Linear Programming Solver Research Articles

GPU-accelerated transportation simplex algorithm

Transportation Problem (TP) is a popular linear program for optimally matching several supply centers to several demand centers at the smallest transportation cost. Recent disruptions in the physical supply chains and the growth of internet marketplaces such as ride-sharing, doorstep delivery, and expedited shipping have engendered a need for efficient algorithms to solve large-scale TPs in near real-time. The Transportation Simplex Algorithm (TSA) is a traditional method to solve TP to optimality. However, TSA is unsuitable for these new applications because of its long run time. The evolution of accelerated computing using Graphics Processing Units (GPUs) has recently attracted some interest in solving optimization problems. In this paper, we develop a GPU-accelerated TSA for large-dimensions. The underlying parallelism in the iterative steps of TSA has been uncovered and exploited. The results show that the accelerated algorithm performs up to 8 times faster on average compared to the known sequential algorithm and up to 4 times faster on average compared to the state-of-the-art commercial Linear Programming solver.

Reachability Analysis of Sigmoidal Neural Networks

This paper extends the star set reachability approach to verify the robustness of feed-forward neural networks (FNNs) with sigmoidal activation functions such as Sigmoid and TanH. The main drawbacks of the star set approach in Sigmoid/TanH FNN verification are scalability, feasibility, and optimality issues in some cases due to the linear programming solver usage. We overcome this challenge by proposing a relaxed star (RStar) with symbolic intervals, which allows the usage of the back-substitution technique in DeepPoly to find bounds when overapproximating activation functions while maintaining the valuable features of a star set. RStar can overapproximate a sigmoidal activation function using four linear constraints (RStar4) or two linear constraints (RStar2), or only the output bounds (RStar0). We implement our RStar reachability algorithms in NNV and compare them to DeepPoly via robustness verification of image classification DNNs benchmarks. The experimental results show that the original star approach (i.e., no relaxation) is the least conservative of all methods yet the slowest. RStar4 is computationally much faster than the original star method and is the second least conservative approach. It certifies up to 40% more images against adversarial attacks than DeepPoly and on average 51 times faster than the star set. Last but not least, RStar0 is the most conservative method, which could only verify two cases for the CIFAR10 small Sigmoid network, δ = 0.014. However, it is the fastest method that can verify neural networks up to 3528 times faster than the star set and up to 46 times faster than DeepPoly in our evaluation.

Solution strategies for integrated distribution, production, and relocation problems arising in modular manufacturing

Recently, there has been a paradigm shift by certain energy companies towards modular manufacturing, whereby transportable modular production units can be relocated between production facilities to meet the spatial and temporal changes in the availabilities, demands, and prices of the underlying commodities. We refer to the optimal distribution, production, and storage of commodities, including intermediary commodities, and the relocation and operation of the modular production units as the dynamic multiple commodity supply chain problem with modular production units. To this end, we present a “flow-based” and a “path-based” mixed-integer linear programming formulation to model the problem. In an effort to solve large-scale instances of the problem, we propose an iterative three-stage matheuristic for the “path-based” formulation. In the first stage of the matheuristic, a feasible solution to the problem is generated by an integrated column generation and Lagrangian relaxation based heuristic. In the second stage of the matheuristic, a path-relinking procedure is utilized as a local search heuristic to further improve the solution. And in the final stage of the matheuristic, the Lagrangian multipliers are updated via a subgradient method. The effectiveness of the matheuristic is illustrated through numerical experiments with a set of randomly generated test instances. For the large-scale test instances, the results show that the matheuristic produces quality solutions orders of magnitude faster than a “state-of-the-art” mixed-integer linear programming solver.

A matheuristic for tri-objective binary integer linear programming

Many real-world optimisation problems involve multiple objectives. When considered concurrently, they give rise to a set of optimal trade-off solutions, also known as efficient solutions. These solutions have the property that neither objective can be improved without deteriorating another objective. Motivated by the success of matheuristics in the single-objective domain, we propose a linear programming-based matheuristic for tri-objective binary integer linear programming. To achieve a high-quality approximation of the optimal set of trade-off solutions, a lower bound set is first obtained using the vector linear programming solver Bensolve. Then, feasibility pump-based ideas in combination with path relinking are applied in novel ways so as to obtain a high-quality upper bound set. Our matheuristic is compared to a recently suggested algorithm that is, to the best of our knowledge, the only existing matheuristic method for tri-objective binary integer linear programming. In an extensive computational study, we show that our method generates a better approximation of the true Pareto front than the state-of-the-art matheuristic on a large set of tri-objective benchmark instances. Since the developed approach starts from a potentially fractional lower bound set, it may also be used as a primal heuristic in the context of linear relaxation-based multi-objective branch-and-bound algorithms.

A Computationally Efficient Benders Decomposition for Energy Systems Planning Problems with Detailed Operations and Time-Coupling Constraints

Energy systems planning models identify least-cost strategies for expansion and operation of energy systems and provide decision support for investment, planning, regulation, and policy. Most are formulated as linear programming (LP) or mixed integer linear programming (MILP) problems. Despite the relative efficiency and maturity of LP and MILP solvers, large scale problems are often intractable without abstractions that impact quality of results and generalizability of findings. We consider a macro-energy systems planning problem with detailed operations and policy constraints and formulate a computationally efficient Benders decomposition separating investments from operations and decoupling operational timesteps using budgeting variables in the master model. This novel approach enables parallelization of operational subproblems and permits modeling of relevant constraints coupling decisions across time periods (e.g., policy constraints) within a decomposed framework. Runtime scales linearly with temporal resolution; tests demonstrate substantial runtime improvement for all MILP formulations and for some LP formulations depending on problem size relative to analogous monolithic models solved with state-of-the-art commercial solvers. Our algorithm is applicable to planning problems in other domains (e.g., water, transportation networks, production processes) and can solve large-scale problems otherwise intractable. We show that the increased resolution enabled by this algorithm mitigates structural uncertainty, improving recommendation accuracy. Funding: Funding for this work was provided by the Princeton Carbon Mitigation Initiative (funded by a gift from BP) and the Princeton Zero-carbon Technology Consortium (funded by gifts from GE, Google, ClearPath, and Breakthrough Energy). Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2023.0005 .

Multiobjective U-Shaped Disassembly Line Balancing Problem Considering Human Fatigue Index and an Efficient Solution

The progress of science and technology speeds up the replacement of products and produces a large number of end-of-life products. Traditional incineration causes a waste of resources and pollution to the environment. Disassembling and recycling end-of-life products are the recommended way to maximize the utilization of resources and reduce environmental pollution. Disassembly performance is affected by many factors, such as the disassembly posture of the human body, the fatigue of workers on a workstation, disassembly profit, and task precedence relationship. In this article, a mixed integer linear programming mathematical model for U-shaped layout disassembly line balancing problems is developed, in which the balance of workers’ fatigue indices is an optimization objective in addition to disassembly profits. An efficient solution to the problem that uses a collaborative resource allocation strategy of the multiobjective evolutionary algorithm is proposed. The linear programming solver CPLEX is used to verify the accuracy of the model and compared with the proposed algorithm. Experiments demonstrate that the algorithm is significantly superior to the CPLEX solver in handling large-scale cases. The proposed algorithm is also compared with two well-known algorithms, which further verifies its superiority.

Polytopal approximation for thrust magnitude constraints in rocket trajectory optimization

This paper proposes a linear programming approach for onboard optimization of rocket trajectories, motivated by the need for reliable and computationally efficient trajectory optimization methods in real-time applications. To reformulate rocket trajectory optimization problems within a linear programming framework, a polytopal approximation method is developed. The polytopal approximation method approximates the thrust magnitude constraint using a set of artificial variables and linear constraints. Compared with a state-of-the-art approximation method, the polytopal approximation method demonstrates a higher accuracy-to-size ratio, enabling rocket trajectory optimization problems to be cast into linear subproblems without significantly increasing problem sizes. As linear subproblems can be readily solved by linear programming solvers, the proposed trajectory optimization approach inherits the solver’s reliability and computational efficiency, making it potentially useful for critical onboard applications. An ascent trajectory optimization problem is provided as an illustrative example to demonstrate the effectiveness of the proposed linear programming approach.

Real-time energy scheduling for home energy management systems with an energy storage system and electric vehicle based on a supervised-learning-based strategy

With rising energy costs and concerns about environmental sustainability, there is a growing need to deploy Home Energy Management Systems (HEMS) that can efficiently manage household energy consumption. This paper proposes a new supervised-learning-based strategy for optimal energy scheduling of an HEMS that considers the integration of energy storage systems (ESS) and electric vehicles (EVs). The proposed supervised-learning-based HEMS framework aims to optimize the energy costs of households by forecasting the energy demand and simultaneously scheduling the charging and discharging operations of ESS and EV. From the scenarios extracted from historical data, the HEMS optimization problem is solved using a mixed-integer linear programming (MILP) solver to collect the datasets on the optimal actions of the ESS and EV. Accordingly, a supervised learning method is used to learn the optimal actions of the MILP solver using deep neural networks (DNNs). Well-trained DNNs act as decision-making tools that are subsequently applied to predict near-optimal actions for ESS and EV based on real-time data. The effectiveness of the proposed method is demonstrated through simulation results and compared with deep reinforcement learning-based and forecasting-based methods. The results show that the proposed method can significantly reduce energy costs and improve the efficiency of ESS and EV operations. Overall, the proposed supervised-learning-based HEMS offers a practical and effective solution for residential energy management.

Development of Heuristic Approaches for Last-Mile Delivery TSP with a Truck and Multiple Drones

Unmanned Aerial Vehicles (UAVs) are gaining momentum in many civil and military sectors. An example is represented by the logistics sector, where UAVs have been proven to be able to improve the efficiency of the process itself, as their cooperation with trucks can decrease the delivery time and reduce fuel consumption. In this paper, we first state a mathematical formulation of the Travelling Salesman Problem (TSP) applied to logistic routing, where a truck cooperates synchronously with multiple UAVs for parcel delivery. Then, we propose, implement, and compare different sub-optimal routing approaches to the formulated mFSTSP (multiple Flying Sidekick Travelling Salesman Problem) since the inherent combinatorial computational complexity of the problem makes it unattractable for commercial Mixed-Integer Linear Programming (MILP) solvers. A local search algorithm, two hybrid genetic algorithms that permutate feasible and infeasible solutions, and an alternative ad-hoc greedy method are evaluated in terms of the total delivery time of the output schedule. For the sake of the evaluation, the savings in terms of delivery time over the well-documented truck-only TSP solution are investigated for each proposed routing solution, and this is repeated for two different scenarios. Monte Carlo simulations corroborate the results.

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Renewable Hydrogen Energy Communities layouts towards off-grid operation

Solar hydrogen technology, i.e. producing green hydrogen by electrolysis fed by solar panels, is gaining attention as studies show it can compete with traditional electric batteries. Contemporary, the interest in the realization of Renewable Energy Community driven by the recent adoption in almost all the EU Countries of the RED II Directive is increasing. To match innovative technologies and new business models making the energy demand and the production closer in terms of time and space, the feasibility of Renewable Hydrogen Energy Community is explored. To determine the profitability of these solutions and size and operate the energy system, energy and economic analyses are needed as well as the optimization of its performance. A building complex with 180 kWp of PhotoVoltaics was studied using hourly real data on energy loads and production. Storage capacities ranging from 500 to 2000 kWh were simulated for hydrogen and electric battery options. The cost of investment and operation was also analyzed for the years 2020, 2030, and 2040 to evaluate changes in technology readiness level as well as accounting for the strong changes in electricity tariff due to the geopolitical issues effects on the market. The analysis was conducted using the DECAPLAN™ digital platform, which employs a mixed integer linear programming solver. Finally, the dispatch on hourly base has been compared for the same two months in 2019, 2020 and 2021 corresponding to pre, during and after pandemic restrictions.

Synthesizing MILP Constraints for Efficient and Robust Optimization

While mixed integer linear programming (MILP) solvers are routinely used to solve a wide range of important science and engineering problems, it remains a challenging task for end users to write correct and efficient MILP constraints, especially for problems specified using the inherently non-linear Boolean logic operations. To overcome this challenge, we propose a syntax guided synthesis (SyGuS) method capable of generating high-quality MILP constraints from the specifications expressed using arbitrary combinations of Boolean logic operations. At the center of our method is an extensible domain specification language (DSL) whose expressiveness may be improved by adding new integer variables as decision variables, together with an iterative procedure for synthesizing linear constraints from non-linear Boolean logic operations using these integer variables. To make the synthesis method efficient, we also propose an over-approximation technique for soundly proving the correctness of the synthesized linear constraints, and an under-approximation technique for safely pruning away the incorrect constraints. We have implemented and evaluated the method on a wide range of benchmark specifications from statistics, machine learning, and data science applications. The experimental results show that the method is efficient in handling these benchmarks, and the quality of the synthesized MILP constraints is close to, or higher than, that of manually-written constraints in terms of both compactness and solving time.

Access mechanism for period flows of non-deterministic end systems for time-sensitive networks

The IEEE 802.1Qbv standard schedules time-triggered (TT) flows (i.e., period flows) in a fixed TT Window each period. However,period flows generated by non-deterministic end systems exhibit significant jitter, whichleads to a mismatch between the generation times and the scheduled TT Windows. In the worst case scenario, this mismatch can cause an additional send delay of approximately one period in the flow. In this study,we model the send delay issue due to the maximum send delay requirement for all frames in a period flow not being satisfied simultaneously, and measure the jitter of period flows in a typical non-deterministic end system. Moreover, we propose an access mechanism for jittered period flows to schedule multiple conflict-free TT Windows in each period of flows at the source end system and control the send delay within the required delay tolerance. This mechanism enables deterministic access to jittered period flows, providing a prerequisite for reliable end-to-end transmission in the network. Moreover, this mechanism adopts a multi-objective integer linear programming (MILP) solver to optimize the TT Windows schedule. Furthermore, we establish the constraints and objective functions for the MILP solver and evaluate the mechanism with actual sampled frames in a real non-deterministic end system. Compared with the conventional fixed single TT Window and worst-case delay analysis mechanisms, the proposed mechanism satisfies the send delay requirements and considerably reduces the buffer usage at the source end system.

A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed-Integer Linear Programming

We propose a hybrid approach for solving problems of mixed-integer linear programming with two types of variables (MILPTTV). The hybrid approach includes three algorithms: (1) Benders decomposition; (2) the ant colony algorithm; (3) the feasible heuristic method. We decompose the programming problem into a master problem and a subproblem using Benders decomposition. The ant colony algorithm generates an initial solution for the master problem. A feasible heuristic is used to obtain feasible solutions as input to the subproblem, whereas the subproblem is solved to optimality by using a linear programming solver that is named Linprog function in MATLAB R2016a. Over successive iterations, the master problem is refined by adding cuts from the subproblem and the master problem is solved by a linear programming solver Cplexmilp function. We compare the performance of the proposed hybrid approach against a general Benders decomposition approach as well as against a mixed-integer programming solver CPLEX (version 12.6) that is invoked by MATLAB on MIPLIB test problem mas76, mas74 and some large-scale numerical examples.

Conditional TMDTO as a MILP Instance

Conditional Time-Memory-Data Trade-off (TMDTO) attack given by Biryukov and Shamir can be reduced to the following problem: “Find the minimum number of state bits that should be fixed in order to recover the maximum number of state bits by utilizing the keystream bits and value of rest of the state bits”. As per our literature survey, existing algorithms search for state bits that should be fixed (as minimum as possible) in order to recover the maximum possible state bits directly through the keystream bits. However, those algorithms are cipher specific and require extensive manual effort in analyzing the keystream bit equations. In this manuscript, we have constructed an automated framework that is easy to implement and solves the above problem (for the case when bits are fixed to 0) for any NLFSR based stream cipher with better complexity, thereby reducing manual efforts. However, we do not claim any global optimum for fixed bits. We tried to reduce the number of fixed bits as much as possible. To show that our algorithm is applicable to a majority of NLFSR based stream ciphers, we implement it on three different stream ciphers: LIZARD, GRAIN-128a and ESPRESSO. It improves all existing TMDTO results on these ciphers. The framework involves modelling keystream bit equations into a set of linear constraints, which is then solved by using a Mixed Integer Linear Programming (MILP) solver, Gurobi. The advantages of our automated framework over other methods are that we can achieve better results with far less effort, and it can be applied to any stream cipher of a similar structure with very ease. To the best of our knowledge, our MILP model is the first work that converts the conditional TMDTO of a stream cipher into a linear optimization problem. As a consequence, for LIZARD cipher, we reduce the number of fixed bits by 20 bits from the previous best result when the number of recovered bits is 18. In the case of GRAIN-128a, the highest reduction in the number of fixed bits is by 34 bits when the number of recovered bits is 35. Lastly, for ESPRESSO cipher, the reduction is by 7 bits when the number of recovered bits is 35.

Minimizing the makespan and carbon emissions in the green flexible job shop scheduling problem with learning effects

One of the most difficult challenges for modern manufacturing is reducing carbon emissions. This paper focuses on the green scheduling problem in a flexible job shop system, taking into account energy consumption and worker learning effects. With the objective of simultaneously minimizing the makespan and total carbon emissions, the green flexible job shop scheduling problem (GFJSP) is formulated as a mixed integer linear multiobjective optimization model. Then, the improved multiobjective sparrow search algorithm (IMOSSA) is developed to find the optimal solution. Finally, we conduct computational experiments, including a comparison between IMOSSA and the nondominated sorting genetic algorithm II (NSGA-II), Jaya and the mixed integer linear programming (MILP) solver of CPLEX. The results demonstrate that IMOSSA has high precision, good convergence and excellent performance in solving the GFJSP in low-carbon manufacturing systems.

Facilities Location in the Supply Chain Network Using an Iterated Local Search Algorithm

The supply chain design problem is one of the important problems in supply chain management. In this problem, the goal is to locate several facilities for distribution centers in different geographical areas to cover customer demands. In the model presented in this paper, customer demand is met through distribution centers, while these distribution centers supply products from manufacturing plants. In this paper, a mathematical model of integer linear programming is presented that, in addition to the optimal supply chain network design, considers the facility's location. This model aims to minimize the costs of sending products from the factory to distribution centers, from distribution centers to customers, and establishing distribution centers. To evaluate the proposed model, different network samples were created randomly on small, medium and large scales for experiments. In order to perform the experiments, a linear programming solver (LP Solver) and an iterated local search algorithm were used to compare the performance of each of these methods to find the optimal solution. These experiments show the superiority of the iterated local search algorithm compared to LP Solver in achieving the optimal solution in the shortest runtime for all instances.

Using Revised Simplex Modelling to Determine the Optimal Loan Portfolio

Purpose: The main aim of this study is to develop Linear Programming (LP) model to help ADB, Sunyani branch in the Brong Ahafo Region to allocate their funds to prospective loan seekers in order to maximize profits. Methodology: To achieve this aim, a secondary data were extracted from the annual reports and financial statements of the bank. Based on these empirical data, LP model was formulated. A LP Solver based on Revised Simplex Algorithm (a computerized software application) was used to solve the problem. Findings: The results from the model showed that ADB, Sunyani branch would be making annual profit of GH¢ 476732.00 on loans alone as against GH¢ 190693.00 in 2011 if they stick to the model. Recommendation: From the study, it was realized that the Scientific method used to develop the proposed model can have a significant increase in the Bank’s profit margin if put into use.

A deterministic bounding algorithm vs. a hybrid meta-heuristic to deal with a bilevel mixed-integer nonlinear optimization model for electricity dynamic pricing

In the electricity retail market, the retailer company aims to determine the optimal time-of-use (ToU) prices to maximize profits resulting from buying energy in organized (long-term, day-ahead, balancing) markets and selling it to consumers. Therefore, the retailer should take into account the consumer’s demand response actions to minimize the electricity bill in face of time varying prices. In this paper, this problem is formulated as a bilevel mixed-integer nonlinear programming model in which the retailer is the leader and the consumer is the follower. The consumer’s problem encompasses the integrated optimization of all home energy resources, considering re-scheduling appliance operation, charging/discharging of electric vehicle and stationary batteries, local microgeneration and (buying and selling) exchanges with the grid. The accurate physical modelling of appliance operation to generate effective load scheduling solutions imposes a high computational burden. Two algorithms are proposed to address this problem: a deterministic bounding algorithm (DBA) using an optimal-value-function approach for bilevel optimization, and a hybrid meta-heuristic using a particle swarm optimization algorithm to tackle the upper-level problem that calls an exact mixed-integer linear programming solver to deal with the lower-level problem. In the framework of the DBA, three different techniques were implemented to deal with the nonlinearities arising from the products of integer and continuous variables (bilinear terms): 1) solving the (non-convex) subproblems of DBA using a mixed-integer nonlinear solver, 2) using the McCormick envelopes to approximate the bilinear terms by linear ones, and 3) expressing the integer variables by binary ones and linearizing the bilinear terms using an exact form. Computational experiments are presented and discussed for real data settings of the problem under study considering a computational budget to compare the different algorithms and techniques employed. The results showed that the DBA with the approximate linearization technique (2) leads to the best solutions.