Standard Commercial Solver Research Articles

The Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station: Compact formulation and a General Variable Neighborhood Search metaheuristic

In this article, we introduce the Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station (VRPCDSD), which integrates pickup and delivery routing decisions with the cross-docking scheduling of the unloading and reloading of requests. We present a compact Mixed Integer Programming (MIP) formulation and a General Variable Neighborhood Search (GVNS) metaheuristic for the problem. The proposed GVNS employs several routing and scheduling neighborhoods that require problem-specific adjustments due to the integrated characteristic of the problem. Computational experiments indicate that the MIP formulation can solve instances with up to 31 nodes (15 requests) within the specified time limit when using a standard commercial solver. Furthermore, the GVNS metaheuristic can obtain high-quality solutions that at least match those encountered by the solver for the tested instances, achieving considerably large improvements over the latter especially as the number of requests increases. In fact, GVNS is able to find the optimal solution for all the instances for which the optimal is known. In addition, GVNS is a viable alternative for tackling larger instances of VRPCDSD and is very robust in that it obtains low deviations from the best-encountered solutions when multiple executions are performed. We also present some insights on how the different cross-docking operation costs impact the optimal solutions.

Mixed-Integer Second-Order Cone Programming Reformulations of a Fractional 0-1 Program for Task Assignment

Fractional 0-1 programming is a subfield of nonlinear integer optimization in which the objective is to optimize the sum of ratios of affine functions subject to a set of linear constraints. It is well-known that fractional 0-1 programs can be formulated as mixed-integer linear programs. Recently, several alternative mixed-integer second-order cone programming reformulations have been proposed for fractional 0-1 programs. These reformulations, which can be solved directly by standard commercial solvers, have been reported to be efficient for certain types of problems. In this paper, we consider a task assignment problem with respect to preferences, where the objective is to maximize total weighted satisfaction while maintaining a fair distribution. The problem’s mathematical model turns out to be a fractional 0-1 program. We investigate three mixed-integer second-order cone programming reformulations thereof, and we compare, by means of a computational study, the performance of these reformulations with a benchmark mixed-integer linear programming formulation that was proposed and analyzed in the literature before. The latter, namely the mixed-integer linear programming formulation, turns out to be significantly better for the problem in question.

Efficient learning of decision-making models: A penalty block coordinate descent algorithm for data-driven inverse optimization

Decision-making problems are commonly formulated as optimization problems, which are then solved to make optimal decisions. In this work, we consider the inverse problem where we use prior decision data to uncover the underlying decision-making process in the form of a mathematical optimization model. This statistical learning problem is referred to as data-driven inverse optimization. We focus on problems where the underlying decision-making process is modeled as a convex optimization problem whose parameters are unknown. We formulate the inverse optimization problem as a bilevel program and propose an efficient block coordinate descent-based algorithm to solve large problem instances. Numerical experiments on synthetic datasets demonstrate the computational advantage of our method compared to standard commercial solvers. Moreover, the real-world utility of the proposed approach is highlighted through two realistic case studies in which we consider estimating risk preferences and learning local constraint parameters of agents in a multiplayer Nash bargaining game.

A Mixed-Integer Quadratically Constrained Programming Model for Network Reconfiguration in Power Distribution Systems with Distributed Generation and Shunt Capacitors

This paper proposes a formulation based on mixed-integer quadratically constrained programming (MIQCP) for the problem of optimally determining network topology aiming at minimum power loss in electrical distribution grids considering distributed generation and shunt capacitors. The proposed optimization model is derived from the originally nonlinear optimization model by leveraging the modified distribution power flow method that is linear. This optimization model can be effectively solved by standard commercial solvers such as CPLEX. Then, the MIQCP-based formulation is verified on an IEEE 33-bus distribution network and a 190-bus real distribution network in Luc Ngan district, Bac Giang province, Vietnam, in 2021. The effects of the load demand level on optimal solutions are analyzed in detail. Furthermore, results of power flow analysis achieved from the modified distribution power flow approach are compared to those from solving nonlinear equations of power flow using the power summation method that gives exact solutions.

Using a Bi-Level Optimization Model for Assessing the Impact of Demand Forecast Uncertainty on the Maximum Profit of Power Distribution Companies Owning Energy Storage Systems

This paper proposes a bi-level optimization model to maximize the net revenue of a power distribution company owning energy storage systems. Furthermore, the proposed model also takes into account the uncertainty of load forecast based on a set of multiple scenarios in order to assess its impact on the maximum net revenue of the power distribution company. This bi-level optimization model is transformed into the single-level mixed-integer linear programming model by using the Karush-Kuhn-Tucker optimality conditions and strong duality theorem. This single-level optimization formulation can be effectively solved by using standard commercial solvers such as CPLEX. The proposed model and the effects of the load forecast uncertainty on the net revenue of the power distribution company are validated and analyzed on an IEEE 24-bus meshed transmission grid.

Maximum weighted induced forests and trees: new formulations and a computational comparative review

AbstractGiven a graph with a weight associated with each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset of the vertices such that induces a forest. This NP‐hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem (MWIT), on the other hand, requires that the subset induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch‐and‐cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial mixed integer programming solver. More specifically, five formulations are compared, two compact ones (i.e., with a polynomial number of variables and constraints) and the three others with an exponential number of constraints. The experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the benchmark instances with up to 81 vertices. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to analyze the difference between the optimal solution values of the two problems when considering different classes of graphs.

Optimizing Cooperative Pallet Loading Robots: A Mixed Integer Approach

We consider a system with a metering conveyor that feeds a palletizing machine. Multiple robots cooperatively move packages to obtain a desired final pallet layout. We address the problem of finding the optimal sequence of manipulations and packages positions on the conveyor that allow to maximize production. Our main contribution is the reformulation of this operation as a mixed integer linear programming (MILP) problem, that we solve with a metaheuristic that exploits a standard commercial solver. We also present some numerical experiments on randomly generated problems.

Time-preference-based on-spot bundled cloud-service provisioning

The cloud computing spot instance is one offering that vendors are leveraging to provide differentiated service to an expanding pay-per-use computing market. Spot instances have cost advantages, albeit at a trade-off of interruptions that can occur when the user's bid price falls below the spot price. The interruptions are often exacerbated since customers often require resources in bundles. For these reasons, customers might have to wait for a long time before their jobs are completed. In this paper, we propose a behavioral-economic model in the form of time-preference-based bids, wherein users are willing to use and bid for services at other times if the vendor cannot provide the resources at the preferred time. Given such bids, we consider the problem of provisioning for such service requests. We develop a time-preference-based optimization model. Since the optimization model is NP-Hard, we develop rule-based genetic algorithms. We have obtained very encouraging results with respect to standard commercial solver as a benchmark. In turn, our results provide evidence for the viability of our approach for online service-provisioning problems.

Airplane Seating Assignment Problem

SARS-CoV-2, the virus that causes COVID-19, began infecting humans in late 2019 and has since spread to over 57 million people and caused over 1.75 million deaths, as of December 27, 2020. In response to reduced demand and travel restrictions as a result of COVID-19, airlines experienced a 94% reduction in passenger capacity worldwide in April and an estimated 60% reduction in passengers transported for all of 2020. SARS-CoV-2 has been shown to spread on airplanes by infected passengers, so minimizing the risk of secondary infections aboard aircraft may save lives. We present the airplane seating assignment problem (ASAP) to minimize transmission risks on airplanes, and we provide two models to solve ASAP. We show that both models can be effectively solved using a standard commercial solver and that seating assignments provided by these models have lower aggregate risk than the strategy of blocking the middle seats, given the same number of passengers. The available risk models for aircraft are based on influenza data, and hence risk models based on SARS-CoV-2 should be developed to maximize the benefits of our research.

Codon optimization: a mathematical programing approach.

Synthesizing proteins in heterologous hosts is an important tool in biotechnology. However, the genetic code is degenerate and the codon usage is biased in many organisms. Synonymous codon changes that are customized for each host organism may have a significant effect on the level of protein expression. This effect can be measured by using metrics, such as codon adaptation index, codon pair bias, relative codon bias and relative codon pair bias. Codon optimization is designing codons that improve one or more of these objectives. Currently available algorithms and software solutions either rely on heuristics without providing optimality guarantees or are very rigid in modeling different objective functions and restrictions. We develop an effective mixed integer linear programing (MILP) formulation, which considers multiple objectives. Our numerical study shows that this formulation can be effectively used to generate (Pareto) optimal codon designs even for very long amino acid sequences using a standard commercial solver. We also show that one can obtain designs in the efficient frontier in reasonable solution times and incorporate other complex objectives, such as mRNA secondary structures in codon design using MILP formulations. http://alpersen.bilkent.edu.tr/codonoptimization/CodonOptimization.zip.

A graph-based formulation for the shift rostering problem

This paper investigates a shift rostering problem – the assignment of staff to shifts over a planning horizon such that work rules are observed. Traditional integer-programming models are not able to solve shift rostering problems effectively for large number of staff and feasible shift patterns. We formulate work rules in terms of newly-proposed prohibited meta-sequences and resource constraints. A graph-based formulation and a specialized graph construction algorithm are proposed where the set of feasible shift patterns is represented by paths of a graph. The formulation size depends on the structure of the work-rule constraints and is independent of the number of staff. This approach results in smaller networks allowing large-scale rostering problems with hard constraints to be solved efficiently using standard commercial solvers. Moreover, it allows finding multiple optimal solutions which are beneficial for managerial decision makers. Computational results show that the proposed approach can obtain new best-known solutions and identify proven optimal solutions for almost all NSPLIB instances at significantly lower CPU times.

Using discrete-time mathematical programming to optimise the extraction rate of a durable non-renewable resource with a single primary supplier

A non-linear discrete-time mathematical program model is proposed to determining the optimal extraction policy for a single primary supplier of a durable non-renewable resource, such as gemstones or some metals. Karush, Kuhn and Tucker conditions allow obtaining analytic solutions and general properties of them in some specific settings. Moreover, provided that the objective function (i.e., the discounted value of the incomes throughout the planning horizon) is concave, the model can be easily solved, even using standard commercial solver. However, the analysis of the solutions obtained for different assumptions of the values of the parameters show that the optimal extraction policies and the corresponding prices do not exhibit a general shape.

Experimental and Numerical Investigations of Thermal Ignition of a Phase Changing Energetic Material

Fortuitous exposure to high temperatures initiates reaction in energetic materials and possibilities of such event are of great concern in terms of the safe and controlled usage of explosive devices. Experimental and numerical investigations on time to explosion and location of ignition of a phase changing polymer bonded explosive material (80 per cent RDX and 20 per cent binder), contained in a metallic confinement subjected to controlled temperature build-up on its surface, are presented. An experimental setup was developed in which the polymer bonded explosive material filled in a cylindrical confinement was provided with a precise control of surface heating rate. Temperature at various radial locations was monitored till ignition. A computational model for solving two dimensional unsteady heat transfer with phase change and heat generation due to multi-step chemical reaction was developed. This model was implemented using a custom field function in the framework of a finite volume method based standard commercial solver. Numerical study could simulate the transient heat conduction, the melting pattern of the explosive within the charge and also the thermal runaway. Computed values of temperature evolution at various radial locations and the time to ignition were closely agreeing with those measured in experiment. Results are helpful both in predicting the possibility of thermal ignition during accidents as well as for the design of safety systems.

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.

Network-flow based algorithms for scheduling production in multi-processor open-pit mines accounting for metal uncertainty

The open-pit mine production scheduling problem (MPSP) deals with the optimization of the net present value of a mining asset and has received significant attention in recent years. Several solution methods have been proposed for its deterministic version. However, little is reported in the literature about its stochastic version, where metal uncertainty is accounted for. Moreover, most methods focus on the mining sequence and do not consider the flow of the material once mined. In this paper, a new MPSP formulation accounting for metal uncertainty and considering multiple destinations for the mined material, including stockpiles, is introduced. In addition, four different heuristics for the problem are compared; namely, a tabu search heuristic incorporating a diversification strategy (TS), a variable neighborhood descent heuristic (VND), a very large neighborhood search heuristic based on network flow techniques (NF), and a diversified local search (DLS) that combines VND and NF. The first two heuristics are extensions of existing methods recently proposed in the literature, while the last two are novel approaches. Numerical tests indicate that the proposed solution methods are effective, able to solve in a few minutes up to a few hours instances that standard commercial solvers fail to solve. They also indicate that NF and DLS are in general more efficient and more robust than TS and VND.

A heuristic algorithm for optimal fleet composition with vehicle routing considerations

This paper proposes a fast heuristic algorithm for solving a combined optimal fleet composition and multi-period vehicle routing problem. The aim of the problem is to determine an optimal fleet mix, together with the corresponding vehicle routes, to minimize total cost subject to various customer delivery requirements and vehicle capacity constraints. The total cost includes not only the fixed, variable, and transportation costs associated with operating the fleet, but also the hiring costs incurred whenever vehicle requirements exceed fleet capacity. Although the problem under consideration can be formulated as a mixed-integer linear program (MILP), the MILP formulation for realistic problem instances is too large to solve using standard commercial solvers such as the IBM ILOG CPLEX optimization tool. Our proposed heuristic decomposes the problem into two tractable stages: in the first (outer) stage, the vehicle routes are optimized using cross entropy; in the second (inner) stage, the optimal fleet mix corresponding to a fixed set of routes is determined using dynamic programming and golden section search. Numerical results show that this heuristic approach generates high-quality solutions and significantly outperforms CPLEX in terms of computational speed.

A Trust Region Method for the Solution of the Surrogate Dual in Integer Programming

We propose an algorithm for solving the surrogate dual of a mixed integer program. The algorithm uses a trust region method based on a piecewise affine model of the dual surrogate value function. A new and much more flexible way of updating bounds on the surrogate dual's value is proposed, in which numerical experiments prove to be advantageous. A proof of convergence is given and numerical tests show that the method performance is better than a state of the art subgradient solver. Incorporation of the surrogate dual value as a cut added to the integer program is shown to greatly reduce solution times of a standard commercial solver on a specific class of problems.

Global Search Metaheuristics for planning transportation of multiple petroleum products in a multi-pipeline system

The objective of this work is to develop several metaheuristic algorithms to improve the efficiency of the MILP algorithm used for planning transportation of multiple petroleum products in a multi-pipeline system. The problem involves planning the optimal sequence of products assigned to each new package pumped through each polyduct of the network in order to meet product demands at each destination node before the end of the planning horizon. All the proposed metaheuristics are combinations of improvement methods applied to solutions resulting from different construction heuristics. These improvements are performed by searching the neighborhoods generated around the current solution by different Global Search Metaheuristics: Multi-Start Search, Variable Neighborhood Search, Taboo Search and Simulated Annealing. Numerical examples are solved in order to show the performance of these metaheuristics against a standard commercial solver using MILP. Results demonstrate how these metaheuristics are able to reach better solutions in much lower computational time.

Pyomo: modeling and solving mathematical programs in Python

We describe Pyomo, an open source software package for modeling and solving mathematical programs in Python. Pyomo can be used to define abstract and concrete problems, create problem instances, and solve these instances with standard open-source and commercial solvers. Pyomo provides a capability that is commonly associated with algebraic modeling languages such as AMPL, AIMMS, and GAMS. In contrast, Pyomo’s modeling objects are embedded within a full-featured high-level programming language with a rich set of supporting libraries. Pyomo leverages the capabilities of the Coopr software library, which together with Pyomo is part of IBM’s COIN-OR open-source initiative for operations research software. Coopr integrates Python packages for defining optimizers, modeling optimization applications, and managing computational experiments. Numerous examples illustrating advanced scripting applications are provided.

Tool sequence optimization based on three logical modes of robot control via CPLEX optimization

Cluster tools have advantages of shorter cycle times, faster process development, and better yield for less contamination. The sequence of dual-arm cluster tools is a complex logistics process during the semiconductor production. Efficient use of cluster tools is naturally very significant to competitive fab operations. Generating an optimized sequence in a computationally efficient manner and assessing the quality of the requirements to improve the fab production are the key factors for semiconductor manufacturing productivity. The Petri net modeling is introduced to minimize the makespan of the process for the three different logical modes and select a better mode after comparing the makespan among the three logical modes. The tool sequence optimization problem is formulated as optimization firing transition sequences based on the Petri net and then the formulation is converted to be linearly solved by the branch-and-cut method in the standard commercial solver CPLEX. Special methods for the linear conversion are highlighted. Due to the limited calculation time requirement for the real production and the large scale of the problem, special methods for the efficiency tuning are applied according to the characteristics of the problem. Numerical testing is supported by one of the most advanced semiconductor enterprises and the computational results show significant improvement compared with the traditional manual sequence results.