Small Problem Instances Research Articles

Two‐Agent Single Machine Order Acceptance Scheduling Problem to Maximize Net Revenue

The paper considers two‐agent order acceptance scheduling problems with different scheduling criteria. Two agents have a set of jobs to be processed by a single machine. The processing time and due date of each job are known in advance. In the order accepting scheduling problem, jobs are allowed to be rejected. The objective of the problem is to maximize the net revenue while keeping the weighted number of tardy jobs for the second agent within a predetermined value. A mixed‐integer linear programming (MILP) formulation is provided to obtain the optimal solution. The problem is considered as an NP-hard problem. Therefore, MILP can be used to solve small problem instances optimally. To solve the problem instances with realistic size, heuristic and metaheuristic algorithms have been proposed. A heuristic method is used to determine and secure a quick solution while the metaheuristic based on particle swarm optimization (PSO) is designed to obtain the near‐optimal solution. A numerical experiment is piloted and conducted on the benchmark instances that could be obtained from the literature. The performances of the proposed algorithms are tested through numerical experiments. The proposed PSO can obtain the solution within 0.1% of the optimal solution for problem instances up to 60 jobs. The performance of the proposed PSO is found to be significantly better than the performance of the heuristic.

Parallel machine scheduling with the total weighted delivery time performance measure in distributed manufacturing

In this paper, we address a parallel machine scheduling problem that is motivated by distributed manufacturing settings. Our objective is to minimize the total weighted delivery time (TWD) after including delivery durations for the jobs. We first analyse several special cases which can be solved to optimality in polynomial time. Based on the gained insights of the analysis, constructive algorithms are proposed for the general problem setting. A greedy randomized adaptive search (GRASP) framework is proposed to guide the subordinate heuristics to further improve algorithm performance for large-sized problem instances. Computational experiments based on randomly generated problem instances are carried out. They demonstrate that the GRASP computes competitive schedules for the special cases. For small problem instances, the GRASP is able to compute optimal solutions. Moreover, it outperforms two genetic algorithms (GAs) that differ in the way how structural properties of optimal solutions are included in terms of both solution quality and computing time.

Modified Harris Hawks Optimizer for Solving Machine Scheduling Problems

Scheduling can be described as a decision-making process. It is applied in various applications, such as manufacturing, airports, and information processing systems. More so, the presence of symmetry is common in certain types of scheduling problems. There are three types of parallel machine scheduling problems (PMSP): uniform, identical, and unrelated parallel machine scheduling problems (UPMSPs). Recently, UPMSPs with setup time had attracted more attention due to its applications in different industries and services. In this study, we present an efficient method to address the UPMSPs while using a modified harris hawks optimizer (HHO). The new method, called MHHO, uses the salp swarm algorithm (SSA) as a local search for HHO in order to enhance its performance and to decrease its computation time. To test the performance of MHHO, several experiments are implemented using small and large problem instances. Moreover, the proposed method is compared to several state-of-art approaches used for UPMSPs. The MHHO shows better performance in both small and large problem cases.

Improved Success Probability with Greater Circuit Depth for the Quantum Approximate Optimization Algorithm

Present-day, noisy, small or intermediate-scale quantum processors---although far from fault-tolerant---support the execution of heuristic quantum algorithms, which might enable a quantum advantage, for example, when applied to combinatorial optimization problems. On small-scale quantum processors, validations of such algorithms serve as important technology demonstrators. We implement the quantum approximate optimization algorithm (QAOA) on our hardware platform, consisting of two superconducting transmon qubits and one parametrically modulated coupler. We solve small instances of the NP-complete exact-cover problem, with 96.6% success probability, by iterating the algorithm up to level two.

Development of a reliable and flexible supply chain network design model: a genetic algorithm based approach

Enhancing the proactive strategic capabilities to withstand the most unfavourable circumstances is always appreciated as a long-term policy rather than incident-based responses. The present research is positioned on this fundamental notion of supply chain risk management with a particular focus on strategic capabilities like reliability and flexibility that often conflict with cost. Accordingly, the authors propose a multi-objective mathematical model for designing a four-echelon supply chain that optimises cost, reliability, and volume flexibility. Interestingly, this research is the maiden effort to optimise the supply chain with these trifold objectives and herein lies the novelty as well as the challenges. Consequently, a genetic algorithm based approach is utilised as the solution methodology. To demonstrate the effectiveness of the proposed method, the small problem instances and the four-echelon problems have also been validated through exact methods and simulated annealing algorithm, respectively. A case study on a footwear supply chain involving three echelons is also presented to showcase the industrial applicability and adaptability of the proposed model. A fuzzy TOPSIS method has been adopted in the case study to incorporate the expert opinion for assigning priorities to the objectives. Supply chain professionals can leverage this methodology to establish a risk resistant supply chain.

Variable neighborhood search-based solution methods for the pollution location-inventory-routing problem

This work presents efficient solution approaches for a new complex NP-hard combinatorial optimization problem, the Pollution Location Inventory Routing problem (PLIRP), which considers both economic and environmental issues. A mixed-integer linear programming model is proposed and first, small problem instances are solved using the CPLEX solver. Due to its computational complexity, General Variable Neighborhood Search-based metaheuristic algorithms are developed for the solution of medium and large instances. The proposed approaches are tested on 30 new randomly generated PLIRP instances. Parameter estimation has been performed for determining the most suitable perturbation strength. An extended numerical analysis illustrates the effectiveness and efficiency of the underlying methods, leading to high-quality solutions with limited computational effort. Furthermore, the impact of holding cost variations to the total cost is studied.

Metaheuristic algorithms with solution encoding mixing for effective optimization of SDM optical networks

Due to the continuous growth of Internet traffic, new technologies are proposed to overcome the capacity crunch problem. One of the most promising solutions for optical backbone networks is space division multiplexing (SDM), which allows parallel transmissions in properly designed optical fibers. In SDM, the problem of routing, space, and spectrum allocation (RSSA) emerges. RSSA is more challenging than corresponding optimization problems in optical networks without spatial multiplexing since the additional spatial dimension significantly increases the number of encodable solutions. The exact optimization methods can be used only for small and unrealistic problem instances. Therefore, effective metaheuristics are needed to solve RSSA for larger and practice-based problem instances. In this paper, we propose a new technique of solution encoding in metaheuristic algorithms. The key novelty and contribution is to mix two different solution encoding types to utilize their advantages and improve the overall method performance. The results of experiments conducted on reference optical network topologies using two representative optimization methods (dedicated evolutionary algorithm and tabu search) show that the proposed approach significantly improves the quality of the results for both method types. The proposed encoding technique is generic and applicable to many different metaheuristics employed in solving various optimization problems.

Scheduling spatially distributed jobs with degradation: Application to pothole repair

This paper considers scheduling spatially distributed jobs with degradation. A mixed integer programming (MIP) model is developed for the linear degradation case in which no new jobs arrive. Properties of the model are analyzed, following which three heuristics are developed, enhanced greedy, chronological decomposition and simulated annealing. Numerical tests are conducted to: (i) establish limits of the exact MIP solution, (ii) identify the best heuristic based on an analysis of performance on small problem instances for which exact solutions are known, (iii) solve large problem instances and obtain lower bounds to establish solution quality, and (iv) study the effect of three key model parameters. Findings from our computational experiments indicate that: (i) exact solutions are limited to instances with less than 14 jobs; (ii) the enhanced greedy heuristic followed by the application of the simulated annealing heuristic yields high quality solutions for large problem instances in reasonable computation time; and (iii) the factors “degradation rate” and “work hours” have a significant effect on the objective function. To demonstrate applicability of the model, a case study is presented based on a pothole repair scenario from Buffalo, New York, USA. Findings from the case study indicate that scheduling spatially dispersed jobs with degradation such as potholes requires: (i) careful consideration of the number of servers assigned, degradation rate and depot location; (ii) appropriate modeling of continuously arriving jobs; and (iii) appropriate incorporation of equity consideration.

Nonprofit peer-to-peer ridesharing optimization

Both for-profit and nonprofit peer-to-peer (P2P) ridesharing services have gained enormous popularity in recent years due to their advantages over solo driving and public transit. We study the rideshare matching and routing problem in a nonprofit P2P ridesharing system consisting of a matching agency, drivers and riders. The matching agency is a government or a not-for-profit organization and its objective is to maximize the societal benefits of ridesharing. The drivers involved are commuters and hence have their own travel plans, which are executed regardless of whether riders are matched with them. We consider both static and dynamic versions of the nonprofit P2P ridesharing problem. Existing modeling and solution approaches for similar P2P ridesharing problems can only solve relatively small problem instances optimally. We propose an exact solution algorithm for the static version of the problem by taking advantage of its special characteristics. This exact solution approach formulates and solves the problem as a set packing formulation using route-based variables, and uses an efficient graph-based approach to generate all necessary vehicle routes in the formulation quickly. We also develop a column generation (CG) based heuristic approach for the static problem. Finally, we propose two dynamic dispatching policies for the dynamic version of the problem. Our proposed exact algorithm solves very large problem instances (e.g., with 600 drivers and 1800 riders) of the static problem and our CG based heuristic can find near-optimal solutions for even larger instances of the static problem in short computation time. Our dynamic dispatching policies can generate near-optimal solutions for the dynamic problem in real-time fashion. We also generate some important insights based on some taxi trip data in Washington DC. First, P2P ridesharing can bring significant cost-saving, especially when the participants have a relatively flexible schedule. For every 10% increase in schedule flexibility, there is an about 4% to 7% increase in cost-saving. Second, the cost-saving due to ridesharing increases with the vehicle capacity, but this increase slows down quickly when the vehicle capacity reaches 4. Third, ridesharing generates more cost savings during peak hours and in urban areas.

Finding Longest Common Subsequences: New anytime A[formula omitted] search results

The Longest Common Subsequence (LCS) problem aims at finding a longest string that is a subsequence of each string from a given set of input strings. This problem has applications, in particular, in the context of bioinformatics, where strings represent DNA or protein sequences. Existing approaches include numerous heuristics, but only a few exact approaches, limited to rather small problem instances. Adopting various aspects from leading heuristics for the LCS, we first propose an exact A∗ search approach, which performs well in comparison to earlier exact approaches in the context of small instances. On the basis of A∗ search we then develop two hybrid A∗–based algorithms in which classical A∗ iterations are alternated with beam search and anytime column search, respectively. A key feature to guide the heuristic search in these approaches is the usage of an approximate expected length calculation for the LCS of uniform random strings. Even for large problem instances these anytime A∗ variants yield reasonable solutions early during the search and improve on them over time. Moreover, they terminate with proven optimality if enough time and memory is given. Furthermore, they yield upper bounds and, thus, quality guarantees when terminated early. We comprehensively evaluate the proposed methods using most of the available benchmark sets from the literature and compare to the current state-of-the-art methods. In particular, our algorithms are able to obtain new best results for 82 out of 117 instance groups. Moreover, in most cases they also provide significantly smaller optimality gaps than other anytime algorithms.

Reliable facility location design with round-trip transportation under imperfect information Part I: A discrete model

In this paper, we propose a discrete model to investigate reliable location design with round-trip transportation under imperfect information. The unique feature of this problem is that under each disruption scenario, a customer’s outbound and inbound trips are different when she travels to obtain a service. The discrete model is formulated as a compact non-linear integer programming problem and solved efficiently by a customized Lagrangian relaxation algorithm. Numerical experiments find that the discrete model performs well for small and medium-scale problem instances. Sensitivity analyses reveal the impacts of several parameters on both the cost components and the optimal facility layouts.

ERPOT: A Quad-Criteria Scheduling Heuristic to Optimize Execution Time, Reliability, Power Consumption and Temperature in Multicores

We investigate multi-criteria optimization and Pareto front generation. Given an application modeled as a Directed Acyclic Graph (DAG) of tasks and a multicore architecture, we produce a set of non-dominated (in the Pareto sense) static schedules of this DAG onto this multicore. The criteria we address are the execution time, reliability, power consumption, and peak temperature. These criteria exhibit complex antagonistic relations, which make the problem challenging. For instance, improving the reliability requires adding some redundancy in the schedule, which penalizes the execution time. To produce Pareto fronts in this 4-dimension space, we transform three of the four criteria into constraints (the reliability, the power consumption, and the peak temperature), and we minimize the fourth one (the execution time of the schedule) under these three constraints. By varying the thresholds used for the three constraints, we are able to produce a Pareto front of non-dominated solutions. We propose two algorithms to compute static schedules. The first is a ready list scheduling heuristic called Execution time, Reliability, POwer consumption and Temperature (ERPOT). ERPOT actively replicates the tasks to increase the reliability, uses Dynamic Voltage and Frequency Scaling to decrease the power consumption, and inserts cooling times to control the peak temperature. The second algorithm uses an Integer Linear Programming (ILP) program to compute an optimal schedule. However, because our multi-criteria scheduling problem is NP-complete, the ILP algorithm is limited to very small problem instances. Comparisons showed that the schedules produced by ERPOT are on average only 10 percent worse than the optimal schedules computed by the ILP program, and that ERPOT outperforms the PowerPerf-PET heuristic from the literature on average by 33 percent.

Quantum Annealing with Longitudinal Bias Fields.

Quantum annealing aims at solving hard computational problems through adiabatic state preparation. Here, I propose to use inhomogeneous longitudinal magnetic fields to enhance the efficiency of the annealing. Such fields are able to bias the annealing dynamics into the desired solution, and in many cases, suitable field configurations can be found iteratively. Alternatively, the longitudinal fields can also be applied as an antibias which filters out unwanted contributions from the final state. This strategy is particularly well suited for instances which are difficult to solve within the standard quantum annealing approach. By numerically simulating the dynamics for small instances of the exact cover problem, the performance of these different strategies is investigated.

Learning Optimal Classification Trees Using a Binary Linear Program Formulation

We provide a new formulation for the problem of learning the optimal classification tree of a given depth as a binary linear program. A limitation of previously proposed Mathematical Optimization formulations is that they create constraints and variables for every row in the training data. As a result, the running time of the existing Integer Linear programming (ILP) formulations increases dramatically with the size of data. In our new binary formulation, we aim to circumvent this problem by making the formulation size largely independent from the training data size. We show experimentally that our formulation achieves better performance than existing formulations on both small and large problem instances within shorter running time.

Metaheuristic approaches to a vehicle scheduling problem in sugar beet transportation

A variant of vehicle scheduling problem (VSP) arising from the sugar beet transportation in a sugar factory in Serbia is introduced. The objective of the considered VSP is to minimize the required transportation time under problem-specific constraints. The problem is formulated as a mixed integer linear program (MILP). Within the framework of commercial CPLEX solver the proposed MILP model was able to produce optimal solutions for small size problem instances. Therefore, two metaheuristic methods, variable neighborhood search (VNS) and greedy randomized adaptive search procedure (GRASP), are designed to solve problem instances of larger dimensions. The proposed GRASP and VNS are evaluated and compared against CPLEX and each other on the set of real-life and generated problem instances. Computational results show that VNS is superior method with respect to the solution quality, while GRASP is able to find high quality solutions within very short running times.

Transit network design with pollution minimization

A critical step in the design of urban transport networks is the determination of the routes and the frequencies of buses. This situation entails a highly combinatorial optimization problem with a complex computational solution, even for small instances. Several studies have addressed such a situation, minimizing travel times as the main objective. However, the growing trend toward the development of sustainable transport operations requires that the design of the network also considers the emissions of toxic gases that result from combustion, which leads to a new variant of this type of problem, called the pollution transit network design problem. In this paper, the problem is formulated as a biobjective mathematical programming model. Complex problem instances are proposed for this problem, and by using a multi-objective genetic algorithm, we approach the unimodal and bimodal version of the problem by taking into account the elastic demand between buses and cars. By using the proposed mathematical programming model and the genetic algorithm for small and large problem instances, respectively, we show that the generated pollutant emissions are drastically reduced without increasing travel times or costs.

Hybrid algorithms for placement of virtual machines across geo-separated data centers

Cloud computing has emerged as a new paradigm for hosting and supplying services over the Internet. This technology has brought many benefits, such as eliminating the need for maintaining expensive computing hardware. With an increasing demand for cloud computing, providing performance guarantees for applications that run over cloud become important. Applications can be abstracted into a set of virtual machines with certain guarantees depicting the quality of service of the application. In this paper, we consider the placement of these virtual machines across multiple data centers (VMPlacement), meeting the quality of service requirements while minimizing the bandwidth cost of the data centers. This problem is a generalization of the NP-hard generalized quadratic assignment problem (GQAP). In this paper, we present a greedy randomized adaptive search procedure and a biased random-key genetic algorithm, both hybridized with a path-relinking strategy and a local search based on variable neighborhood descent for solving this problem. The hybrid heuristics are also tested on instances of the GQAP. We show that both algorithms are effective in quickly solving small and large instances of VMPlacement problem, especially when the path-relinking is used. For GQAP, the results outperform the previous state-of-the-art algorithms.

SINGLE-MACHINE SCHEDULING OF THE INDIVISIBLE MULTI-OPERATION JOBS

This paper considers a single-machine scheduling problem of multi- operation jobs where each job consists of several operations processed contiguously, rather than being intermingled with the operations of different jobs. That is, the jobs are indivisible. A sequence-independent setup is required if the machine switches from one operation to another. However, no setup is necessary before the first operation of a job if this first operation is the same as the last operation of the immediately previous job. A job is complete when all of its operations have been processed. We investigate the problem for two cases. Makespan, which is the time needed to complete all jobs, is minimised in the first case; whereas the total completion time, which is the sum of the job completion times, is minimised in the second case. We show that the makespan problem is solvable in polynomial time. For the problem of minimising total completion time, we develop a mixed integer linear programming (MILP) model, which is capable of solving small and medium-sized problem instances optimally, and obtain a very small gap between the solution found and the best possible solution for the unsolved large-sized problem instances.

A general variable neighborhood search-based solution approach for the location-inventory-routing problem with distribution outsourcing

This work presents a Mixed Integer Programing (MIP) formulation for a new complex NP-hard combinatorial optimization problem, the Location Inventory Routing with Distribution Outsourcing (LIRPDO). Due to its computational complexity, only small problem instances can be solved by exact solvers. Therefore, a General Variable Neighborhood Search (GVNS)-based metaheuristic algorithm for solving large LIRPDO instances is presented. The proposed approach has been tested on 20 new randomly generated LIRPDO instances, 20 existing benchmark LIRP instances from the literature and 30 new large-scale random generated LIRP instances. An extended numerical analysis illustrates the efficiency of the underlying method, leading to acceptable solutions requiring limited computational effort.

A hybrid genetic algorithm for operating room scheduling.

In this research, we studied operating room scheduling problem of assigning a set of surgeries to several multifunctional operating rooms. The objectives are to maximize the utilization of the operating rooms, to minimize the overtime-operating cost, and to minimize the wasting cost for the unused time. To begin with, a revised mathematical model is constructed to assign surgeries to operating rooms within one week. Then, we proposed four easy-to-implement heuristics that can guarantee to find feasible solutions for the studied problem efficiently. Furthermore, we presented four local search procedures that can improve a given solution significantly. Finally, a hybrid genetic algorithm (HGA) that incorporated with initial solutions, local search procedures and elite search procedure is applied to the studied problem. Computational results show that for small problem instances, the HGA can find near optimal solutions efficiently while for large problem instances, the HGA performs significantly better than the four proposed heuristics. We concluded that surgery schedules obtained by using HGA has less wasting cost for the unused time, much higher utilization of operating rooms, and produce less overtime-operating cost.