Integrated Parametric Graph Closure and Branch-and-Cut Algorithm for Open Pit Mine Scheduling under Uncertainty

Open pit mine production scheduling is a computationally expensive large-scale mixed-integer linear programming problem. This research develops a computationally efficient algorithm to solve open pit production scheduling problems under uncertain geological parameters. The proposed solution approach for production scheduling is a two-stage process. The stochastic production scheduling problem is iteratively solved in the first stage after relaxing resource constraints using a parametric graph closure algorithm. Finally, the branch-and-cut algorithm is applied to respect the resource constraints, which might be violated during the first stage of the algorithm. Six small-scale production scheduling problems from iron and copper mines were used to validate the proposed stochastic production scheduling model. The results demonstrated that the proposed method could significantly improve the computational time with a reasonable optimality gap (the maximum gap is 4%). In addition, the proposed stochastic method is tested using industrial-scale copper data and compared with its deterministic model. The results show that the net present value for the stochastic model improved by 6% compared to the deterministic model.

Consistency Cuts for Dantzig-Wolfe Reformulations

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

The Electric Vehicle Routing Problem with Capacitated Charging Stations

Electric vehicle routing problems (E-VRPs) deal with routing a fleet of electric vehicles (EVs) to serve a set of customers while minimizing an operational criterion, for example, cost or time. The feasibility of the routes is constrained by the autonomy of the EVs, which may be recharged along the route. Much of the E-VRP research neglects the capacity of charging stations (CSs) and thus implicitly assumes that an unlimited number of EVs can be simultaneously charged at a CS. In this paper, we model and solve E-VRPs considering these capacity restrictions. In particular, we study an E-VRP with nonlinear charging functions, multiple charging technologies, en route charging, and variable charging quantities while explicitly accounting for the number of chargers available at privately managed CSs. We refer to this problem as the E-VRP with nonlinear charging functions and capacitated stations (E-VRP-NL-C). We introduce a continuous-time model formulation for the problem. We then introduce an algorithmic framework that iterates between two main components: (1) the route generator, which uses an iterated local search algorithm to build a pool of high-quality routes, and (2) the solution assembler, which applies a branch-and-cut algorithm to combine a subset of routes from the pool into a solution satisfying the capacity constraints. We compare four assembly strategies on a set of instances. We show that our algorithm effectively deals with the E-VRP-NL-C. Furthermore, considering the uncapacitated version of the E-VRP-NL-C, our solution method identifies new best-known solutions for 80 of 120 instances.

Implications, conflicts, and reductions for Steiner trees

The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization. In the past 10 years, there have been significant advances concerning approximation and complexity of the SPG. However, the state of the art in (practical) exact solution of the SPG has remained largely unchallenged for almost 20 years. While the DIMACS Challenge 2014 and the PACE Challenge 2018 brought renewed interest into Steiner tree problems, even the best new SPG solvers cannot match the state of the art on the vast majority of benchmark instances. The following article seeks to advance exact SPG solution once again. The article is based on a combination of three concepts: Implications, conflicts, and reductions. As a result, various new SPG techniques are conceived. Notably, several of the resulting techniques are (provably) stronger than well-known methods from the literature that are used in exact SPG algorithms. Finally, by integrating the new methods into a branch-and-cut framework, we obtain an exact SPG solver that is not only competitive with, but even outperforms the current state of the art on an extensive collection of benchmark sets. Furthermore, we can solve several instances for the first time to optimality.

Imposing Contiguity Constraints in Political Districting Models

For nearly 60 years, operations research techniques have assisted in the creation of political districting plans, beginning with an integer programming model. This model, which seeks compactness as its objective, tends to generate districts that are contiguous, or nearly so, but provides no guarantee of contiguity. In the paper “Imposing contiguity constraints in political districting models” by Hamidreza Validi, Austin Buchanan, and Eugene Lykhovyd, the authors consider and analyze four different contiguity models (two old and two new). Their computer implementation can handle redistricting instances as large as Indiana (1,511 census tracts). Their fastest approach uses a branch-and-cut algorithm, where contiguity constraints are added in a callback. Critically, many variables can be fixed to zero a priori by Lagrangian arguments. All test instances and source code are publicly available.