Mixed integer linear programming formulations for open pit production scheduling

Optimal operation of water-energy microgrids; a mixed integer linear programming formulation

The redundancy allocation problem (RAP) aims to find an allocation of redundant components that maximizes system reliability under resource constraints. In this paper, we consider the RAP for complex systems with heterogeneous components, which may have bridges or interconnecting subsystems and are generally more complicated in structure than commonly studied series-parallel systems. A major challenge of the RAP is that the reliability function is nonlinear. To tackle this, we develop six exact mixed integer linear programming (MILP) formulations, which, to our knowledge, are the first of such formulations for the problem. The key idea is to use binary variables and big-M constraints to successively linearize each product term in the reliability function. In addition, we propose a novel tree representation of the reliability function to establish the interconnections between different product terms, which is then used to reduce the number of decision variables and constraints needed in the linearization. We also propose an alternative form of the reliability function that only involves the failure probabilities of subsystems, and leverage this to derive better variable bounds, smaller big-M constants, and simpler formulations. We compare the six MILP formulations by solving them directly using an MILP solver based on three sets of test instances of varying complexity. Results indicate that all formulations achieve optimal solutions quickly for the first set and within reasonable computation times for the second set. For the third set, two formulations can always be solved within a given time limit. Overall, the MILP formulation derived based on the alternative form of the reliability function and a specific type of tree representation scheme performs the best. We also demonstrate that our approach is more robust and scalable than a benchmark branch-and-bound method. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by National Natural Science Foundation of China [Grants 72310107003, 72471189, 71901176]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0842 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0842 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In the literature, e.g. [10], one can find the so-called basic UAV mission target assignment in which m UAVs each with a capacity limit q visit n targets in a cooperative manner (and return to their departure points) such that the cost incurred by each UAV’s travel is minimized. In [10], we proposed a mixed integer linear program (MILP) formulation which exactly solves the problem, as well as four alternative MILP formulations which are computationally less intensive (and therefore suited for real-time purposes) yet yield a theoretically guaranteed sub-optimal solution. In this chapter, we further consider timing constraints imposed on some p of the targets, so-called prime targets. This consideration is often required for scenarios in which prime targets must be visited in a pre-defined time interval, and mathematically results in the addition of several integer linear constraints to the previous MILP formulation making the problem computationally intractable. We propose a novel procedure of adding these cumbersome timing constraints to the previous MILP formulation, in order to avoid increasing too much computational cost under practically valid assumptions. We first show that the proposed procedure still guarantees the previously claimed theoretical solution quality associated with the basic mission. We then show through extensive numerical simulations that under certain conditions, our algorithms return solutions which are still computationally manageable.

An MILP formulation for optimizing detailed schedules of a multiproduct pipeline network

Joint scheduling and power control schemes have previously been proposed to reduce power dissipation in wireless ad hoc networks. However, instead of power consumption, throughput is a more important performance concern for some emerging multihop wireless networks, such as wireless mesh networks. This paper examines joint link scheduling and power control with the objective of throughput improvement. The MAximum THroughput link Scheduling with Power Control (MATH-SPC) problem is first formulated and then a mixed integer linear programming (MILP) formulation is presented to provide optimal solutions. However, simply maximizing the throughput may lead to a severe bias on bandwidth allocation among links. To achieve a good tradeoff between throughput and fairness, a new parameter called the demand satisfaction factor (DSF) to characterize the fairness of bandwidth allocation and formulate the MAximum Throughput fAir link Scheduling with Power Control (MATA-SPC) problem is defined. An MILP formulation and an effective polynomial-time heuristic algorithm, namely, the serial linear programming rounding (SLPR) heuristic, to solve the MATA-SPC problem are also presented. Numerical results show that bandwidth can be fairly allocated among all links/flows by solving the MILP formulation or by using the heuristic algorithm at the cost of a minor reduction of network throughput. In addition, extensions to end-to-end throughput and fairness and multiradio wireless multihop networks are discussed.

In this paper, we seek to jointly minimize the network power consumption and the user transmission delays in green wireless access networks. We recently studied the case of a WLAN, where we evaluated the tradeoffs between these two minimization objectives using a Mixed Integer Linear Programming (MILP) formulation. However, the MILP formulation could not deliver solutions in a reasonable amount of time due to computational complexity issues. As a result, we propose here a heuristic algorithm for the power-delay minimization problem. The proposed heuristic aims to compute the transmit power level of the Access Points (APs) deployed in the network and associate users with these APs in a way that jointly minimizes the total network power and the total network delay. The simulation results show that the proposed algorithm has a low computational complexity, which makes it advantageous compared with the optimal scheme, particularly in dense networks. Moreover, the heuristic algorithm performs close to optimally and provides power savings of up to 45% compared with legacy networks.

Autonomous vehicles have emerged as transformative technologies with vast implications across various domains, including surveillance. However, they are constrained by battery limitations, which can hinder their usability and efficiency. Designing algorithms to efficiently utilize them optimizing their battery usage, is crucial to unlocking their full potential. This paper addresses this challenge by studying an extension of the rural postman problem with multiple depots and rechargeable and reusable vehicles that can make multiple trips but have limited battery capacity. It presents a new Mixed Integer Linear Programming (MILP) formulation to find optimal solutions to the problem and proposes a new heuristic called the multi-trip (MT) algorithm. The paper describes the results of a study that tested the MT algorithm, other heuristics, and the MILP formulation on both benchmark instances and larger, real-world instances generated from road maps. Our results show that the MT algorithm found feasible solutions for every instance and produced better solutions than heuristics from the literature on 38 of the 39 instances that were tested. Due to its requirements for memory and computing time, the Gurobi optimizer used for solving the MILP formulation could not find optimal solutions for the largest instances that we tested. This research contributes to enhancing the efficiency and effectiveness of battery-operated unmanned by developing new heuristics to address the challenge of battery limitations.

The optimal scheduling of crude-oil operation in refineries has been studied by various groups during the past decade leading to different mixed integer linear programming or mixed nonlinear programming formulations. This paper presents a new formulation with oil residency time constraint based on single-operation sequencing (SOS). It is different from previous formulations as it considers oil residency time constraint and pipeline transfer and does not require to postulate the number of priority-slots in which operations take place. This model is also based on the representation of a crude-oil scheduling by a single sequence of transfer operations. A simple MILP procedure has been used to solve this model leading to an satisfactory optimal result.

The purpose of this research is to solve a general job shop problem with alternative machine routings. We consider four performance measures: mean flow time, makespan, maximum lateness, and total absolute deviation from the due dates. We first develop mixed-integer linear programming (MILP) formulations for the problems. The MILP formulations can be used either to compute optimal solutions for small-sized problems or to test the performance of existing heuristic algorithms. In addition, we have developed a genetic algorithm that can be used to generate relatively good solutions quickly. Further, computational experiments have been performed to compare the solution of the MILP formulations with that of existing algorithms.

An elastic optical network (EON) plays an important role in transport technology for virtualization of networks. A key aspect of EONs is to establish lightpaths (virtual links) with exactly the amount of spectrum that is needed and with the possibility of grooming, the process of grouping many small traffic flows into larger units, creating a super-lightpath. Grooming eliminates the need for many guard bands between lightpaths and also saves transceivers; however, it often leads to the need to perform optical–electrical–optical conversions to multiple-data-rate optical signals at intermediate nodes. The aim of this paper is to provide a mixed-integer linear programming (MILP) formulation, as well as heuristic and meta-heuristic approaches, for the design of multiple virtual optical networks (VONs) in an elastic optical substrate network with bandwidth-variable lightpaths, modulation format constraints, and virtual elastic regenerator placement. Traffic grooming is allowed inside each VON, and a distance-adaptive modulation format technique is employed to guarantee efficiency in terms of bandwidth for a physical substrate, subject to several virtual topologies. A reduced MILP formulation without grooming capability is also proposed for comparison. The complete MILP formulation jointly solves the virtual topology design, regenerator placement, and grooming problems, as well as the routing, modulation, and spectrum assignment (RMSA) problem. The reduced MILP formulation, heuristics, and meta-heuristic, on the other hand, separate the virtual topology design problem from the RMSA problem. It is shown that the grooming approach can provide good results, since it solves the problem for a complete design when compared to the approach without grooming. Furthermore, heuristic solutions for large networks are proposed, which present good performance (in terms of saving spectrum) for the design with large instances.

We present an efficient polynomial-time approximation scheme (EPTAS) for the problem of scheduling jobs on parallel uniform machines that improves all previously known EPTAS for the problem. Our algorithm uses a Mixed Integer Linear Program (MILP) formulation for a relaxed version of the problem. We simplify the MILP by repeatedly removing carefully selected sets of jobs and machines until all integer variables have been removed. Then, we build up the solution through the use of a dynamic programming approach, enumeration, and linear program solving. Notable about our approach is that our algorithm only uses an LP solver, different from previous EPTAS approaches where MILP solvers are necessary.

Throughput is an important performance consideration for multihop wireless networks. In this paper, we study the joint link scheduling and power control problem, focusing on maximizing the network throughput. We formulate the maximum throughput link scheduling with power control (MATH-SPC) problem, and present a mixed integer linear programming (MILP) formulation to provide optimal solutions. However, simply maximizing the throughput leads to a severe bias on bandwidth allocation among all links. In order to enhance both throughput and fairness, we define a new parameter, the demand satisfaction factor (DSF), to characterize the fairness of bandwidth allocation. We formulate the maximum throughput fair link scheduling with power control (MATA-SPC) problem and present an MILP formulation for this problem. We also present an effective polynomial time heuristic algorithm, namely, the serial LP rounding (SLPR) heuristic. Our numerical results show that bandwidth can be fairly allocated among all links/flows by solving our MATA-SPC formulation or using our heuristic algorithm at the cost of a minor reduction of network throughput

An important problem in WDM network design is to construct a logical topology and determine an optimal routing over that topology. Mixed integer linear program (MILP) formulations to generate optimal solutions for this problem are computationally intractable, even for moderate sized networks. A standard approach is to decouple the problem of logical topology design and the problem of routing on this logical topology. Heuristics for finding the logical topology exist and a straightforward linear program (LP), based on the node-arc formulation is normally used to solve the routing problem over a given logical topology. Such LP formulations become computationally infeasible for large networks. In this paper, we present a new formulation for optimally routing traffic over a given logical topology. Our formulation is based on the arc-chain representation and minimizes the congestion of the network. We have used an implicit column generation technique and exploited the special generalized upper bounding structure of our formulation to efficiently solve the problem.

This paper proposes a framework for scheduling the observation and download tasks of multiple agile satellites with practical considerations such as attitude transition time, onboard data capacity, and stereoscopic image acquisition. A mixed integer linear programming (MILP) formulation for optimal scheduling that can address these practical considerations is introduced. A heuristic algorithm to obtain a near-optimal solution of the formulated MILP based on the time windows pruning procedure is proposed. A comprehensive case study demonstrating the validity of the proposed formulation and heuristic is presented.

Multienergy systems (MES) can optimally deploy their internal operational flexibility to use combinations of different energy vectors to meet the needs of end-users and potentially support the wider system. Key relevant applications of MES are multienergy districts (MEDs) with, for example, integrated electricity and gas distribution and district heating networks. Simulation and optimization of MEDs is a grand challenge requiring sophisticated techno-economic tools that are capable of modeling buildings and distributed energy resources (DERs) across multienergy networks. This article provides a tutorial-like overview of the state-of-the-art concepts for techno-economic modeling and optimization of integrated electricity-heat-gas systems in flexible MEDs, also considering operational uncertainty and multiple grid support services. Relevant mixed integer linear programming (MILP) formulations for two-stage stochastic scheduling of buildings and DER, iteratively soft-coupled to nonlinear network models, are then presented as the basis of a practical network-constrained MED energy management tool developed in several projects. The concepts presented are demonstrated through real-world applications based on The University of Manchester MED case study, the details of which are also provided as a testbed for future research.