Exact Algorithm Research Articles

Exact and heuristic algorithms for minimizing the makespan on a single machine scheduling problem with sequence-dependent setup times and release dates

This paper proposes efficient exact and heuristic approaches for minimizing the makespan on a single machine scheduling problem with sequence-dependent setup times and release dates. The exact procedure consists of a branch-and-price (B&P) algorithm implemented over an arc-time-indexed formulation with a pseudo-polynomial number of variables and constraints. Our B&P algorithm includes several modern features, such as a dynamic ng-path relaxation and a bidirectional labeling method for efficiently solving the pricing subproblem. The proposed heuristic algorithm is based on the iterated local search framework that employs a beam search approach adapted from the literature for generating initial solutions and an efficient scheme to perform move evaluations in amortized constant time. Computational experiments were carried out on benchmark instances containing 1800 instances ranging from 25 to 150 jobs. The results obtained attest the high performance of both the exact and heuristic algorithms in obtaining high-quality bounds when compared to existing methods. We report improved lower and upper bounds for the vast majority of the instances, as well as optimal solutions for 42.7% of the instances.

On Exact Computation of Tukey Depth Central Regions

The Tukey (or halfspace) depth extends nonparametric methods toward multivariate data. The multivariate analogues of the quantiles are the central regions of the Tukey depth, defined as sets of points in the d-dimensional space whose Tukey depth exceeds given thresholds k. We address the problem of fast and exact computation of those central regions. First, we analyze an efficient Algorithm (A) from Liu, Mosler, and Mozharovskyi, and prove that it yields exact results in dimension d = 2, or for a low threshold k in arbitrary dimension. We provide examples where Algorithm (A) fails to recover the exact Tukey depth region for d > 2, and propose a modification that is guaranteed to be exact. We express the problem of computing the exact central region in its dual formulation, and use that viewpoint to demonstrate that further substantial improvements to our algorithm are unlikely. An efficient C++ implementation of our exact algorithm is freely available in the R package TukeyRegion.

An Improved Hashing Approach for Biological Sequence to Solve Exact Pattern Matching Problems

Pattern matching algorithms have gained a lot of importance in computer science, primarily because they are used in various domains such as computational biology, video retrieval, intrusion detection systems, and fraud detection. Finding one or more patterns in a given text is known as pattern matching. Two important things that are used to judge how well exact pattern matching algorithms work are the total number of attempts and the character comparisons that are made during the matching process. The primary focus of our proposed method is reducing the size of both components wherever possible. Despite sprinting, hash-based pattern matching algorithms may have hash collisions. The Efficient Hashing Method (EHM) algorithm is improved in this research. Despite the EHM algorithm’s effectiveness, it takes a lot of time in the preprocessing phase, and some hash collisions are generated. A novel hashing method has been proposed, which has reduced the preprocessing time and hash collision of the EHM algorithm. We devised the Hashing Approach for Pattern Matching (HAPM) algorithm by taking the best parts of the EHM and Quick Search (QS) algorithms and adding a way to avoid hash collisions. The preprocessing step of this algorithm combines the bad character table from the QS algorithm, the hashing strategy from the EHM algorithm, and the collision-reducing mechanism. To analyze the performance of our HAPM algorithm, we have used three types of datasets: E. coli, DNA sequences, and protein sequences. We looked at six algorithms discussed in the literature and compared our proposed method. The Hash-q with Unique FNG (HqUF) algorithm was only compared with E. coli and DNA datasets because it creates unique bits for DNA sequences. Our proposed HAPM algorithm also overcomes the problems of the HqUF algorithm. The new method beats older ones regarding average runtime, number of attempts, and character comparisons for long and short text patterns, though it did worse on some short patterns.

Exact algorithms for a parallel machine scheduling problem with workforce and contiguity constraints

We address a real-world scheduling problem where the objective is to allocate a set of tasks to a set of machines and to a set of workers in such a way that the total weighted tardiness is minimized. Our case study encompasses four types of constraints: precedence, resource, eligibility, and contiguity. While the first three constraints are common in the scheduling literature, contiguity constraints, which can be defined as a form of precedence constraints that requires both a predecessor and its successor to be processed on the same machine with no intermediate jobs in-between (but idle time is allowed), have never been studied in the literature. We present four exact methods to solve the problem: two methods use integer linear programming, one uses constraint programming, and one uses a combinatorial Benders’ decomposition. We introduce method-specific strategies to model the contiguity constraints for each of the proposed methods. We empirically evaluate, through an extensive set of computational experiments, the performance of the four methods on a heterogeneous dataset composed of real, realistic, and random instances, and outline that every method offers a competitive advantage on a targeted subset of instances. We also show that our algorithms can be generalized to solve related scheduling problems with contiguity constraints.

Features of mild duration asthma diagnostics in childbearing aged females

Asthma diagnostics especially of it’s mild forms, is considered to be a serious problem within primary health care unit due to the absence of exact evaluation algorithm of subjective, clinical and functional manifestations. Objective. The goal of our study was to create recommendation on early asthma diagnostics in childbearing aged females, based on particular analysis of complaints, clinical and functional test results and to make them available to all primary medical care units. Materials and methods. 2460 females of childbearing age with various severity of asthma duration were studied with complex of clinic and functional methods. Results. Patients with various grades of asthma severity could be compliant free, same time the can appear signs of bronchial obstruction during auscultation. The marked discordance leads to mismatch in therapy administration. PEF50 decrease while initial spirometry and its’ increase after probe with bronchodilator is considered to be the earliest sign of bronchial obstruction. Asthma diagnostics at prehospital stage is to based upon complete chart collection, complaints analysis, auscultation signs while forced expiration as well as on spirography data on volume and forced expiration dynamics and PEF50. Conclusion. Results of large population study on clinical feature of asthma duration and functional diagnostics of mild asthma duration could be recommended for primary health care physicians as method of early asthma diagnostics and therapy administration

Portfolio choice algorithms, including exact stochastic dominance

Assume data on Nj stock (asset) returns are available for p stocks, allowing us to construct approximate density functions f(xj) for (j=1, 2, …, p) from p empirical cumulative distribution functions (ECDFs). Our portfolio choice is designed to rank ECDF-induced, ill-behaved f(xj) densities subject to multiple modes, asymmetric fat tails, dips, turns, and numerous overlaps. Older portfolio theory assumes that parameters like the mean, variance, and percentiles fully describe f(xj). All six of our algorithms avoid (expected) utility theory. The only available algorithm by Anderson for order-k Stochastic Dominance (SDk) needs a trapezoidal approximation. Our new exact algorithm for SDk is based on ECDFs and overcomes pairwise comparisons. We include algorithms for statistical inference using the bootstrap and one for “pandemic proof” out-of-sample portfolio performance comparisons from our R package ‘generalCorr’. We suggest a test for “zero cost profitable arbitrage” and illustrate our algorithms in action by using two sets of recent 169-month stock returns. We do not claim to suggest new optimal portfolios.

An exact projection pursuit-based algorithm for multivariate two-sample nonparametric testing applicable to retrospective and group sequential studies

Nonparametric tests for equality of multivariate distributions are frequently desired in research. It is commonly required that test-procedures based on relatively small samples of vectors accurately control the corresponding Type I Error (TIE) rates. Often, in the multivariate testing, extensions of null-distribution-free univariate methods, e.g., Kolmogorov-Smirnov and Cramér-von Mises type schemes, are not exact, since their null distributions depend on underlying data distributions. The present paper extends the density-based empirical likelihood technique in order to nonparametrically approximate the most powerful test for the multivariate two-sample (MTS) problem, yielding an exact finite-sample test statistic. We rigorously apply one-to-one-mapping between the equality of vectors' distributions and the equality of distributions of relevant univariate linear projections. We establish a general algorithm that simplifies the use of projection pursuit, employing only a few of the infinitely many linear combinations of observed vectors' components. The displayed distribution-free strategy is employed in retrospective and group sequential manners. A novel MTS nonparametric procedure in the group sequential manner is proposed. The asymptotic consistency of the proposed technique is shown. Monte Carlo studies demonstrate that the proposed procedures exhibit extremely high and stable power characteristics across a variety of settings. Supplementary materials for this article are available online.

An Enhanced Ant Colony System Algorithm Based on Subpaths for Solving the Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) is regarded as an NP-hard problem. Moreover, the CVRP is described as a model that can be used in many applications such as transport, logistics, and distribution. The exact algorithms can find exact optimal solutions on the small-sized problem instances; however, for large-sized instances it is difficult to find the exact optimal solutions in polynomial time. This reason motivated the researchers to present heuristic/metaheuristic algorithms to solve large-sized problem instances within a reasonable computational time. One of the good algorithms that deal with the CVRP is the ant colony optimization (ACO) algorithm. Several ACO algorithms have been suggested in the literature, such as the ant system (AS) algorithm, ant colony system (ACS) algorithm, and so on. On the other hand, ACO is designed to solve the path problem that finds the best way. However, this algorithm still lacks exploratory mechanisms, which results in premature convergence and stagnation issues. Therefore, we propose to develop an enhanced ACS (EACS) algorithm for solving the CVRP based on subpaths. In our proposed algorithm, we propose to utilize the K-nearest neighbour (KNN) algorithm for finding the best initial solution and then enhance the diversity mechanism of the proposed algorithm by avoiding the generation of the same solution using subpaths. This uses the diversity of the generated solution to find a better solution with a shorter route in a reasonable amount of computational time. Furthermore, we propose to apply the three-opt algorithm to the completed subtour and the k-opt algorithm to the subpath gained from the experience of the subpath. Finally, to verify the effectiveness of the proposed EACS algorithm, the algorithm is tested on some CVRP instances and is compared with one of the state-of-the-art methods, namely, the enhanced simulated annealing algorithm. The comparative study showed a better performance of our EACS compared to the enhanced simulated annealing algorithm.

Traveling itinerary problem in a scheduled multimodal transportation network for a fixed sequence of cities.

Developing an efficient and economical journey plan in multimodal transportation networks is of significant and fast-growing importance, but it is still an annoying experience for a traveler. This paper aims to find the journey plan at a combined cross-border and inter-regional level when visiting a sequence of cities while utilizing several transport modes to reduce travel costs and planning time. We study a traveling itinerary problem in a scheduled multimodal transportation network with constraints on both arcs and nodes as a new extension of the shortest path problem. We formulate a 0-1 integer linear programming model for the traveling itinerary problem and develop an exact algorithm that finds a combined cross-border and inter-regional low costs journey plan. We present case studies based on real-world transport data to illustrate the usefulness and computational efficiency of the proposed approaches. We compare the results with the previously proposed approach to demonstrate the benefits of multimodal journeys. Finally, we compare the results with the solution obtained by the general-purpose 0-1 integer linear programming solver to evaluate the computational time.

A hybrid meta-heuristic approach to design a Bi-objective cosmetic tourism supply chain: A case study

In recent decades, medical tourism especially cosmetic tourism has increased in many countries. In this study, a bi-objective mathematical model is designed for the cosmetic tourism supply chain for the first time. In the first objective function, the profit margin in the supply chain is maximized. In contrast, the distance between the medical and accommodation centers is minimized in the second objective function. The general revised multi-choice goal programming (GRMCGP) method is an exact method for optimizing the problem. To solve the problem, model is run by Lingo software. Then, a new hybrid genetic algorithm named genetic algorithm-general revised multi-choice goal programming (GA-GRMCGP), to solve large-scale instances, is utilized. Moreover, a case study is considered to validate the proposed mathematical model. Furthermore, sensitivity analyses are carried out to evaluate the effectiveness of the research. Finally, the results of the exact and meta-heuristic algorithms are compared and discussed.

Transmission reliability evaluation of wireless sensor networks considering channel capacity randomness and energy consumption failure

The reliable transmission of data packets is the basis of ensuring the Internet of Things applications. Therefore, how to ensure the transmission reliability of wireless sensor networks has already become a hotspot. However, the existing researches ignore the effects of multiple fault modes and random factors on transmission reliability. Firstly, in the node reliability modeling, few studies have considered the effects of random failure and energy consumption failure comprehensively. Secondly, in the link reliability modeling, existing studies ignore the effect of the channel capacity randomness, which can lead to the erroneous estimation of transmission reliability. To overcome the above shortcomings, we propose a more realistic transmission reliability model. Firstly, considering the comprehensive effects of two failure modes, we establish a new node reliability model to better characterize the node performance. Secondly, considering the channel capacity randomness, we propose an improved SNR-Capacity connection model by deriving the one-hop connectivity probability, which can better characterize the link reliability. Then, we propose a transmission reliability evaluation model, and give the corresponding exact and approximate algorithms for its solution. Finally, taking the military Internet as an example, the effectiveness of the proposed method is verified.

An exact algorithm for Two-Echelon Location-Routing problem with simultaneous pickup and delivery

This study considers an extension of the two-Echelon Location-Routing Problem (2E-LRP) namely two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD). A Mixed Integer Programming (MIP) formulation is developed and strengthened by the use of valid inequalities to solve the 2E-LRPSPD, and then a Branch and Cut-based exact algorithm (B&C) is presented. To improve the upper bounds, Iterated Local Search and Variable Neighborhood Search-based metaheuristic algorithms, namely, ILS-VNS and VNS, are developed and embedded into the B&C. Computational results, conducted on instances produced from the literature, demonstrate that the B&C yields tight lower and upper bounds, and also is capable of solving some instances optimally with up to 50 customers and 10 satellites in a reasonable time. Furthermore, sensitivity analysis reveals that the heuristic approaches and an increase in the runtime limit assist the algorithm to reach tight the upper bounds, whereas the number of potential depots does not have any effect on the performance of the algorithm. Besides, further computational analysis on the 2E-LRPSPD with a single depot, which is studied in the literature, shows that the B&C improves some best-known solutions while proving the optimality of some instances. Finally, the B&C is applied to a two-echelon supermarket distribution system in Türkiye. The computational results show that simultaneous pickup and delivery operations, as well as depot location decisions are critical elements to consider in the design of cost-effective distribution networks.

Dynamics between warehouse operations and vehicle routing

When scheduling the distribution of ordered items from a warehouse to customers, the transportation planning is generally done first and serves as input for planning warehouse operations. Such a sequential approach can lead to substantial inefficiencies when the customer deliveries are restricted by time windows, and the warehouse has limited resources available (both order pickers and space in the staging area). This paper studies the trade‐offs between warehouse operations and transportation planning. The goal is to understand the impact of three specific managerial interventions: adopting an integrated planning approach, expanding the available staging space, and expanding the delivery time windows. To this end, we propose a mathematical model for a general vehicle routing problem that incorporates order batching, order picker scheduling, staging, and vehicle loading. We introduce a novel idea to express the picking time of an order batch as a function of the batch size and develop a metaheuristic to solve this integrated problem. Furthermore, we develop exact algorithms to provide optimal solutions for the individual warehouse and transportation problems in a sequential planning approach. Managerial insights are distilled from case studies in two warehouses, one for ambient products and the other for refrigerated products, of a leading grocery retailer in the Netherlands. Our results show that integrated planning outperforms the other managerial interventions and generates cost savings between 9% and 11%. Savings are generally realized by executing larger order batch sizes to be picked in the warehouses at the expense of additional routing cost (around 2–3%). The second intervention in the form of time window expansions of only 15 minutes for customer deliveries can lead to cost savings between 4% and 6%, which results from a reduction in both transportation and warehousing cost. Expanding the capacity of the staging area is only meaningful when the staging space is highly utilized and only results in cost savings for the warehouse operations.

A mathematical model for optimal turnaround maintenance planning and scheduling for a network of plants in process industry supply chain

Turnaround (TAM) is a periodic maintenance in which plants are shutdown to conduct inspections, repairs, preventive maintenance and overhauls. The shutdown periods range from three to four weeks and prolonging this period is expensive and therefore requires effective planning. The primary purpose of this paper is to develop a mathematical programming model to coordinate and plan TAM for several plants in an integrated supply chain network. A secondary objective is to develop a heuristic algorithm for solving the model when the network has large number of plants. The model and the algorithm are developed, tested and applied on a case study petrochemical supply chain. The exact algorithm takes 8.3 times the time of the heuristic. In addition the heuristic algorithm provides near optimal solution in much shorter time. The developed model provides a tool to plan TAM for several plants in an integrated supply in an optimal fashion.

Electric Vehicle Charge Scheduling with Flexible Service Operations

Operators who deploy large fleets of electric vehicles often face a challenging charge scheduling problem. Specifically, time-ineffective recharging operations limit the profitability of charging during service operations such that operators recharge vehicles off duty at a central depot. Here, high investment cost and grid capacity limit available charging infrastructure such that operators need to schedule charging operations to keep the fleet operational. In this context, flexible service operations, that is, allowing delayed or expedited vehicle departures, can potentially increase charger utilization. Beyond this, jointly scheduling charging and service operations promises operational cost savings through better utilization of time-of-use energy tariffs and carefully crafted charging schedules designed to minimize battery wear. Against this background, we study the resulting joint charging and service operations scheduling problem accounting for battery degradation, nonlinear charging, and time-of-use energy tariffs. We propose an exact branch-and-price algorithm, leveraging a custom branching rule and a primal heuristic to remain efficient during the branch-and-bound phase. Moreover, we develop an exact labeling algorithm for our pricing problem, constituting a resource-constrained shortest path problem that considers variable energy prices and nonlinear charging operations. We benchmark our algorithm in a comprehensive numerical study and show that it can solve problem instances of realistic size with computational times below one hour, thus enabling its application in practice. Additionally, we analyze the benefit of jointly scheduling charging and service operations. We find that our integrated approach lowers the amount of charging infrastructure required by up to 57% besides enabling operational cost savings of up to 5%. Funding: This work was supported by the German Federal Ministry for Economic Affairs and Energy [Grant 01MV21020B]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0272 .

Evolving ensembles of heuristics for the travelling salesman problem

AbstractThe Travelling Salesman Problem (TSP) is a well-known optimisation problem that has been widely studied over the last century. As a result, a variety of exact and approximate algorithms have been proposed in the literature. When it comes to solving large instances in real-time, greedy algorithms guided by priority rules represent the most common approach, being the nearest neighbour (NN) heuristic one of the most popular rules. NN is quite general but it is too simple and so it may not be the best choice in some cases. Alternatively, we may design more sophisticated heuristics considering the particular features of families of instances. To do that, we have to consider problem attributes other than the proximity of the next city to build priority rules. However, this process may not be easy for humans and so it is often addressed by some learning procedure. In this regard, hyper-heuristics as Genetic Programming (GP) stands as one of the most popular approaches. Furthermore, a single heuristic, even being good in average, may not be good for a number of instances of a given set. For this reason, the use of ensembles of heuristics is often a good alternative, which raises the problem of building ensembles from a given set of heuristic rules. In this paper, we study the application of two kinds of ensembles to the TSP. Given a set of TSP instances having similar characteristics, we firstly exploit a GP to build a set of heuristics involving a number of problem attributes, and then we build ensembles combining these heuristics by means of a Genetic Algorithm (GA). The experimental study provided valuable insights into the construction and utilisation of single rules and ensembles. It clearly demonstrated that the performance of ensembles justifies the time invested when compared to using individual heuristics.

Quantum algorithms for learning hidden strings with applications to matroid problems

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of learning a pair of n-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string s,s′∈{0,1}n satisfying the above constraints, for any x∈{0,1}n the max inner product oracle Omax(x) returns the max value between s⋅x and s′⋅x, and the sub-set oracle Osub(x) indicates whether the index set of the 1s in x is a subset of that in s or s′. We present an exact quantum algorithm consuming O(1) queries to the max inner product oracle for identifying the pair {s,s′}, and prove that any randomized algorithm requires Ω(n/log2⁡n) queries. Also, we present a quantum algorithm consuming n2+O(n) queries to the subset oracle, and prove that any randomized algorithm requires at least n−3+Ω(log2⁡n) queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called k-bases if it has k bases.

Designing a two-stage model for the resilient agri-food supply chain network under dynamic competition

PurposeSupply chain risk management can effectively reduce the loss of retailers. In this regard, retailers need to consider the competition risks of competitors in addition to the disruption risks. This paper designs a resilient retail supply chain network for perishable foods under the dynamic competition to maximize retailer's profits.Design/methodology/approachA two-stage mixed-integer non-linear model is presented for designing the supply chain network. In the first stage, an equilibrium model that considers the characteristics of perishable foods is developed. In the second stage, a mixed integer non-linear programming model is presented to deal with the strategic decisions. Finally, an efficient memetic algorithm is designed to deal with large-scale problems.FindingsThe optimal the selection of suppliers, distribution centers and the order allocation are found among the supply chain entities. Considering the perishability of agri-food products, the equilibrium retail price and selling quantity are determined. Through a numerical example, the optimal inventory period under different maximum shelf life and the impact of three resilient strategies on retailer's profit, selling price and selling quantity are analyzed.Research limitations/implicationsAs for future research, the research can be extended in a number of directions. First, this paper studies the retail supply chain network design problem under competition among retailers. It can be an interesting direction to consider retailers competing with suppliers. Second, the authors can try to linearize the non-linear model and solve the large-scale integer programming problem by exact algorithm. Finally, the freshness of perishable foods gradually declines linearly to zero as the maximum shelf life approaches, and it would be a meaningful attempt to consider the freshness of perishable foods declines exponentially.Originality/valueThis paper innovatively designs the resilient supply chain network for perishable foods under dynamic competition. The retailer's dynamic competition and resilient strategies are considered simultaneously when designing supply chain network for perishable foods. In addition, this paper gives insights into how to obtain the optimal inventory period and compare the retailer's resilient strategies.

Gaussian Boson Sampling with Pseudo-Photon-Number-Resolving Detectors and Quantum Computational Advantage.

We report new Gaussian boson sampling experiments with pseudo-photon-number-resolving detection, which register up to 255 photon-click events. We consider partial photon distinguishability and develop a more complete model for the characterization of the noisy Gaussian boson sampling. In the quantum computational advantage regime, we use Bayesian tests and correlation function analysis to validate the samples against all current classical spoofing mockups. Estimating with the best classical algorithms to date, generating a single ideal sample from the same distribution on the supercomputer Frontier would take ∼600 yr using exact methods, whereas our quantum computer, Jiǔzhāng 3.0, takes only 1.27 μs to produce a sample. Generating the hardest sample from the experiment using an exact algorithm would take Frontier∼3.1×10^{10} yr.

Γ-robust optimization of project scheduling problems

In this paper, we investigate the problem of finding a robust baseline schedule for the project scheduling problem under uncertain process times. We assume that the probability distribution for the duration is unknown, but an estimate is given along with an interval in which this time can vary. At most Γ of the tasks will deviate from the estimated time.We present two approaches to solving this problem. The first approach treats the problem of determining the earliest guaranteed finish times and can be solved in polynomial time by an extension of the critical path method. The second is a two-stage approach that determines the baseline schedule in the first stage and an adaptation to the scenario in the second stage. We show strong NP-hardness of the second stage problem and introduce a novel formulation. From this formulation we derive an exact algorithm and three heuristics. A computational study on benchmark instances shows that the heuristics perform well on larger instances.