Relaxed Master Problem Research Articles

Robust optimization algorithm for integrated crane assignment and scheduling in slab yard with uncertain arrival time

A slab yard served as a key component of the iron and steel plant. Effective slab yard crane assignment and scheduling directly affect the overall efficiency of the steel production process. This work addresses an uncertain slab yard crane assignment and scheduling problem (SYCAS) with the fluctuating arrival time. A stochastic programming model is established to obtain a robust solution that minimises the completion time of slab groups under uncertainty. Due to its NP-hardness, we developed an improved column and cut generation method (IC&CG), which divides the problem into a relaxed master problem (RMP) and a slave problem (SP). A new initialisation strategy for obtaining high-quality solutions and an acceleration strategy for discarding irrelevant scenarios are proposed. We use small-size instances to compare the performance of IC&CG with CPLEX and a generalised column and cut generation method in both deterministic and uncertain environments to validate the effectiveness of the method. For large-size instances, a lower bound (LB) of the optimal objective function is proposed to show the effectiveness of IC&CG. Furthermore, we prove the robustness of the method by sensitive analysis.

An accelerated benders decomposition algorithm for the solution of the multi-trip time-dependent vehicle routing problem with time windows

The logistics companies executing the last mile delivery of goods in urban areas deal every day with the problem of routing their vehicles, while taking into account multiple trips per vehicle, time-dependent travel time, customers’ time windows and loading time at the depot simultaneously. This paper addresses this problem, known as Multi-Trip Time-Dependent Vehicle Routing Problem with Time Windows, aiming at its exact solution, minimizing the total travelled distance of a company's fleet. Based on a literature model, a new reformulation with reduced size is suggested. This reformulation is decomposed by applying the Benders method in an effective way, resulting in a subproblem with no duality gap. By exploiting the special features of the problem and the particular structure of the decomposition made, several novel valid inequalities are introduced, in order to both tighten the non-decomposed formulations and warm start the relaxed master problem to achieve less infeasible solutions and higher lower bounds. For the solution of the problem, an innovative algorithm is proposed, including suboptimal master solutions and a multi-cut generation procedure, which is based on the careful observation of the values of the Benders dual subproblem variables. The impact of the valid inequalities as well as two variants of the suggested algorithm are tested on benchmark data and they are compared with the non-decomposed models and a heuristic introduced in the literature. The computational results indicate improved efficiency and stronger bounds for the proposed algorithm.

A stabilizing benders decomposition method for the accessibility-oriented charging station location problem

Electric vehicles are a promising selection to deal with climate change and energy crises. However, the development of electric vehicles is impeded by various barriers including shortened driving range, lack of charging facilities, and long recharging periods. Setting additional charging facilities is essential to construct an efficient charging network. Previous studies neglected accessibility-oriented indicators to locate charging stations. This study locates charging stations by maximizing the space-time-electricity accessibility of electric vehicle travelers. For the accessibility-oriented charging station location problem, we formulate a multi-commodity network flow model in the space-time-electricity network. Then, we reformulate this model into a two-stage integer program and develop a Benders decomposition method with stabilizing techniques. At each iteration, a relaxed Benders master problem (RMP) and a set of dual routing subproblems (DSPs) are solved. The RMP provides an ascending lower bound and a feasible charging station location scheme. These DSPs generate a set of Benders optimality cuts for RMP and a feasible solution to update the upper bound. All instances in three transportation networks are solved optimally, showing that the stabilizing techniques can significantly improve the performance of the standard Benders decomposition method.

Limiting the Failure Impact of Interdependent Power-Communication Networks via Optimal Partitioning

The mutual dependency between the power grid and communication network increases the risk of large-scale cascading failure, making the mitigation of outages a difficult task. In this paper, we use the concept of joint partitioning of the power and communication subsystems by identifying the most vulnerable typed-graphlets. This results in higher-order partitions, which prevent cascading failures from propagating globally in the system. For this purpose, we first formulate the problem of partitioning the interdependent system as a large-scale optimization problem with the objective of minimizing the overall load shedding subject to: i) power flow convergence, ii) power stability, iii) partition connectivity, iv) communication congestion/delay control, and v) a high partition quality. Then, using Benders’ method, we solve the problem by decomposing it into a relaxed master problem that consists of constrained spectral clustering, and a linear subproblem that consists of minimizing the load shedding subject to power flow convergence, stability, and congestion/delay control. Simulations are conducted on an IEEE 118-bus supported by a 118-node communication network. Our investigations reveal an average of 62% decrease in damage when the system is optimally partitioned.

Online Distributed Routing Problem of Electric Vehicles

Considering the penetration of numerous electric vehicles (EV) into the transportation sector, the EV routing problem that jointly optimizes the charging and routing process of EVs is becoming increasingly popular, which should be solved in online fashion for the practical requirements. First, we start with an offline EVs routing and charging (EVRP) problem which is a large-scale mixed integer nonlinear program (MINLP). Considering the NP hardness of MINLP, solving the offline EVRP directly is a time-consuming task that is not suitable for the online setting. Hence, a Benders decomposition based method is proposed to decompose the offline EVRP into a master problem and a set of sub-problems which allows for distributed implementation. To further speed up the computation, we relax the mixed integer master problem to one that can be equivalently solved as a linear program. Moreover, a novel kind of valid cut is added to the relaxed master problem to further reduce the number of iterations. In order to adapt the offline EVRP to the online setting, by introducing the virtual depot, we utilize the rolling-horizon framework to tackle the uncertainty of future information, where the offline EVRP is solved in real-time repeatedly. Finally, simulations using real road map in Belgium are performed. Besides, numerical results validate that the computation speed of proposed algorithm is faster than the state-of-the-art algorithms by several orders of magnitude, and showcase the capability of proposed algorithm to solve large size instances up to 350 nodes with 35 EVs within 100 seconds.

A generalized Benders decomposition approach for the mean-standard deviation shortest path problem

ABSTRACT This paper concentrates on the mean-standard deviation shortest path problem, which is an important extension of traditional shortest path problem. Due to the standard deviation term, the general formulation of this problem is nonlinear and concave. We transform this formulation into a mixed-integer conic quadratic program and develop a generalized Benders decomposition approach. The Benders master problem is a continuous conic quadratic program about travel time mean and standard deviation. The subproblem is a least expected travel time path problem with the variance limit. At each iteration, the subproblem generates a generalized Benders optimality cut for the relaxed Benders master problem. The relaxed Benders master problem provides an ascending lower bound and the subproblem produces a feasible solution to update the upper bound. In the numerical experiments, all instances in four transportation networks are solved optimally. This paper provides a novel solving scheme for the mean-standard deviation shortest path problem.

Integrated bin packing and lot-sizing problem considering the configuration-dependent bin packing process

This paper addresses a practical production problem in an aviation manufacturing factory that focuses on the production of composite aeronautic products, where packing and assembling are two major operations conducted in the production process. We extract a novel integer programming model that integrates bin packing with the multi-level lot-sizing problem, which is characterized by re-configurable bins and configuration-dependent processing time. To obtain tighter lower bounds, we decompose the original model into a master problem and several sub-problems based on the Dantzig–Wolfe framework. To accelerate the column generation process, we develop a bi-level dynamic programming algorithm to solve the sub-problems exactly. A hybrid algorithm that combines the column generation and the fix-and-optimize heuristic is proposed to generate high-quality upper bounds. We first evaluate the quality of the lower bounds obtained from the linearly relaxed restricted master problem and then demonstrate the competitiveness of the proposed heuristic algorithm in terms of the solution quality and computational efficiency by comparing it with CPLEX, a commercial solver. In addition, we conduct sensitivity analysis of the bi-level dynamic programming to investigate the impact of the processing time required by the configuration-based bins. Moreover, a case study was conducted, which demonstrated the effect on cost reduction by shortening the length of unit time period in the planning horizon.

Distributionally robust front distribution center inventory optimization with uncertain multi-item orders

<p style='text-indent:20px;'>As a new retail model, the front distribution center (FDC) has been recognized as an effective instrument for timely order delivery. However, the high customer demand uncertainty, multi-item order pattern, and limited inventory capacity pose a challenging task for FDC managers to determine the optimal inventory level. To this end, this paper proposes a two-stage distributionally robust (DR) FDC inventory model and an efficient row-and-column generation (RCG) algorithm. The proposed DR model uses a Wasserstein distance-based distributional set to describe the uncertain demand and utilizes a robust conditional value at risk decision criterion to mitigate the risk of distribution ambiguity. The proposed RCG is able to solve the complex max-min-max DR model exactly by repeatedly solving relaxed master problems and feasibility subproblems. We show that the optimal solution of the non-convex feasibility subproblem can be obtained by solving two linear programming problems. Numerical experiments based on real-world data highlight the superior out-of-sample performance of the proposed DR model in comparison with an existing benchmark approach and validate the computational efficiency of the proposed algorithm.</p>

Reliable closed-loop supply chain design problem under facility-type-dependent probabilistic disruptions

Closed-loop supply chains (CLSCs) have received considerable attention because of various economic and regulatory factors. A CLSC is characterized by more complicated network structures and higher uncertainties compared to traditional supply chain networks. Therefore, reliable CLSCs are being increasingly emphasized in academic circles due to the vast impacts of disruptions such as natural disasters and terrorist attacks. This paper studies a reliable location-inventory problem in a CLSC considering the mutual effects between failures of forward and reverse distribution centers (DCs) when they are co-located. The disruption probability of a co-located forward DC is different from that of a standalone forward DC, i.e., probabilistic disruptions are dependent on facility type. The problem is formulated as a nonconvex mixed-integer programming problem. A decomposition approach based on the outer approximation (DOA) algorithm is proposed to address the resulting model. The algorithm alternately solves relaxed master problems (mixed-integer linear programs, MILPs) and two nonlinear programming (NLPs) problems. Extensive numerical experiments are conducted to evaluate the performance of the proposed solution approach, after which managerial insights are explored.

Logic-based Benders decomposition for gantry crane scheduling with transferring position constraints in a rail–road container terminal

ABSTRACT Rail–road container terminals can be seen as buffers within a large logistics network encompassing hinterland distribution systems. They are used as temporary storage points for containers such that unloading operations from a train and loading operations onto a truck need not be synchronized. Container transferring operation implemented by gantry cranes is one of the most important issues to ensure efficient logistics. To enable efficient container transfer, this article considers the special features of container movement, and integrates the starting position and the ending position of container movement into the gantry crane scheduling problem. In order to solve the proposed problem, this article develops two models: first using mixed integer programming, and secondly using constraint programming (CP). Since the problem is computationally intractable for realistic sizes, this article also proposes a logic-based Benders decomposition (LBBD) methodology to solve it. This method exploits a natural decomposition of the studied problem into a relaxed master problem (RMP) solved by mixed integer programming (MIP), and a series of sub-problems, solved separately by MIP and CP. Computational experiments are performed on generated instances, the structures of which are designed to follow the practical application. The computational results show that the CP approach is better suited to solve the proposed problem with small-sized and medium-sized instances compared with MIP and LBBD. In addition, the LBBD framework is far superior to mixed integer programming on all but a few small instances, and it is highly effective in finding good-quality solutions for larger cases with up to 100 tasks within a reasonable runtime. Moreover, the results find that LBBD using constraint programming to solve the sub-problems usually outperforms LBBD with mixed integer programming on the instances tested.

A joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programs

This paper proposes a joint decomposition method that combines Lagrangian decomposition and generalized Benders decomposition, to efficiently solve multiscenario nonconvex mixed-integer nonlinear programming (MINLP) problems to global optimality, without the need for explicit branch and bound search. In this approach, we view the variables coupling the scenario dependent variables and those causing nonconvexity as complicating variables. We systematically solve the Lagrangian decomposition subproblems and the generalized Benders decomposition subproblems in a unified framework. The method requires the solution of a difficult relaxed master problem, but the problem is only solved when necessary. Enhancements to the method are made to reduce the number of the relaxed master problems to be solved and ease the solution of each relaxed master problem. We consider two scenario-based, two-stage stochastic nonconvex MINLP problems that arise from integrated design and operation of process networks in the case study, and we show that the proposed method can solve the two problems significantly faster than state-of-the-art global optimization solvers.

Reformulation, linearization, and decomposition techniques for balanced distributed operating room scheduling

We study the balanced distributed operating room (OR) scheduling (BDORS) problem as a location-allocation model, encompassing two levels of balancing decisions: (i) daily macro imbalance among collaborating hospitals in terms of the number of allocated ORs and (ii) daily micro imbalance among open ORs in each hospital in terms of the total caseload assigned. BDORS is formulated as a novel mixed-integer nonlinear programming (MINLP) in which the macro and micro imbalance are penalized using absolute value and quadratic functions. We develop various reformulation-linearization techniques (RLTs) for the MINLP models, leading to three mathematical modelling variants: (i) a mixed-integer quadratically constrained program (MIQCP) and (ii) two mixed-integer programs (MIPs) for the absolute value penalty function and an MIQCP for the quadratic penalty function. Two novel exact techniques based on reformulation-decomposition techniques (RDTs) are developed to solve these models: a uni- and a bi-level logic-based Benders decomposition (LBBD). We motivate the LBBD methods with an application to BDORS in the University Health Network (UHN), consisting of three collaborating hospitals: Toronto General Hospital, Toronto Western Hospital, and Princess Margaret Cancer Centre in Toronto, Ontario, Canada. The uni-level LBBD method decomposes the model into a surgical suite location, OR allocation, and macro balancing master problem (MP) and micro OR balancing sub-problems (SPs) for each hospital-day. The bi-level approach uses a relaxed MP, consisting of a surgical suite location and relaxed allocation/macro balancing MP and two optimization SPs. The primary SP is formulated as a bin-packing problem to allocate patients to open operating rooms to minimize the number of ORs, while the secondary SP is the uni-level micro balancing SP. Using UHN datasets consisting of two datasets, hard MP/easy SPs and easy MP/hard SPs, we show that both LBBD approaches and both MIP models solved via Gurobi converge to ≈ 2% and ≈ 1–2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas the MIQCP solved via Gurobi could not solve any instance of the UHN datasets given the same runtime. The uni- and bi-level LBBD approaches solved all instances of hard MP/easy SPs dataset to ≈ 11% and ≈ 2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas MIQCP solved via Gurobi could not solve any of these instances. Additionally, we show that convergence of each LBBD varies depending on where in the decomposition the actual computational complexity lies.

A branch and price algorithm for EOS constellation imaging and downloading integrated scheduling problem

Earth observation satellite (EOS) constellation imaging and downloading integrated scheduling (EOSCIDIS) is a challenging optimization problem with several engineering applications. Given a large number of imaging requests and a constellation consisting of a group of EOSs, a workable schedule must be developed for each EOS to allow for imaging and downloading in the most efficient manner given a certain scheduling horizon. No studies have proposed exact algorithms for the EOSCIDIS problem, although it has considerable potential for applications. In this paper, we developed a branch and price (B&P) algorithm for the EOSCIDIS problem. First, we decomposed the problem by extracting the sub-structure of the problem, and the decomposition resulted in a master problem and multiple pricing problems. Solving the linear relaxed master problem with column generation (CG) method provides a very tight upper bounds for the primal problem. We then embedded the CG process into a branch and bound (B&B) framework to find optimal integer solutions. In order to accelerate the convergence of the algorithm, we developed a heuristic utilizing the generated columns to construct feasible integer solutions to prune B&P tree branches earlier. We tested the B&P algorithm on highly-simulated instances that are constructed according to the Chinese Gaofen-1 satellite and 3 types of random instances which correspond to short, medium and long term scheduling respectively. Both results show that our method could find high-quality solutions with optimality gap less than 10% for most instances using reasonable computational costs .

Coordination of directional over‐current relays in active distribution networks using generalised benders decomposition

Coordination of directional over-current relays (DOCRs) as the primary protection of meshed distribution networks is formulated as a mixed-integer non-linear programming (MINLP) problem. The coordination of DOCRs has been mainly accomplished using heuristic methods. Therefore, this study aims at proposing an efficient analytical tool for this problem. A two-stage analytical approach based on the generalised Benders decomposition is proposed to find the time and current settings of DOCRs with minimum operating time, while satisfying coordination constraints. The standard curves and the discrete current settings of DOCRs are determined in the first stage using a relaxed MINLP master problem. The obtained settings are then passed to the second stage to determine the time dial settings. In this stage, two linear sub-problems are solved to check feasibility and optimality of the solution. In each sub-problem another linear optimisation problem is developed to construct new constraints that are then backed to the first stage to remove the infeasibility and reduce the optimality gap, respectively. The iterative process continues till all the infeasibilities are removed and the optimality gap decreases below a given threshold. The obtained results on test cases are compared with two well-known MINLP solvers and some previous approaches.

A new cross decomposition method for stochastic mixed-integer linear programming

Two-stage stochastic mixed-integer linear programming (MILP) problems can arise naturally from a variety of process design and operation problems. These problems, with a scenario based formulation, lead to large-scale MILPs that are well structured. When first-stage variables are mixed-integer and second-stage variables are continuous, these MILPs can be solved efficiently by classical decomposition methods, such as Dantzig/Wolfe decomposition (DWD), Lagrangian decomposition, and Benders decomposition (BD), or a cross decomposition strategy that combines some of the classical decomposition methods. This paper proposes a new cross decomposition method, where BD and DWD are combined in a unified framework to improve the solution of scenario based two-stage stochastic MILPs. This method alternates between DWD iterations and BD iterations, where DWD restricted master problems and BD primal problems yield a sequence of upper bounds, and BD relaxed master problems yield a sequence of lower bounds. The method terminates finitely to an optimal solution or an indication of the infeasibility of the original problem. Case study of two different supply chain systems, a bioproduct supply chain and an industrial chemical supply chain, show that the proposed cross decomposition method has significant computational advantage over BD and the monolith approach, when the number of scenarios is large.

Decomposition algorithms for optimizing multi-server appointment scheduling with chance constraints

This paper investigates a problem of scheduling appointments with random service durations on multiple servers with operating time limits. We minimize the cost of operating servers and serving appointments, subject to a joint chance constraint limiting the risk of server running overtime. With finite samples of random service time, we consider a mixed-integer linear programming extended formulation and propose a two-stage decomposition framework with cutting planes. The first stage considers a relaxed master problem as a variant of the chance-constrained binary packing problem discussed in Song et al. (INFORMS J Comput 26(4):735---747, 2014), which packs appointments into servers under chance-constrained server overtime. Given appointment-to-server assignments, the second stage seeks feasible schedules on individual servers. We propose strengthening, bounding, and branch-and-cut methods for computing problems in both stages. Via testing instances with diverse sizes, we compare different decomposition schemes. In particular, we demonstrate the efficacy of our branch-and-cut algorithm that incorporates server-based decomposition for optimizing the problem.

Relay Requirement and Traffic Assignment of Electric Vehicles

AbstractThis article defines, formulates, and solves a new equilibrium traffic assignment problem with side constraints—the traffic assignment problem with relays. The relay requirement arises from the driving situation that the onboard fuel capacity of vehicles is lower than what is needed for accomplishing their trips and the number and distribution of refueling infrastructures over the network are under the expected level. We proposed this problem as a modeling platform for evaluating congested regional transportation networks that serve plug‐in electric vehicles (in addition to internal combustion engine vehicles), where battery‐recharging or battery‐swapping stations are scarce. Specifically, we presented a novel nonlinear integer programming formulation, analyzed its mathematical properties and paradoxical phenomena, and suggested a generalized Benders decomposition framework for its solutions. In the algorithmic framework, a gradient projection algorithm and a labeling algorithm are adopted for, respectively, solving the primal problem and the relaxed master problem—the shortest path problem with relays. The modeling and solution methods are implemented for solving a set of example network problems. The numerical analysis results obtained from the implementation clearly show how the driving range limit and relay station location reshape equilibrium network flows.

On the tuning of predictive controllers: Application of generalized Benders decomposition to the ELOC problem

This work investigates the computational procedures used to obtain global solution to the economic linear optimal control (ELOC) problem. The proposed method employs the generalized Benders decomposition (GBD) algorithm. Compared to the previous branch and bound approach, a naive application of GBD to the ELOC problem will improve computational performance, due to less frequent calls to computationally slow semi-definite programming (SDP) routines. However, the reverse-convex constraints of the original problem will reappear in the relaxed master problem. In response, a convexification of the relaxed master constraints has been developed and proven to preserve global solution characteristics. The result is a multi-fold improvement in computational performance. A technological benefit of decomposing the problem into steady-state and dynamic parts is the ability to utilize nonlinear steady-state models, since the relaxed master problem is free of SDP type constraints and can be solved using any global nonlinear programming algorithm.

Truss topology optimization with discrete design variables by outer approximation

Several variants of an outer approximation method are proposed to solve truss topology optimization problems with discrete design variables to proven global optimality. The objective is to minimize the volume of the structure while satisfying constraints on the global stiffness of the structure under the applied loads. We extend the natural problem formulation by adding redundant force variables and force equilibrium constraints. This guarantees that the designs suggested by the relaxed master problems are capable of carrying the applied loads, a property which is generally not satisfied for classical outer approximation approaches applied to optimal design problems. A set of two- and three-dimensional benchmark problems are solved and the numerical results suggest that the proposed approaches are competitive with other special-purpose global optimization methods for the considered class of problems. Numerical comparisons indicate that the suggested outer approximation algorithms can outperform standard approaches suggested in the literature, especially on difficult problem instances.

Optimizing the supply chain configuration for make-to-order manufacturing

We consider a make-to-order (MTO) manufacturer who has won multiple contracts with specified quantities to be delivered by certain due dates. Before production starts, the company must configure its supply chain and make sourcing decisions. It also needs to plan the starting time for each production task under limited availability of resources such as machines and workforce. We develop a model for simultaneously optimizing such sourcing and planning decisions while exploiting their tradeoffs. The resulting multi-mode resource-constrained project scheduling problem (MMRCPSP) with a nonlinear objective function is NP-complete. To efficiently solve it, a hybrid Benders decomposition (HBD) algorithm combining the strengths of both mathematical programming and constraint programming is developed. The HBD exploits the structure of the model formulation and decomposes it into a relaxed master problem handled by mixed-integer nonlinear programming (MINLP), and a scheduling feasibility sub-problem handled by constraint programming (CP). Cuts are iteratively generated by solving the feasibility sub-problem and added back to the relaxed master problem, until an optimal solution is found or infeasibility is proved. Computational experiments are conducted to examine performance of the model and algorithm. Insights about optimal configuration of MTO supply chains are drawn and discussed.