Logic-based Benders Decomposition Research Articles

Automatic Logic-Based Benders Decomposition with MiniZinc

Logic-based Benders decomposition (LBBD) is a powerful hybrid optimisation technique that can combine the strong dual bounds of mixed integer programming (MIP) with the combinatorial search strengths of constraint programming (CP). A major drawback of LBBD is that it is a far more involved process to implement an LBBD solution to a problem than the "model-and-run" approach provided by both CP and MIP. We propose an automated approach that accepts an arbitrary MiniZinc model and solves it using LBBD with no additional intervention on the part of the modeller. The design of this approach also reveals an interesting duality between LBBD and large neighborhood search (LNS). We compare our implementation of this approach to CP and MIP solvers on 4 different problem classes where LBBD has been applied before.

Closed-loop coordination of inland vessels operations in large seaports using hybrid logic-based benders decomposition

This paper addresses two coordination problems that exist in the waterborne transport in large seaports, the long time of stay of inland vessels and insufficient terminal and quay crane planning with respect to their sailing schedules. A novel coordination model is proposed and tackled using logic-based Benders decomposition and Large Neighborhood Search. In addition, a closed-loop perspective is taken, in which possible disturbances that may occur are considered. Simulation results show that our approach can scale to real-world sizes and provide better schedules for inland vessels within the port. The potential of using inland vessels for inter-terminal transport is also extensively investigated.

Logic-based Benders decomposition for planning and scheduling: a computational analysis

AbstractLogic-based Benders decomposition (LBBD) has improved the state of the art for solving a variety of planning and scheduling problems, in part by combining the complementary strengths of constraint programming and mixed integer programming (MIP). We undertake a computational analysis of specific factors that contribute to the success of LBBD, to provide guidance for future implementations. We study a problem class that assign tasks to multiple resources and poses a cumulative scheduling problem on each resource. We find that LBBD is at least 1000 times faster than state-of-the-art MIP on larger instances, despite recent advances in the latter. Further, we conclude that LBBD is most effective when the planning and scheduling aspects of the problem are roughly balanced in difficulty. The most effective device for improving LBBD is the inclusion of a subproblem relaxation in the master problem. The strengthening of Benders cuts also plays an important role when the master and subproblem complexity are properly balanced. These findings suggest future research directions.

Propagating logic-based Benders’ decomposition approaches for distributed operating room scheduling

We develop three novel logic-based Benders’ decomposition (LBBD) approaches and a cut propagation mechanism to solve large-scale location-allocation integer programs (IPs). We show that each LBBD can be implemented in four different possible ways, yielding distinct LBBD variants with completely different computational performances. LBBDs decompose the IP model into an integer location and knapsack-based allocation master problem and multiple packing IP sub-problems. We illustrate the performance of our LBBDs on the distributed operating room (OR) scheduling (DORS) problem, where patients and ORs are collaboratively scheduled across a network of hospitals, and the goal is to select patients with the highest priority scores and schedule them in the current planning horizon, while determining the number of surgical suites and operating rooms required to accommodate the schedule at minimum cost. We quantitatively demonstrate that the new Benders’ cuts, cut propagation, and implementation can individually or collectively yield a computational time impact of at least two orders of magnitude. We also demonstrate that our LBBDs are 10–100x faster than IP+Gurobi and are more successful at finding optimal solutions. Finally, we show that DORS increases the average OR utilization across the network to more than 95%.

A Logic-Based Benders Decomposition Approach for Mapping Applications on Heterogeneous Multicore Platforms

The development of efficient methods for mapping applications on heterogeneous multicore platforms is a key issue in the field of embedded systems. In this article, a novel approach based on the Logic-Based Benders decomposition principle is introduced for mapping complex applications on these platforms, aiming at optimizing their execution time. To provide optimal solutions for this problem in a short time, a new hybrid model that combines Integer Linear Programming (ILP) and Constraint Programming (CP) models is introduced. Also, to reduce the complexity of the model and its solution time, a set of novel techniques for generating additional constraints called Benders cuts is proposed. An extensive set of experiments has been performed in which synthetic applications described by Directed Acyclic Graphs (DAGs) were mapped to a number of heterogeneous multicore platforms. Moreover, experiments with DAGs that correspond to two real-life applications have also been performed. Based on the experimental results, it is proven that the proposed approach outperforms the pure ILP model in terms of the solution time and quality of the solution. Specifically, the proposed approach is able to find an optimal solution within a time limit of 2 hours in the vast majority of performed experiments, while the pure ILP model fails. Also, for the cases where both methods fail to find an optimal solution within the time limit, the solution of the proposed approach is systematically better than the solution of the ILP model.

Decomposition Methods for the Parallel Machine Scheduling Problem with Setups

We study the unrelated parallel machine scheduling problem with sequence and machine-dependent setup times and the objective of makespan minimization. Two exact decomposition-based methods are proposed based on logic-based Benders decomposition and branch and check. These approaches are hybrid models that make use of a mixed-integer programming (MIP) master problem and a specialized solver for travelling salesman subproblems. The master problem is used to assign jobs to machines, whereas the subproblems find optimal schedules on each machine given the master problem assignments. Computational results show that the decomposition models are able to find optimal solutions up to four orders of magnitude faster than the existing state of the art as well as solve problems six times larger than an existing MIP model. We further investigate the solution quality versus runtime trade-off for large problem instances for which the optimal solutions cannot be found and proved in a reasonable time. We demonstrate that the branch-and-check hybrid algorithm is able to produce better schedules in less time than the state-of-the-art metaheuristic, while also providing an optimality gap.

Analysis of a parallel machine scheduling problem with sequence dependent setup times and job availability intervals

In this study, we propose constraint programming (CP) model and logic-based Benders algorithms in order to make the best decisions for scheduling non-identical jobs with availability intervals and sequence dependent setup times on unrelated parallel machines in a fixed planning horizon. In this problem, each job has a profit, cost and must be assigned to at most one machine in such a way that total profit is maximized. In addition, the total cost has to be less than or equal to a budget level. Computational tests are performed on a real-life case study prepared in collaboration with the U.S. Army Corps of Engineers (USACE). Our initial investigations show that the pure CP model is very efficient in obtaining good quality feasible solutions but, fails to report the optimal solution for the majority of the problem instances. On the other hand, the two logic-based Benders decomposition algorithms are able to obtain near optimal solutions for 86 instances out of 90 examinees. For the remaining instances, they provide a feasible solution. Further investigations show the high quality of the solutions obtained by the pure CP model.

Projection, consistency, and George Boole

Although best known for his work in symbolic logic, George Boole made seminal contributions in the logic of probabilities. He solved the probabilistic inference problem with a projection method, leading to the insight that inference (as well as optimization) is essentially a projection problem. This unifying perspective has applications in constraint programming, because consistency maintenance is likewise a form of inference that can be conceived as projection. Viewing consistency in this light suggests a concept of J-consistency, which is achieved by projection onto a subset J of variables. We show how this projection problem can be solved for the satisfiability problem by logic-based Benders decomposition. We also solve it for among, sequence, regular, and all-different constraints. Maintaining J-consistency for global constraints can be more effective than maintaining traditional domain and bounds consistency when propagating through a richer structure than a domain store, such as a relaxed decision diagram. This paper is written in recognition of Boole's 200th birthday.

Sequencing Operator Counts

Operator-counting is a recently developed framework for analysing and integrating many state-of-the-art heuristics for planning using Linear Programming. In cost-optimal planning only the objective value of these heuristics is traditionally used to guide the search. However the primal solution, i.e. the operator counts, contains useful information. We exploit this information using a SAT-based approach which given an operator-count, either finds a valid plan; or generates a generalized landmark constraint violated by that count. We show that these generalized landmarks can be used to encode the perfect heuristic, h*, as a Mixed Integer Program. Our most interesting experimental result is that finding or refuting a sequence for an operator-count is most often empirically efficient, enabling a novel and promising approach to planning based on Logic-Based Benders Decomposition (LBBD).

Logic-based Benders decomposition for an inventory-location problem with service constraints

We study an integrated inventory-location problem with service requirements faced by an aerospace company in designing its service parts logistics network. Customer demand is Poisson distributed and the service levels are time-based leading to highly non-linear, stochastic service constraints and a nonlinear, mixed-integer optimization problem. Unlike previous work in the literature, which propose approximations for the nonlinear constraints, we present an exact solution methodology using logic-based Benders decomposition. We decompose the problem to separate the location decisions in the master problem from the inventory decisions in the subproblem. We propose a new family of valid cuts and prove that the algorithm is guaranteed to converge to optimality. This is the first attempt to solve this type of problem exactly. Then, we present a new restrict-and-decompose scheme to further decompose the Benders master problem by part. We test on industry instances as well as random instances. Using the exact algorithm and restrict-and-decompose scheme we are able to solve industry instances with up to 60 parts within reasonable time, while the maximum number of parts attempted in the literature is 5.

Boosting an Exact Logic-Based Benders Decomposition Approach by Variable Neighborhood Search

Logic-based Benders decomposition (BD) extends classic BD by allowing more complex subproblems with integral variables. Metaheuristics like variable neighborhood search are becoming useful here for faster solving the subproblems' inference duals in order to separate approximate Benders cuts. After performing such a purely heuristic BD approach, we continue by exactly verifying and possibly correcting each heuristic cut to finally obtain a proven optimal solution. On a bi-level vehicle routing problem, this new hybrid approach exhibits shorter overall runtimes and yields excellent intermediate solutions much earlier than the classical exact method.

A two-stage coupled algorithm for an integrated maintenance planning and flowshop scheduling problem with deteriorating machines

We address a novel integrated maintenance and production scheduling problem in a multi-machine and multi-period production system, considering maintenance as a long-term decision. Deterioration of machines over time decreases production capacity. Since maintenance activities not only improve machine conditions, increasing production capacity, but also take time that cannot be used for production, the challenge is to assign maintenance to periods and to schedule maintenance and production activities within each period to minimize the combined cost of maintenance and lost production over the planning horizon. Motivated by logic-based Benders decomposition, we design an integrated two-stage algorithm to solve the problem. The first stage assigns maintenance to machines and time periods, abstracting the scheduling problem, while the second stage creates a schedule for the current time period. The first stage is then re-solved using feedback from the schedule. This iteration between maintenance planning and scheduling continues until the solution costs in two stages converge. The integrated approach models the interdependencies between maintenance and scheduling decisions in highly coupled processes such as wafer fabrication in the semiconductor manufacturing. Our results demonstrate that the benefit of integrated decision making increases when maintenance is less expensive relative to lost production cost and that a longer horizon for maintenance planning is beneficial when maintenance cost increases.

시맨틱 웹과 SWCL하의 제품설계 최적 공통속성 선택을 위한 의사결정 지원 시스템

논문투고일:2014년 04월 14일 논문수정완료일:2014년 05월 26일 논문게재확정일:2014년 05월 29일 * 연세대학교 경영대학, 교신저자 ** 연세대학교 경영대학 It is unavoidable to provide products that meet customers' needs and wants so that firms may survive under the competition in this globalized market. This paper focuses on how to provide levels for attributes that compse product so that firms may give the best products to customers. In particular, its main issue is how to determine common attributes and the others with their appropriate levels to maximize firms' profits, and how to construct a decision support system to ease decision makers' decisons about optimal common attribute selection using the Semantic Web and SWCL technologies. Parameter data in problems and the relationships in the data are expressed in an ontology data model and a set of constraints by using the Semantic Web and SWCL technologies. They generate a quantitative decision making model through the automatic process in the proposed system, which is fed into the solver using the Logic-based Benders Decomposition method to obtain an optimal solution. The system finally provides the generated solution to the decision makers. This presentation suggests the opportunity of the integration of the proposed system with the broader structured data network and other decision making tools because of the easy data shareness, the standardized data structure and the ease of machine processing in the Semantic Web technology. Keyword:Product Design, the Semantic Web, SWCL, Decision Support System 韓國IT서비스學會誌 第13卷 第2號 2014年 6月, pp.133-149

Upper and lower bounds for the permutation flowshop scheduling problem with minimal time lags

In this paper, we consider the problem of scheduling \(n\) jobs in an \(m\)-machine permutation flowshop with minimal time lags between consecutive operations of each job. The processing order of jobs is to be the same for each machine. The time lag is defined as the waiting time between two consecutive operations of each job. Upper bounds for the problem are provided by applying heuristic procedures based on known and new rules. Lower bounds based on Moore’s algorithm and logic-based Benders decomposition are developed. For the last one, we define a long time horizon on the last machine divided into many segments of time. We combine a mixed integer linear programming to allocate jobs to time segments and scheduled using the constraint programming. Then, computational results are reported.

Scheduling a Dynamic Aircraft Repair Shop with Limited Repair Resources

We address a dynamic repair shop scheduling problem in the context of military aircraft fleet management where the goal is to maintain a full complement of aircraft over the long-term. A number of flights, each with a requirement for a specific number and type of aircraft, are already scheduled over a long horizon. We need to assign aircraft to flights and schedule repair activities while considering the flights requirements, repair capacity, and aircraft failures. The number of aircraft awaiting repair dynamically changes over time due to failures and it is therefore necessary to rebuild the repair schedule online. To solve the problem, we view the dynamic repair shop as successive static repair scheduling sub-problems over shorter time periods. We propose a complete approach based on the logic-based Benders decomposition to solve the static sub-problems, and design different rescheduling policies to schedule the dynamic repair shop. Computational experiments demonstrate that the Benders model is able to find and prove optimal solutions on average four times faster than a mixed integer programming model. The rescheduling approach having both aspects of scheduling over a longer horizon and quickly adjusting the schedule increases aircraft available in the long term by 10% compared to the approaches having either one of the aspects alone.

Using Logic-Based Benders Decomposition to Solve the Capacity- and Distance-Constrained Plant Location Problem

We address an optimization problem that requires deciding the location of a set of facilities, the allocation of customers to those facilities under capacity constraints, and the allocation of customers to trucks at those facilities under truck travel-distance constraints. We present a hybrid approach that combines integer and constraint programming using logic-based Benders decomposition. Computational experiments demonstrate that the Benders model is able to find and prove optimal solutions up to three orders-of-magnitude faster than an existing integer programming approach; it also finds better feasible solutions in less time when compared with an existing tabu search algorithm.

The stop-and-drop problem in nonprofit food distribution networks

In this paper, we introduce the stop-and-drop problem (SDRP), a new variant of location-routing problems, that is mostly applicable to nonprofit food distribution networks. In these distribution problems, there is a central warehouse that contains food items to be delivered to agencies serving the people in need. The food is delivered by trucks to multiple sites in the service area and partner agencies travel to these sites to pick up their food. The tactical decision problem in this setting involves how to jointly select a set of delivery sites, assign agencies to these sites, and schedule routes for the delivery vehicles. The problem is modeled as an integrated mixed-integer program for which we delineate a two-phase sequential solution approach. We also propose two Benders decomposition-based solution procedures, namely a linear programming relaxation based Benders implementation and a logic-based Benders decomposition heuristic. We show through a set of realistic problem instances that given a fixed time limit, these decomposition based methods perform better than both the standard branch-and-bound solution and the two-phase approach. The general problem and the realistic instances used in the computational study are motivated by interactions with food banks in southeastern United States.

Single-facility scheduling by logic-based Benders decomposition

Logic-based Benders decomposition can combine mixed integer programming and constraint programming to solve planning and scheduling problems much faster than either method alone. We find that a similar technique can be beneficial for solving pure scheduling problems as the problem size scales up. We solve single-facility non-preemptive scheduling problems with time windows and long time horizons. The Benders master problem assigns jobs to predefined segments of the time horizon, where the subproblem schedules them. In one version of the problem, jobs may not overlap the segment boundaries (which represent shutdown times, such as weekends), and in another version, there is no such restriction. The objective is to find feasible solutions, minimize makespan, or minimize total tardiness.

Scheduling an Aircraft Repair Shop

We address a scheduling problem in the context of military aircraft maintenance where the goal is to meet the aircraft requirements for a number of missions in the presence of breakdowns. The assignment of aircraft to a mission must consider the requirements for the mission, the probability of aircraft failure, and capacity of the repair shop that maintains the aircraft. Therefore, a solution both assigns aircraft to missions and schedules the repair shop to meet the assignments. We propose a dispatching heuristic algorithm; three complete approaches based on mixed integer programming, constraint programming, and logic-based Benders decomposition; and a hybrid heuristic-complete approach. Experiments demonstrate that the logic-based Benders variation combining mixed integer programming and constraint programming outperforms the other approaches, that the dispatching heuristic can feasibly schedule the repair shop in a very short time, and that using the dispatching solution as a bound marginally improves the complete approaches.

Task scheduling algorithm based on model decomposition for multi-machine with time windows

The algorithm based on model decomposition was proposed for the problem of multi-machine task scheduling with time windows. The classical Benders decomposition was extended into the field of mixed integer linear programming model by introducing the logic based Benders decomposition. The states of art software MOSEK and GECODE were deployed to solve the master and sub-problems respectively. The method of generating Benders cuts was presented. The solution space was converged to a satisfied feasible solution through running the algorithm iteratively. The algorithm was implemented and tested by testing cases, and its effectiveness was verified.