Set Partitioning Problem Research Articles

The static ridesharing routing problem with flexible locations: A Norwegian case study

The municipalities in the Bergen region in Norway have recently announced a pilot project for ridesharing in the region as a means to reduce traffic congestion. As part of this project, we study the Static Ridesharing Routing Problem with Flexible Locations (SRRPFL), which aims at determining efficient routes and schedules for a set of drivers to pick up and deliver passengers at different, flexible pickup and delivery locations. We present a bi-objective mixed integer programming (MIP) model for the SRRPFL where we (lexicographically) first maximize the number of passengers serviced and then minimize the total travel times. To solve real-life instances of the SRRPFL, we propose a new Adaptive Large Neighborhood Search (ALNS) heuristic. To further improve its performance, we extend the ALNS heuristic with a local search, as well as with a set partitioning problem (denoted the Route Combination Problem) that optimally recombines the routes previously encountered in the search. The ALNS heuristic is tested on a number of test instances based on real trip data and the results demonstrate its effectiveness. The results also provide a number of insights regarding the potential benefits of ridesharing in our case study.

New advances for quantum‐inspired optimization

AbstractAdvances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.

On the Multilayer Planning of Filterless Optical Networks With OTN Encryption

With enhanced cost-effectiveness, filterless optical networks (FONs) have been considered as a promising candidate for future optical infrastructure. However, as the transmission in FON relies on the “select-and-broadcast” scenario, it is more vulnerable to eavesdropping. Therefore, encrypting the communications in FONs will be indispensable, and this can be realized by introducing the optical transport network (OTN) encryption technologies that leverage high-speed encryption cards (ECs) to protect the integrity of OTN payload frames. In this paper, we study the problem of security-aware multilayer planning of FONs with OTN encryption. We first formulate a mixed integer linear programming (MILP) model (i.e., w-MILP) to solve the problem exactly. Then, to reduce the time complexity of problem-solving, we transform w-MILP into two correlated MILP models for establishing fiber trees for an FON (t-MILP) and planning flows in the fiber trees (s-MILP), respectively. The optimization in t-MILP is further transformed into a weighted set partitioning problem, which can be solved time-efficiently. As for s-MILP, we propose a polynomial-time approximation algorithm based on linear programming (LP) relaxation and randomized rounding. Extensive simulations verify the performance of our proposals.

Complexity and algorithm of setting optimal location for data sink in real-time NOMA-based IIoTs

Real-time performance is one of the most vital metrics in Non-Orthogonal Multiple Access (NOMA) based Industrial Internet of Things (IIoTs) applications. Since the relative geographic relationship between data sink and wireless sensors affects the transmission parallelism and thus the real-time performance, setting a reasonable location for the data sink is thus a feasible way to high real-time performance for applications where the locations of wireless sensors are fixed. In this paper, we consider how to find an optimal location for the data sink to minimize average access delay. We first formulate the problem and prove it to be NP-Complete by presenting an original reduction proof from classic set partition problem. Second, by tightening the NOMA decoding constraint of the original problem, a heuristic algorithm, which is designed based on Apollonian Circle Theorem and Dilworth Theorem, obtains a feasible location for the data sink. Simulation results reveal that the real-time performance loss of the heuristic algorithm is no more than 50% of the best one. Compared with the classic TDMA scheme, the real-time performance increases by more than 60% for some typical settings, and it can even reach 70% for the linear network topology, owing to the full exploitation of NOMA parallelism.

What is a decision problem?

This paper presents a general framework about what is a decision problem. Our motivation is related to the fact that decision analysis and operational research are structured (as disciplines) around classes of methods, while instead we should first characterise the decision problems our clients present us. For this purpose, we define a decision problem as a set partitioning problem and we introduce a new framework, independent from any existing method, based upon primitives provided by (or elicited from) the client and four fundamental problem statements. We show that constructing the set to partition is a decision problem itself and that the types of primitives are finite. The result is that the number of archetypal decision problems are finite and so the archetypal decision support methods.

Coding Application

Key words: message code, coding equivalence, subset code, discrete optimization problem, set partitioning, knapsack problem, variant observation algorithm The work is devoted to the development of the algorithm for observing all subsets of the set. For that purpose, coding for subsets of the set is considered, in such a way as to create a 1-1 correspondence between the set of subsets of the set and the set of codes. A convenient arrangement of the observation of all those codes allows the observation of all subsets of the set in a convenient way. Subsets of a set are coded in the binary number system (with binary numbers). The problem of considering subsets of a set arises when solving the knapsack problem, set partitioning problems, and other problems. To the mentioned general problems are included a number of specific problems arising in many fields. In the work, an algorithm for observing all subsets of the set is developed, which is suitable for solving all the mentioned problems. The developed algorithm can be applied to the optimal solution of non-large NP-complete problems. All the known optimal solution algorithms for solving large-scale problems run in practically unacceptable time (have very long execution time). As a consequence, for the solution of large NP-complete problems, approximate algorithms with polynomial time complexity are developed, by means of which approximate solutions of the problem are obtained.

The set partitioning problem in a quantum context

The set partitioning problem in a quantum context

Optimal unlabeled set partitioning with application to risk-based quarantine policies

We consider the problem of partitioning a set of items into unlabeled subsets so as to optimize an additive objective, i.e., the objective function value of a partition is equal to the sum of the contribution of each subset. Under an arbitrary objective function, this family of problems is known to be an -complete combinatorial optimization problem. We study this problem under a broad family of objective functions characterized by elementary symmetric polynomials, which are “building blocks” to symmetric functions. By analyzing a continuous relaxation of the problem, we identify conditions that enable the use of a reformulation technique in which the set partitioning problem is cast as a more tractable network flow problem solvable in polynomial-time. We show that a number of results from the literature arise as special cases of our proposed framework, highlighting its generality. We demonstrate the usefulness of the developed methodology through a novel and timely application of quarantining heterogeneous populations in an optimal manner. Our case study on real COVID-19 data reveals significant benefits over conventional measures in terms of both spread mitigation and economic impact, underscoring the importance of data-driven policies.

Patient appointment scheduling at hemodialysis centers: An exact branch and price approach

Scheduling patient appointments at a hemodialysis center presents a unique setting. Unlike other appointment scheduling problems in healthcare systems, patients are scheduled for a series of dialysis treatment appointments instead of a single appointment. In this study, we formulate this multiple-appointment system as a set-partitioning problem that allows partial schedules to be feasible. We employ a Branch and Price (BP) algorithm to solve the problem, however, the pricing sub-problem proves to be exceedingly challenging for state-of-the-art dynamic programming algorithms. Therefore, we propose a novel decomposition of the sub-problem and design an efficient embedded Column Generation (CG) algorithm to find the optimal solution. We further design a greedy heuristic that enhances the computational efficiency of the BP algorithm. Our proposed BP algorithms efficiently solve challenging instances that are simulated based on the data from a collaborating hemodialysis center. Specifically, the Enhanced CG-embedded BP algorithm accelerates the CPU time on average by 78% compared to the Base BP algorithm (32% and 59% compared to the Enhanced and the CG-embedded BP algorithms, respectively). We also compare the optimal results with the current scheduling policy at the center. Our proposed Enhanced CG-embedded BP algorithm improves the percentage of leftover appointments by 98% on average and the hours of deviations per patient by 46% on average, compared to the current policy.

Pattern-Avoiding Ascent Sequences of Length 3

Pattern-avoiding ascent sequences have recently been related to set-partition problems and stack-sorting problems. While the generating functions for several length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100, 110, 120 are not known. We have generated extensive series expansions for these four cases, and analysed them in order to conjecture the asymptotic behaviour.
 We provide polynomial time algorithms for the $000$ and $110$ cases, and exponential time algorithms for the $100$ and $120$ cases. We also describe how the $000$ polynomial time algorithm was detected somewhat mechanically given an exponential time algorithm.
 For 120-avoiding ascent sequences we find that the generating function has stretched-exponential behaviour and prove that the growth constant is the same as that for 201-avoiding ascent sequences, which is known.
 The other three generating functions have zero radius of convergence, which we also prove. For 000-avoiding ascent sequences we give what we believe to be the exact growth constant. We give the conjectured asymptotic behaviour for all four cases.

Robust and Fault-Tolerant Fog Design and Dimensioning for Reliable Operation

Internet of Things (IoT) applications depend on reliable external storage and processing such as cloud data centers. In response to high latency from cloud, fog computing has been introduced as a network of microdata centers closer to IoT devices that provides a geo-distributed low-latency response. Current contributions regarding design and dimensioning of fog infrastructures are developed to service a static set of IoT traffic and a reliable fog network. However, these designs are not fault tolerant. This article explores the implementation of reliable and fault-tolerant fog infrastructures via dynamically available fog nodes—standby nodes that activate when a nearby fog node fails. We formulate the design and dimensioning of dynamically available nodes as a set partitioning problem, which is solved via a mixed-integer linear program (MILP). This MILP formulation proves to be intractable; we therefore introduce a column generation approach to increase scalability with little loss to optimal design and dimensioning cost. Compared to other benchmark heuristic methods, our column generation approach yields reduced cost, with proportional solution time.

A decision support tool for forwarding operations with sequence-dependent loading

High productivity in forest harvesting requires efficient forwarding. Planning is complicated by multiple choices of routes and their order, the number and types of assortments, the loading sequence, and pile organization at the landing. This paper develops and tests a decision support tool for forwarder routing with sequential co-loading of assortments. Input data are the harvester production file (including Global Navigation Satellite Systems tracking), placement of landing, and machine specifications. The trail network is generated from the harvester production data when devising routes to pick up all log piles, including specific assortments and volumes. Multiple assortments can be loaded along each route, and a certain loading sequence of assortments is preferred and (or) required. Sorting time during co-loading varies, depending on the assortment combinations and bunk loading pattern. Route planning is modeled using a set-partitioning problem, and the solution method is a metaheuristic based on repeated matching. In addition to routes and loading sequences, solutions include the organization of assortment piles at the landing, depending on the total volume of each assortment. The tool produces similar results to those attained by skilled forwarder operators on five clearcuts (3–11 ha) in northern Sweden, when the results are compared with data from actual forwarder production files.

A constraint programming based column generation approach for crew scheduling: A case study for the Kayseri railway

Although railway systems have been widely used in public transportation for several years, crew scheduling research focuses on airline operations. Since the crew costs constitute the critical part of the operational costs in railway public transportation, the crew scheduling problem gains more importance. Crew scheduling aims to assign all trips to duty packages in a timeline while satisfying several labour working rules, company requirements, and operational restrictions. Because of the complexity of real-world conditions, crew scheduling problems are NP-complete. In this study, the crew scheduling problem was addressed, and a constraint programming based column generation algorithm was proposed as a solution approach. A column generation algorithm was employed to solve problems having a considerable number of variables. In this study, the master problem was designed as a set partitioning problem that aimed to assign all trips to duty while minimizing the cost. In the subproblem, new column(s) satisfying all constraints were generated and added to the master problem to reduce the cost. To analyse the performance of the proposed approach, an integer programming model of the problem was developed. An extensive computational study is presented to compare the results of these two modelling approaches via different real-life problem instances.

Batching and scheduling in a continuous-discrete hybrid flowshop: Lagrangian relaxation-based heuristic algorithms

ABSTRACT We study a two-stage hybrid flow shop problem arising from a fine chemicals production facility, where the first stage is a continuous chemical reaction process and the second stage is a discrete filling-packaging process. The objective is to minimise the total weighted completion time through batching of the jobs and scheduling of the batches at first stage and scheduling of the jobs at second stage. We formulate the problem as a mixed integer programming model and develop a Lagrangian relaxation-based framework for solving it, where the original problem is decomposed into family-level subproblems, and each subproblem is transformed into a set-partitioning problem. The subproblems are solved to optimality via branch and price algorithm. We also propose two heuristic algorithms for reducing computational efforts without much loss of solution quality. Finally, we conduct computational experiments with both randomly generated and real data sets to test the performance of the three proposed algorithms and demonstrate that the algorithms perform efficiently.

Heterogeneous donor circles for fair liver transplant allocation.

The United States (U.S.) Department of Health and Human Services is interested in increasing geographical equity in access to liver transplant. The geographical disparity in the U.S. is fundamentally an outcome of variation in the organ supply to patient demand (s/d) ratios across the country (which cannot be treated as a single unit due to its size). To design a fairer system, we develop a nonlinear integer programming model that allocates the organ supply in order to maximize the minimum s/d ratios across all transplant centers. We design circular donation regions that are able to address the issues raised in legal challenges to earlier organ distribution frameworks. This allows us to reformulate our model as a set-partitioning problem. Our policy can be viewed as a heterogeneous donor circle policy, where the integer program optimizes the radius of the circle around each donation location. Compared to the current policy, which has fixed radius circles around donation locations, the heterogeneous donor circle policy greatly improves both the worst s/d ratio and the range between the maximum and minimum s/d ratios. We found that with the fixed radius policy of 500 nautical miles (NM), the s/d ratio ranges from 0.37 to 0.84 at transplant centers, while with the heterogeneous circle policy capped at a maximum radius of 500 NM, the s/d ratio ranges from 0.55 to 0.60, closely matching the national s/d ratio average of 0.5983. Our model matches the supply and demand in a more equitable fashion than existing policies and has a significant potential to improve the liver transplantation landscape.

An integrated local-search/set-partitioning refinement heuristic for the Capacitated Vehicle Routing Problem

In this paper, an effective heuristic algorithm for large-scale instances of the Capacitated Vehicle Routing Problem is proposed. The technique consists in a local search method entangled with a restricted Set Partitioning problem optimization. Helsgaun’s LKH-3 algorithm has been used for the local search phase, with a number of implementation improvements. The restricted Set Partitioning formulation is solved by means of an exact commercial Integer Liner Programming solver. The resulting algorithm is able to consistently improve the solutions obtained by a state-of-the-art heuristic from the literature, as well as some of the best-know solutions maintained by the CVRPLIB website.

Distributed Integral Column Generation for Set Partitioning Problems

Distributed Integral Column Generation for Set Partitioning Problems

A variable neighborhood search approach to solve the order batching problem with heterogeneous pick devices

In this paper, we discuss the order batching problem in manual order picking systems. In such systems, order pickers move through a warehouse to collect items that are requested by customers. Customer orders have to be grouped into picking orders of limited size to ensure that the total length of the picker tours to collect all items is minimized. Motivated by real-world restrictions, we assume that the pick devices are heterogeneous, i.e., it is not necessarily possible to transport each customer order with each device. Since the order batching problem is NP-hard, we propose both serial and parallel versions of a variable neighborhood search (VNS) scheme to tackle large-sized problem instances in a reasonable amount of time. Moreover, the VNS scheme is hybridized with a set partitioning problem formulation. Based on computational experiments, we show that the proposed VNS scheme is a very fast approach that is able to provide solutions for benchmark instances with homogeneous pick devices that are of comparable quality or even partly superior to the best performing approaches from the literature. New benchmark instances for heterogeneous pick devices are proposed. High-quality solutions are obtained for these instances. We also consider large-scale problem instances from a real-world e-commerce warehouse. The VNS scheme outperforms the heuristic applied for order batching in this warehouse. Moreover, we demonstrate that a warehouse setup with heterogeneous pick devices is highly beneficial with respect to operating cost savings.

Integral Column Generation for Set Partitioning Problems with Side Constraints

The integral column generation algorithm (ICG) was recently introduced to solve set partitioning problems involving a very large number of variables. This primal algorithm generates a sequence of integer solutions with decreasing costs, leading to an optimal or near-optimal solution. ICG combines the well-known column generation algorithm and a primal algorithm called the integral simplex using decomposition algorithm (ISUD). In this paper, we develop a generalized version of ICG, denoted I2CG, that can solve efficiently large-scale set partitioning problems with side constraints. This new algorithm can handle the side constraints in the reduced problem of ISUD, in its complementary problem, or in both components. Computational experiments on instances of the airline crew pairing problem (CPP) and the multidepot vehicle routing problem with time windows show that the latter strategy is the most efficient one and I2CG significantly outperforms basic variants of two popular column generation heuristics, namely, a restricted master heuristic and a diving heuristic. For the largest tested CPP instance with 1,761 constraints, I2CG can produce in less than one hour of computational time more than 500 integer solutions leading to an optimal or near-optimal solution. Summary of Contribution: In this paper, we develop a new integral column generation algorithm that can solve efficiently large-scale set partitioning problems with side constraints. The latter alter the quasi-integrality property needed for primal integral algorithms. The paper adds a methodological contribution remedying this issue. This remedy should, in our opinion, boost the use of primal exact methods, especially in the column generation context. The paper also has a computational contribution. Effectively, computational experiments on instances of the airline crew pairing problem and the multidepot vehicle routing problem with time windows are extensively discussed. We compare the proposed algorithm to basic variants of two popular column generation heuristics.

The Multiport Berth Allocation Problem with Speed Optimization: Exact Methods and a Cooperative Game Analysis

We consider a variant of the berth allocation problem—that is, the multiport berth allocation problem—aimed at assigning berthing times and positions to vessels in container terminals. This variant involves optimizing vessel travel speeds between multiple ports, thereby exploiting the potentials of a collaboration between carriers (shipping lines) and terminal operators. Using a graph representation of the problem, we reformulate an existing mixed-integer problem into a generalized set partitioning problem, in which each variable refers to a sequence of feasible berths in the ports that the vessel visits. By integrating column generation and cut separation in a branch-and-cut-and-price procedure, our proposed method is able to outperform commercial solvers in a set of benchmark instances and adapt better to larger instances. In addition, we apply cooperative game theory methods to efficiently distribute the savings resulting from a potential collaboration and show that both carriers and terminal operators would benefit from collaborating.