Commercial Mixed-integer Linear Programming Solver Research Articles

A branch-price-and-cut algorithm for the local container drayage problem with controllable vehicle interference

This paper investigates a local container drayage problem with controllable vehicle interference (LCDP&CVI) under the tractor-and-trailer separation mode in which a tractor can be coupled or decoupled with a trailer at a customer location or a terminal. The container drayage requests proposed by customers should be fulfilled by a set of vehicle fleets, and each vehicle fleet comprises of a certain number of tractors and trailers that should be matched to each other. The controllable vehicle interference (CVI) means that the routes of any tractors from different vehicle fleets are independent of each other, implying that the interference between tractors is restricted only to the tractors from the same vehicle fleets. A mixed-integer linear programming (MILP) model is proposed for the LCDP&CVI. To exactly solve the problem, we develop a branch-price-and-cut algorithm based on the reformation of the MILP model, including a master problem and a pricing sub-problem. The pricing sub-problem that deals with the temporal synchronization constraints is formulated as the multi-vehicle shortest path problem with interference. By adding the tractor to the temporal-spatial network, a new representation of the pricing sub-problem is constructed, which enables us to solve the sub-problem by the multi-label method. In addition, the symmetrical elimination and dominance rules are incorporated into the multi-label method to enhance its effectiveness. Extensive numerical experiments are conducted to validate the proposed model and solution algorithm. For small-scale instances, the proposed solution algorithm is able to solve instances with up to 24 requests to optimality within two hours, which outperforms significantly the commercial MILP solver in terms of efficiency and effectiveness. For large-scale instances, the proposed algorithm can yield competitive solutions that are superior to the heuristic algorithms reported in the literature. Our experimental results also show that the controllable vehicle mechanism can strike a good balance between computational efficiency and operational costs by meticulously designing the structure of vehicle fleets.

Integrated optimization of train stopping plan and seat allocation scheme for railway systems under equilibrium travel choice and elastic demand

This paper examines the integrated optimization of train stopping plan and seat allocation scheme in railway systems, where equilibrium passenger travel choice under elastic demand is considered. The integrated optimization problem is formulated as a non-concave and non-linear mixed-integer mathematical model, where the objective is to maximize the system net benefit considering ticket revenue, consumer surplus, and cost associated with train stoppings. The integrated optimization model can be reformulated into a mixed-integer linear programming (MILP) model based on a series of linearization, relaxation, and outer-approximation techniques, which can then be solved by commercial MILP solvers (e.g., GUROBI). We also compare the integrated optimization approach with that when the train stopping plan and seat allocation are optimized separately and identify the potential benefits. Numerical studies have been conducted on a small-scale example, the Zhengzhou-Xi’an and Shanghai-Beijing high-speed railway corridors to illustrate the proposed model and solution approach.

Regional Constellation Reconfiguration Problem: Integer Linear Programming Formulation and Lagrangian Heuristic Method

A group of satellites, with either homogeneous or heterogeneous orbital characteristics and/or hardware specifications, can undertake a reconfiguration process due to variations in operations pertaining to Earth observation missions. This paper investigates the problem of optimizing a satellite constellation reconfiguration process against two competing mission objectives: 1) the maximization of the total coverage reward, and 2) the minimization of the total cost of the transfer. The decision variables for the reconfiguration process include the design of the new configuration and the assignment of satellites from one configuration to another. We present a novel biobjective integer linear programming formulation that combines constellation design and transfer problems. The formulation lends itself to the use of generic mixed-integer linear programming (MILP) methods such as the branch-and-bound algorithm for the computation of provably optimal solutions; however, these approaches become computationally prohibitive even for moderately sized instances. In response to this challenge, this paper proposes a Lagrangian relaxation-based heuristic method that leverages the assignment problem structure embedded in the problem. The results from the computational experiments attest to the near-optimality of the Lagrangian heuristic solutions and a significant improvement in the computational runtime as compared to a commercial MILP solver.

Reformulations to improve the Lagrangian relaxation approach for the capacitated multi-product dynamic lot sizing problem with batch ordering

In this work, we study the multi-product dynamic lot-sizing problem with capacity constraints and batch ordering. This problem arises in short to medium range production scheduling for several products over a finite number of periods to meet known demand. Each period has a capacity for placing orders, and every order for each product must have a fixed quantity, or batch size, though multiple orders can be placed for each product. We define three mixed-integer linear programming (MILP) models and apply Lagrangian relaxation to formulate the corresponding dual problems by relaxing the capacity constraints. The aim is to identify the dual problem that is the easiest to solve and provides the solution with the smallest duality gap. Subgradient optimisation is applied to solve the preferred Lagrangian dual model, which uses one of two heuristics to find good feasible solutions. We also show that the special case, where the batch sizes for all products are the same, can be modeled as a transportation problem. A set of numerical experiments is designed to compare the performance of the Lagrangian relaxation approach with a commercial MILP solver to identify the version of the subgradient algorithm and the MILP model that provide the best solutions.

Efficient models for the liner shipping hub location problem with spatial structure

This paper investigates the hub location problem arising in liner shipping, which is called liner shipping hub location problem. In contrast to most other hub location problems, many such problems in liner shipping have a special spatial structure where ships mainly follow certain main waterways, e.g., via the Suez Canal between Asia and Europe. In this paper, we utilize this spatial structure to propose three efficient mixed-integer linear programming (MILP) models for the liner shipping hub location problem, where we also explore the effect of the opening of the Arctic sea route on hub location. We further investigate the analytical properties of our proposed models. Numerical experiments are carried out to account for the effectiveness of our proposed models. As compared with the conventional hub location model, our proposed models can be solved by commercial MILP solvers, such as CPLEX, more efficiently.

Unsupervised Learning-Driven Matheuristic for Production-Distribution Problems

In this paper, we study a capacitated production-distribution problem where facility location, production, and distribution decisions are tightly coupled and simultaneously considered in the optimal decision making. Such an integrated production-distribution problem is complicated, and the current commercial mixed-integer linear programming (MILP) solvers cannot obtain favorable solutions for the medium- and large-sized problem instances. Therefore, we propose an unsupervised learning-driven matheuristic that uses easily obtainable solution values (e.g., solutions associated with the linear programming relaxation) to build clustering models and integrates the clustering information with a genetic algorithm to progressively improve feasible solutions. Then we verify the performance of the proposed matheuristic by comparing its computational results with those of the rolling horizon algorithm, a non-cluster-driven matheuristic, and a commercial MILP solver. The computational results show that, under the same computing resources, the proposed matheuristic can deliver better production-distribution decisions. Specifically, it reduces the total system costs by 15% for the tested instances when compared with the ones found by the commercial MILP solver. Additionally, we apply the proposed matheuristic to a related production-distribution problem in the literature and obtain 152 equivalent or new best-known solutions out of 200 benchmark test instances.

Two-stage design optimization based on artificial immune system and mixed-integer linear programming for energy supply networks

A two-stage optimization approach based on an artificial immune system (AIS) and mixed-integer linear programming (MILP) was developed to efficiently solve large-scale structural design problems of energy supply networks and obtain multiple and diverse design candidates. By focusing on a hierarchical relationship between design and operation variables, a structural design problem, formulated using MILP, is decomposed into an upper-level design problem and a lower-level operation problem. The upper-level design problem is solved using an AIS, in which multiple and diverse sets of suboptimal solutions are searched in a short computation time. In the lower-level optimization, design variables are fixed at the values searched in the upper-level optimization and operation variables are optimized using MILP. Moreover, the lower-level optimization for multiple sets of design variables is separately conducted using parallel computing. The developed approach was applied to the structural design of an energy supply network, consisting of candidates of cogeneration units and heat pump water heating units under power and heat interchange, for a housing complex with four dwellings. The diversity and energy-saving performance of multiple design candidates were analyzed. The computational efficiency was also demonstrated in comparison to the results obtained using only a commercial MILP solver.

An Efficient Mixed Integer Linear Programming Model for the Minimum Spanning Tree Problem

Finding a minimum spanning tree in a given network is a famous combinatorial optimization problem that appears in different engineering applications. Even though this problem is solvable in polynomial time, having efficient mathematical programming models is important as they can provide insights for formulating larger models that integrate other decisions in more complex applications. In the literature, there are ten different integer and mixed integer linear programming (MILP) models for this problem. They are variants of set packing, cuts, network flow and node level formulations. In addition, this paper introduces an efficient node level MILP model. Comparisons for the eleven models are provided. First, the models are compared in terms of the number of decision variables and the number of constraints. Then, computational comparisons using a commercial MILP solver on sets of randomly generated instances of different sizes are conducted. Results provide evidence that the proposed MILP model is competitive in terms of the computational time needed for proving optimality of generated solutions for instances with up to 50 nodes. Meanwhile, the LP relaxation of a multi-commodity flow MILP model which has integer polyhedron provides stable computational times despite its larger size.

An Improved Model for Parallel Machine Scheduling Under Time-of-Use Electricity Price

A recent study has led to an interesting mixed-integer linear programming (MILP) model for parallel machine scheduling under time-of-use (TOU) tariffs, which assumes great importance in achieving sustainable economic development. In this paper, we provide an improved MILP model by significantly reducing the number of decision variables. The computational results show that the performance of the improved model is superior to that of the existing one. Note to Practitioners —Time-of-use (TOU) pricing is widely implemented in many countries around the world, which provides variable electricity price over time. An appropriate response to the TOU policy can significantly reduce energy expense. This paper proposes an improved mixed-integer linear programming (MILP) approach for the parallel machine scheduling problem under TOU tariffs to minimize the total electricity cost based on a recently published model. The proposed approach can be easily implemented by using a commercial MILP solver such as IBM ILOG CPLEX. The optimal schedule could help decision makers reduce energy cost while maintaining high production efficiency.

Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units

Private investors, flexibility, efficiency and environmental requirements from deregulated markets have led the existence and building of a significant number of combined-cycle gas turbines (CCGTs) in many power systems. These plants represent a complex optimization problem for the short-term planning unit commitment (UC) carried out by independent system operators due to their multiple operating configurations. Accordingly, this paper proposes a mixed-integer linear programming (MIP) formulation of the configuration-based model of CCGTs, which is commonly utilized for bid/offering market processes. This formulation is simultaneously tighter and more compact than analogous MIP-based models; hence, it presents a lower computational burden. The computational efficiency of the proposed formulation is demonstrated by solving network-constrained UC case studies, of different size and complexity, using three of the leading commercial MIP solvers: CPLEX, GUROBI, and XPRESS.

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. This paper summarizes the potential drawbacks of nested-form risk measure issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler’s choice of objective function. This technique leads to models that are time consistent even while using time-inconsistent risk measures and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant, and the associated models thus simplify to a more familiar single-objective form. Unfortunately, we also show that our proposed approach leads to 𝒩𝒫-hard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard average value-at-risk and first-order mean-semideviation risk measures. Finally, we show that for a simple but reasonably realistic test application, the kind of models we propose, although drawn from an 𝒩𝒫-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial mixed-integer linear programming solver. Therefore, there seems some promise of our proposed modeling approach being useful despite its computational complexity properties.

A Mixed-Integer Linear Programming Model for the Electric Vehicle Charging Coordination Problem in Unbalanced Electrical Distribution Systems

This paper presents a novel mixed-integer linear programming (MILP) model for the electric vehicle charging coordination (EVCC) problem in unbalanced electrical distribution systems (EDSs). Linearization techniques are applied over a mixed-integer nonlinear programming model to obtain the proposed MILP formulation based on current injections. The expressions used to represent the steady-state operation of the EDS take into account a three-phase representation of the circuits, as well as the imbalance of the loads, leading to a more realistic model. Additionally, the proposed formulation considers the presence of distributed generators and operational constraints such as voltage and current magnitude limits. The optimal solution for the mathematical model was found using commercial MILP solvers. The proposed formulation was tested in a distribution system used in the specialized literature. The results show the efficiency and the robustness of the methodology, and also demonstrate that the model can be used in the solution of the EVCC problem in EDSs.

Models and Lagrangian heuristics for a two-level lot-sizing problem with bounded inventory

We consider a two-level dynamic lot-sizing problem where the first level consists of N finished products competing for a single type of purchased raw material in the second level. While the procurement and production capacities are unlimited, the storage capacity of the raw material is limited and must be carefully managed. The goal is to simultaneously determine a replenishment plan for the raw material and optimal production plans for the finished products on a horizon of T periods while minimizing production, purchasing, setup and inventory holding costs. The problem is modeled using mixed-integer linear programs and solved using both a Lagrangian relaxation-based heuristic and a commercial mixed-integer linear programming solver. Learning capabilities are integrated in the Lagrangian relaxation to update step size in the subgradient algorithm. The computational results show that the Lagrangian heuristic outperforms the solver on different formulations, in particular for large problems with long time horizons.

A MILP formulation for generalized geometric programming using piecewise-linear approximations

Generalized geometric programming (GGP) problems are converted to mixed-integer linear programming (MILP) problems using piecewise-linear approximations. Our approach is to approximate a multiple-term log-sum function of the form log (x1 + x2 + ⋅⋅⋅ + xn) in terms of a set of linear equalities or inequalities of log x1, log x2, …, and log xn, where x1, …, xn are strictly positive. The advantage of this approach is its simplicity and readiness to implement and solve using commercial MILP solvers. While MILP problems in general are no easier than GGP problems, this approach is justified by the phenomenal progress of computing power of both personal computers and commercial MILP solvers. The limitation of this approach is discussed along with numerical tests.

Finished-vehicle transporter routing problem solved by loading pattern discovery

This work addresses a new transportation problem in outbound logistics in the automobile industry: the finished-vehicle transporter routing problem (FVTRP). The FVTRP is a practical routing problem with loading constraints, and it assumes that dealers have deterministic demands for finished vehicles that have three-dimensional irregular shapes. The problem solution will identify optimal routes while satisfying demands. In terms of complex packing, finished vehicles are not directly loaded into the spaces of transporters; instead, loading patterns matching finished vehicles with transporters are identified first by mining successful loading records through virtual and manual loading test procedures, such that the packing problem is practically solved with the help of a procedure to discover loading patterns. This work proposes a mixed-integer linear programming (MILP) model for the FVTRP considering loading patterns. As a special class of routing models, the FVTRP is typically difficult to solve within a manageable computing time. Thus, an evolutionary algorithm is designed to solve the FVTRP. Comparisons of the proposed algorithm and a commercial MILP solver demonstrate that the proposed algorithm is more effective in solving medium- and large-scale problems. The proposed scheme for addressing the FVTRP is illustrated with an example and tested with benchmark instances that are derived from well-studied vehicle routing datasets.

Optimization of energy supply systems by MILP branch and bound method in consideration of hierarchical relationship between design and operation

To attain the highest performance of energy supply systems, it is necessary to rationally determine types, capacities, and numbers of equipment in consideration of their operational strategies corresponding to seasonal and hourly variations in energy demands. In the combinatorial optimization method based on the mixed-integer linear programming (MILP), integer variables are used to express the selection, numbers, and on/off status of operation of equipment, and the number of these variables increases with those of equipment and periods for variations in energy demands, and affects the computation efficiency significantly. In this paper, a MILP method utilizing the hierarchical relationship between design and operation variables is proposed to solve the optimal design problem of energy supply systems efficiently: At the upper level, the optimal values of design variables are searched by the branch and bound method; At the lower level, the values of operation variables are optimized independently at each period by the branch and bound method under the values of design variables given tentatively during the search at the upper level; Lower bounds for the optimal value of the objective function to be minimized are evaluated, and are utilized for the bounding operations at both the levels. This method is implemented into open and commercial MILP solvers. Illustrative and practical case studies on the optimal design of cogeneration systems are conducted, and the validity and effectiveness of the proposed method are clarified.

ICOPE-15-1077 Generation of multiple best solutions in multiobjective optimal design of energy supply systems

Optimization approaches based on the mixed-integer linear programming (MILP) have been utilized to design energy supply systems. In this paper, an MILP method utilizing the hierarchical relationship between design and operation is extended to search not only the optimal solution but also suboptimal ones which follow the optimal one without any omissions, what are called K-best solutions, efficiently in a multiobjective optimal design problem. At the upper level, the values of design variables for the K-best solutions are searched by the branch and bound method. At the lower level, the values of operation variables are optimized independently at each period by the branch and bound method under the values of design variables given tentatively. Incumbents for the K-best solutions and an upper bound for all the values of the objective function for the K-best solutions are renewed if necessary between both the levels. This method is implemented into a commercial MILP solver. A practical case study on the multiobjective optimal design of a cogeneration system is conducted, and the validity and effectiveness of the method are clarified.

Integer programming for the generalized high school timetabling problem

Recently, the XHSTT format for high school timetabling was introduced. It provides a uniform way of modeling problem instances and corresponding solutions. The format supports a wide variety of constraints, and currently 38 real-life instances from 11 different countries are available. Thereby, the XHSTT format serves as a common ground for researchers within this area. This paper describes the first exact method capable of handling an arbitrary instance of the XHSTT format. The method is based on a mixed-integer linear programming (MIP) model, which is solved in two steps with a commercial general-purpose MIP solver. Computational results show that our approach is able to find previously unknown optimal solutions for 2 instances of XHSTT and proves optimality of 4 known solutions. For the instances not solved to optimality, new non-trivial lower bounds were found in 11 cases, and new best known solutions were found in 9 cases. Furthermore, the approach is compared with the finalists of Round 2 of the International Timetabling Competition 2011 and is shown to be competitive with one of the finalists.

European Electricity Market Integration With Mixed Market Designs—Part I: Formulation

The integration of the day-ahead electricity markets in Europe shall lead to one multi-area market that will substitute the local/national ones. In view of the “target model” that will be enforced in all European markets in conjunction with their forthcoming coupling/integration, a centralized market splitting algorithm is implemented in this paper, respecting the standard market regulatory framework of power exchanges (PXs) and power pools, including 1) block offers/bids, linked block offers/bids, flexible hourly offers/bids and convertible block offers in power exchanges and 2) unit technical/commitment constraints and system operating constraints in power pools. The problem is formulated as a mixed integer linear programming (MILP) model, which can be solved using commercial MILP solvers. The performance and computational requirements of an implementation in pan-European level is presented in the companion paper; the model is tested in terms of problem solvability to its limits, considering a day-ahead market clearing time threshold equal to one hour.

Impact of price‐based demand response on market clearing and locational marginal prices

This study quantitatively analyses the impact of price-based demand response (DR) on market clearing and locational marginal prices (LMPs) of power systems using the network-constrained unit commitment (NCUC) model. The price-based DR is formulated via linear price-sensitive demand bidding curves, while considering DR participation levels and energy payback rates. DR is formulated as load shifting as well as load curtailment by adopting different energy payback rates to represent the amount of energy shifted among different hours. The objective is to maximise the system social welfare, which includes the revenue from price-sensitive DR loads minus the energy production costs from generating units. The NCUC problem is formulated as a mixed-integer linear programming (MILP) problem and solved by the commercial MILP solver. Numerical case studies are performed to analyse the impacts of DR on unit commitment, LMPs, energy production costs and DR load payments. Results indicate that with the current market clearing mechanism which maximises the system social welfare, DR dispatch may augment LMPs by displacing less flexible generating units and/or triggering additional transmission congestions and, in turn, increase DR loads' payments, which will ultimately influence the financial stability of DR providers and the sustainable deployment of DR programs.