the decisions involved. The first MO-MILP problem concerns only the longer term decisions. After the corresponding decisions are fixed, the second MO-MILP problem referring to shorter term decisions is solved respecting the ranking of objectives followed for each solution obtained in the first problem. The difference among the heuristics lies in what is considered as longer or shorter term decisions. Besides solving large instances of the problem, which the exact method used could not do, CPU time savings generated by the heuristics range from 80% to 97% of the time required by the exact method. Compromise between the computational effort and the quality of solutions obtained by each heuristic is discussed. The solving methodology proposed in this study can be adapted to other large multiobjective problems.

Network design with route planning for battery electric high-speed passenger vessel services

This paper studies the Zero Emission passenger Vessel Service Network Design Problem (ZEVSNDP) in order to investigate how technical and economic challenges related to diffusion of battery electric vessels can be alleviated by appropriate planning of services. The ZEVSNDP considers decisions that are strategic (i.e., vessel fleet and charging locations), tactical (i.e., routes, whether to omit servicing ports, fleet deployment, and operating frequencies), as well as operational (i.e., passenger flow, sailing speeds, and scheduling decisions). A novel Mixed Integer Programming (MIP) model considering operator and passenger costs is proposed for the ZEVSNDP. Since the MIP model cannot be solved to optimality by a commercial solver except for tiny instances, we implement a heuristic Decomposition Based (DB) solution method. The DB solution method is applied to a real complex passenger vessel service in Florø, Norway, as well as two other test instances focusing on short-range transport and dense markets, respectively. Except for the short-range test instance, abatement costs (i.e., the costs of removing CO2 emissions by introducing battery electric vessels) are found to be significant. This is attributed to limited reach and time used for charging of battery electric vessels. Routes should consequently accommodate range limitations: omitting ports from the current route can be a cost-effective strategy when the cost of alternative transport for the passengers is moderate.

Compositional Safe Approximation of Response Time Probability Density Function of Complex Workflows

We evaluate a stochastic upper bound on the response time Probability Density Function (PDF) of complex workflows through an efficient and accurate compositional approach. Workflows consist of activities having generally distributed stochastic durations with bounded supports, composed through sequence, choice/merge, and balanced/unbalanced split/join operators, possibly breaking the structure of well-formed nesting. Workflows are specified using a formalism defined in terms of Stochastic Time Petri Nets that permits decomposition into a hierarchy of subworkflows with positively correlated response times, guaranteeing that a stochastically larger end-to-end response time PDF is obtained when intermediate results are approximated by stochastically larger PDFs and when dependencies are simplified by replicating activities appearing in multiple subworkflows. In particular, an accurate stochastically larger PDF is obtained by combining shifted truncated Exponential terms with positive or negative rates. Experiments are performed on sets of manually and randomly generated models with increasing complexity, illustrating under which conditions different decomposition heuristics work well in terms of accuracy and complexity and showing that the proposed approach outperforms simulation having the same execution time.

Multi-period distribution networks with purchase commitment contracts

Retailers which deliver products directly to their customer locations often rely on Logistics Service Intermediaries (LSI) for order management, warehousing, transportation and distribution services. Usually, the LSI acts as a shipper and subcontracts the transportation to carriers for long-haul and last-mile delivery services. All agents interact and are connected through cross-docking facilities. As the demand from customers may vary significantly over time, the shipper’s requirements for transportation evolve accordingly at the tactical level. This creates opportunities for the shipper to take advantage of medium-term contracts with the carriers at prices lower than those offered by the spot market. The study focuses on the tactical design, through dynamic contracts, of a suitable network of cross-docking facilities and related transportation capacities (belonging to different carriers) to reduce the shipper’s operational costs. In this article, we propose an MILP formulation for the multi-period planning problem with minimum purchase commitment contracts faced by the shipper. We propose exact and heuristic decomposition methods for the the model, respectively, based on combinatorial Benders cuts and on relax-and-repair approaches. The performance of these algorithms is experimentally compared to that of commercial solvers (branch-and-cut and classical Benders). The numerical results show that our methods perform comparatively well for the solution of large size instances and brings economic benefits to the shipper.

Heuristic decomposition planning for fast spacecraft reorientation under multiaxis constraints

Spacecraft are required to achieve fast attitude reorientation in many space missions. However, the attitude maneuver will be limited by the bounded and pointing constraints. The presence of these complex multiaxis constraints will largely reduce the feasible attitude space and greatly degrade the solving efficiency of the planning algorithm. For the time-optimal spacecraft reorientation problem under multiaxis constraints, a new heuristic decomposition planning on the virtual domain (HDPV) method is proposed in this paper. The parameterized attitude path is first determined on the virtual domain. Then the virtual-domain-based variable-step approach is presented to rapidly check the pointing constraints and generate the sets of candidate rotational-path decomposition nodes. The composite-position heuristics is developed to select the best node during the decomposition optimization, which considers the maneuver time and connection smoothness of paths. The total continuous parameterized path under keep-out and keep-in constraints is yielded through recursive planning. Finally, the time-optimal path parameterization approach is adopted to minimize the maneuver time along the total path and generate a fast attitude trajectory under bounded constraints on the time domain. Simulation results demonstrate the high computational efficiency and good optimization effect of the proposed method.

Optimized planning of nursing curricula in dual vocational schools focusing on the German health care system

We investigate a problem in vocational school planning for nurses in countries with a dual vocational system, which closely combines theoretical and practical education and is highly regulated by federal legislation. The apprentices rotate through vocational school-blocks followed by assignments to hospital units, where they receive practical education. This program is regulated in high detail. Hospital units offer some slots for apprentices but expect just enough apprentices to be trained and educated. We create two mixed-integer programming models to optimally solve the underlying planning problems of (1) scheduling classes to theoretical and practical education blocks and (2) assigning apprentices to hospital units. The first model determines the number and length of school- and work-blocks on a class level, where its result is input to the second model, which finds individual unit-assignments for every apprentice fulfilling detailed curriculum requirements. Furthermore, it tries to exploit the units' educational capacities as well as possible. To solve the second model, we develop a heuristic decomposition procedure that enables good feasible solutions in short time. Our computational study is based on real-world data of our cooperation partner and provides valuable insights for management. The dataset consists of manually created schedules over the full 3-year program horizon and information on hospital units and their respective capacities. We test different parameter settings for our heuristic procedure and how they influence solution quality and runtime. Finally, we test, if students can be enabled to request individual vacations and evaluate benefits and drawbacks of different degrees of flexibility.

A general surplus decomposition principle in life insurance

ABSTRACT In with-profit life insurance, the prudent valuation of future insurance liabilities leads to systematic surplus that mainly belongs to the policyholders and is redistributed as bonus. For a fair and lawful redistribution of surplus, the insurer needs to decompose the total portfolio surplus with respect to the contributions of individual policies and with respect to different risk sources. For this task, actuaries have a number of heuristic decomposition formulas, but an overarching decomposition principle is still missing. This paper fills that gap by introducing a so-called ISU decomposition principle that bases on infinitesimal sequential updates of the insurer's valuation basis. It is shown that the existing heuristic decomposition formulas can be replicated as ISU decompositions. Furthermore, alternative decomposition principles and their relation to the ISU decomposition principle are discussed. The generality of the ISU concept makes it a useful tool also beyond classical surplus decompositions in life insurance.

Один із способів декомпозиції задачі оптимізації режимів прокатки на товстолистовому стані.

In control theory, decomposition is a method according to which the studied system is divided into subsystems, the problem – into subproblems, etc., each of which is solved independently. The procedure of strict decomposition is based on a clearly formulated mathematical apparatus of decomposition of problems with block structure, as well as on various parametric methods. Heuristic decomposition methods allow an informal analysis of the specified tasks and make it possible to divide them on the basis of intuitive predictions about the interaction of individual tasks. Such a division can be based on various criteria, for example, the minimum of broken ties, technological or economic considerations, the division by stages of the management process, etc. The problem of optimization of rolling mills on a heavy-plate mill is multidimensional and characterized by several output indicators, each of which is a function of several controls. To solve this task it is necessary to replace a vector criterion with a scalar one. After that, there arises the problem of determining the weight coefficients of various indicators, which in the general case is solved at the level of expert assessments, and the efficiency of optimization of automatic control by this criterion is significantly determined by the experience of experts. In this situation, you can use the methods of heuristic decomposition, which allows you to analyze informally set tasks and make it possible to divide them on the basis of intuitive predictions about the interaction of individual tasks. The paper considers the formulation of the problem of optimization of rolling modes on heavy-plate mills by decomposing the general automation problem into a number of single-criteria problems using heuristic methods, which allows ensuring maximum efficiency of automation. The above scientific and technical solutions can be used in the development of automated control systems for rolling modes on the heavy-plate mill.

Minimizing setups and waste when printing labels of consumer goods

A real-world planning problem of a printing company is presented where different sorts of a consumer goods’ label are printed on a roll of paper with sufficient length. The printer utilizes a printing plate to always print several labels of same size and shape (but possibly different imprint) in parallel on adjacent lanes of the paper. It can be decided which sort is printed on which (lane of a) plate and how long the printer runs using a single plate. A sort can be assigned to several lanes of the same plate, but not to several plates. Designing a plate and installing it on the printer incurs fixed setup costs. If more labels are produced than actually needed, each surplus label is assumed to be “scrap”. Since demand for the different sorts may be heterogeneous and since the number of sorts is usually much higher than the number of lanes, the problem is to build “printing blocks”, i.e., to decide how many and which plates to design and how long to run the printer with a certain plate so that customer demand is satisfied with minimum costs for setups and scrap. This industrial application is modeled as an extension of a so-called job splitting problem which is solved exactly and by various decomposition heuristics, partly basing on dynamic programming. Numerical tests compare both approaches with further straightforward heuristics and demonstrate the benefits of decomposition and dynamic programming for large problem instances.

Treewidth-aware reductions of normal ASP to SAT – Is normal ASP harder than SAT after all?

Answer Set Programming (ASP) is a paradigm for modeling and solving problems for knowledge representation and reasoning. There are plenty of results dedicated to studying the hardness of (fragments of) ASP. So far, these studies resulted in characterizations in terms of computational complexity as well as in fine-grained insights presented in form of dichotomy-style results, lower bounds when translating to other formalisms like propositional satisfiability (SAT), and even detailed parameterized complexity landscapes. A generic parameter in parameterized complexity originating from graph theory is the so-called treewidth, which in a sense captures structural density of a program. Recently, there was an increase in the number of treewidth-based solvers related to SAT. While there are translations from (normal) ASP to SAT, no reduction that preserves treewidth or at least keeps track of the treewidth increase is known. In this paper we propose a novel reduction from normal ASP to SAT that is aware of the treewidth, and guarantees that a slight increase of treewidth is indeed sufficient. Further, we show a new result establishing that, when considering treewidth, already the fragment of normal ASP is slightly harder than SAT (under reasonable assumptions in computational complexity). This also confirms that our reduction probably cannot be significantly improved and that the slight increase of treewidth is unavoidable. Finally, we present an empirical study of our novel reduction from normal ASP to SAT, where we compare treewidth upper bounds that are obtained via known decomposition heuristics. Overall, our reduction works better with these heuristics than existing translations.

Shape decomposition algorithms for laser capture microdissection

BackgroundIn the context of biomarker discovery and molecular characterization of diseases, laser capture microdissection is a highly effective approach to extract disease-specific regions from complex, heterogeneous tissue samples. For the extraction to be successful, these regions have to satisfy certain constraints in size and shape and thus have to be decomposed into feasible fragments.ResultsWe model this problem of constrained shape decomposition as the computation of optimal feasible decompositions of simple polygons. We use a skeleton-based approach and present an algorithmic framework that allows the implementation of various feasibility criteria as well as optimization goals. Motivated by our application, we consider different constraints and examine the resulting fragmentations. We evaluate our algorithm on lung tissue samples in comparison to a heuristic decomposition approach. Our method achieved a success rate of over 95% in the microdissection and tissue yield was increased by 10–30%.ConclusionWe present a novel approach for constrained shape decomposition by demonstrating its advantages for the application in the microdissection of tissue samples. In comparison to the previous decomposition approach, the proposed method considerably increases the amount of successfully dissected tissue.

An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem

This paper presents a new class of multiple-follower bilevel problems and a heuristic approach to solving them. In this new class of problems, the followers may be nonlinear, do not share constraints or variables, and are at most weakly constrained. This allows the leader variables to be partitioned among the followers. We show that current approaches for solving multiple-follower problems are unsuitable for our new class of problems and instead we propose a novel analytics-based heuristic decomposition approach. This approach uses Monte Carlo simulation and k-medoids clustering to reduce the bilevel problem to a single level, which can then be solved using integer programming techniques. The examples presented show that our approach produces better solutions and scales up better than the other approaches in the literature. Furthermore, for large problems, we combine our approach with the use of self-organising maps in place of k-medoids clustering, which significantly reduces the clustering times. Finally, we apply our approach to a real-life cutting stock problem. Here a forest harvesting problem is reformulated as a multiple-follower bilevel problem and solved using our approach.

A fix‐and‐optimize heuristic for the minmax regret shortest path arborescence problem under interval uncertainty

AbstractThe minmax regret shortest path arborescence (M‐SPA) problem under interval uncertainty is a robust counterpart of the shortest path (SP) arborescence problem, where arc costs are modeled as intervals of possible values. This problem finds applications in the design of topologies for low‐power wireless personal area networks. The previous work presented a mixed‐integer linear programming formulation and heuristics for this problem. In this paper, we propose a new heuristic for M‐SPA, called the fix‐and‐optimize through heuristic decomposition (FO‐HD). It consists of two steps. The first step solves multiple instances of the minmax regret SP problem under interval uncertainty, while the second step combines their solutions into an arborescence. As far as we know, this is the first work in the literature to apply the fix‐and‐optimize metaheuristic to minmax regret problems. We also show that a lower bound for the optimal solution of M‐SPA can be computed from the first step of FO‐HD. This lower bound allows us to show that the maximum average optimality gap of this heuristic was 0.2% among all classical and novel instances used in the computational experiments.

Optimizing Assemble-to-Order Systems: Decomposition Heuristics and Scalable Algorithms

We consider continuous-review assemble-to-order (ATO) systems with a general bill of materials (BOM) and general leadtimes. ATO systems have the advantage of mass customization and are widely adopted in practice. However, characterizing the optimal inventory policy is notoriously challenging and computational intractable, especially in large-scale systems. We propose effective and computational scalable heuristics through asymptotics, sample-path, linear programming and primal-dual analyses. First, we characterize the asymptotically optimal policy for the M-system. The policy consists of a periodic review priority (PRP) allocation rule and a coordinated base-stock (CBS) replenishment policy. We then construct heuristic policies using insights from the asymptotically optimal policy. In particular, we adopt the PRP allocation rule and develop a decomposition approach for inventory replenishment. This approach decomposes a general system into assembly subsystems and a linear program is constructed to compute policy parameters. However, both the CBS and the assembly decomposition approach are limited to simple systems. We then consider a second approach, which decomposes a system into distribution subsystems and each subsystem has a straightforward solution, which is similar to the newsvendor problem. We use the primal-dual analysis to show that the expected cost under the optimal independent base-stock policy in a general system could be bounded by two newsvendor systems with properly set parameters. Finally, we examine the effectiveness and scalability of these two decomposition approaches in numerical tests. We find that the assembly decomposition is very effective but computationally expensive and thus only good for small-scale systems; the distribution decomposition performs as effective as the optimal independent base-stock (IBS) policy, but is highly scalable for large-scale systems. Numerical tests also provide some interesting insights on the impact of system parameters on the value of past demand information.

A heuristic benders-decomposition-based algorithm for transient stability constrained optimal power flow

Security Constrained Optimal Power Flow (SCOPF) is traditionally performed to obtain the optimal generation levels to achieve the minimum generation cost under operational constraints in normal and contingency conditions. In this paper the SCOPF problem with DC load flow equations is extended to consider the transient stability margin of the generating units as a heuristic decomposition algorithm. However, the proposed algorithm is compatible with any SCOPF formulation. Due to computational complexities of security constraints and transient stability assessment, Benders decomposition algorithm is utilized to handle the transient stability margin of generating units as separate sub-problems. A transient energy function is developed to assess the transient stability of generating units under credible faults using the cut-set energy function criterion. In case of violation of the required transient stability margin, Benders cuts are constructed and added to the master problem. The proposed decomposed Transient Stability Constrained OPF (TSCOPF) is converged in few iterations. The developed TSCOPF algorithm is implemented on the IEEE 39-bus dynamic test system to validate its accuracy and efficiency.

A Relax-and-Fix Decomposition Strategy Based on Adjacent Nodes Applied to the Periodic Capacitated Arc Routing Problem (PCARP)

Periodic Capacitated Arc Routing Problem (PCARP) is a combinatorial problem of hard resolution and due to its intrinsic intractability, researchers have concentrated on designing heuristic methods to obtain good quality solutions. In this work, the aim is to use the Relax-and-Fix heuristic (R and F) as a method of resolution of PCARP. The R and F heuristic consists of decomposing the original model in smaller submodels that are probably easier to be solved. Some factors that contribute to the success of the heuristics are applied decompositions. In this work we introduce a new form of decomposition which is not found in the literature yet as far as we know called Relax-and-Fix Adjacent Nodes (R and F-AN). This new strategy considers the characteristics of the PCARP and the decomposition of the model occurs by exploring in adjacencies in the associated graph. The efficiency of this new approach is evaluated on a set of 23 instances in the literature. Results have shown a reduction of 45% in the number of infeasible problems, when it is compared with other classic strategies of R and F heuristic decomposition. The R and F-AN has obtained 6 new better quality solutions than those already known, being 4 times faster in mean than the attempt of the resolution of the PCARP in the exact form.

Classification and Evaluation of Concepts for Improving the Performance of Applied Energy System Optimization Models

Energy system optimization models used for capacity expansion and dispatch planning are established tools for decision-making support in both energy industry and energy politics. The ever-increasing complexity of the systems under consideration leads to an increase in mathematical problem size of the models. This implies limitations of today’s common solution approaches especially with regard to required computing times. To tackle this challenge many model-based speed-up approaches exist which, however, are typically only demonstrated on small generic test cases. In addition, in applied energy systems analysis the effects of such approaches are often not well understood. The novelty of this study is the systematic evaluation of several model reduction and heuristic decomposition techniques for a large applied energy system model using real data and particularly focusing on reachable speed-up. The applied model is typically used for examining German energy scenarios and allows expansion of storage and electricity transmission capacities. We find that initial computing times of more than two days can be reduced up to a factor of ten while having acceptable loss of accuracy. Moreover, we explain what we mean by “effectiveness of model reduction” which limits the possible speed-up with shared memory computers used in this study.

Decomposition heuristics for parallel-machine multiple orders per job scheduling problems with a common due date

Scheduling problems for identical parallel machines with earliness-tardiness objective are studied that are motivated by manufacturing processes in 300-mm wafer fabs. Wafers are transferred by front opening unified pods (FOUPs) in such fabs. Only a limited number of FOUPs is allowed since a large number of FOUPs results in a highly congested automated material handling system. A FOUP can contain a group of orders. A nonrestrictive common due date is assumed for all orders. Only orders of the same family can be grouped together in a FOUP. The lot and the item processing mode are differentiated in this article. Mixed integer linear programming (MILP) models are provided for both modes. It is shown that the two scheduling problems are NP-hard. Simple decomposition heuristics based on list scheduling and bin packing procedures are proposed. Biased random-key genetic algorithm (BRKGA)-based decomposition schemes are designed for the two scheduling problems. The BRKGAs are hybridised with the simple heuristics and an integer programming-based job formation approach in the lot processing mode. Results of computational experiments based on randomly generated problem instances are analysed and discussed for both scheduling problems. The results show that the proposed heuristics perform well with respect to solution quality and computing time. The BRKGA-type approaches clearly outperform the other heuristics.

New results for single-machine scheduling with past-sequence-dependent setup times and due date-related objectives

Abstract We show that the single-machine scheduling problem with past-sequence-dependent (p-s-d) setup times and either the minimum maximum lateness or the minimum maximum tardiness objective is solvable in O(n2) time by an index priority rule followed by backward insertions of certain qualifying jobs. We also show that the problem with the minimum number of tardy jobs objective is solvable in O(n2) time by a variant of Moore's algorithm. We then show how to modify a general purpose dynamic programming algorithm to solve the problem with other due-date related objectives such as the total weighted tardiness, the weighted number of tardy jobs and, in the case of an unrestricted common due date, the total weighted earliness/ tardiness. We also present heuristic decomposition algorithms for the NP-hard scheduling problems with p-s-d setup times and the objectives of total weighted tardiness, weighted number of tardy jobs and total weighted earliness/tardiness around an unrestricted common due date. These algorithms decompose the respective problem heuristically into smaller sub-problems which are then solved optimally by dynamic programming. We show experimentally that the heuristic performs well with extended job due dates close to the makespan because, in those cases, the heuristic is capable of locating the optimal solution with either zero total weighted tardiness or with only a few tardy jobs. In the remaining cases, the quality of the DS solutions is superior with normally distributed data compared to the corresponding solutions with uniformly distributed data. This is because sampling from the normal distribution reduces the variability among processing times and weights which in turn decreases the hardness of the problem. Finally, we show that the heuristic's performance deteriorates as the problem size increases.