A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

Cancer chemotherapy is a form of treatment in which both healthy and diseased tissues are adversely affected by dosing; this requires the clinician make decisions , which balance the effectiveness of treatment with toxicity effects, financial, and logistical constraints. This work presents a model-based approach for chemotherapy treatment scheduling that accounts for both drug toxicity constraints and the dosing constraints associated with clinical practice. A mixed-integer linear programming (MILP) approach is applied to a nonlinear system considered previously in an optimal control framework by Martin and Teo (1994). The nonlinearities in the pharmacodynamic PD model were eliminated by logarithmic transform and both the pharmacokinetic (PK) and PD models were converted into algebraic constraints by discretizing the system over the treatment window. This formulation also accounts for the toxicity constraints and allows for the easy incorporation of dosing constraints, which would be encountered during the course of treatment. MILP results are shown for three cases considered in R. Martin and K.L. Teo (1994): highly effective drug, moderately effective drug, and a drug with very little effect on the tumor. In the limit of the discretization timestep and delivery interval approaches zero, the MILP algorithm returns the optimal control solution. Like the optimal control solution, most of the drug is delivered at the end of the treatment cycle, which results in the minimal final cancer cell population. The MILP approach results in a problem which can be solved to optimality and can easily be augmented with constraints which give the problem more clinical relevance.

This paper proposes a deterministic two-stage mixed integer linear programming (TSMILP) approach to solve the reserve constrained dynamic economic dispatch (DED) problem considering valve-point effect (VPE). In stage one, the nonsmooth cost function and the transmission loss are piecewise linearized and consequently the DED problem is formulated as a mixed integer linear programming (MILP) problem, which can be solved by commercial solvers. In stage two, based on the solution obtained in stage one, a range compression technique is proposed to make a further exploitation in the subspace of the whole solution domain. Due to the linear approximation of the transmission loss, the solution obtained in stage two dose not strictly satisfies the power balance constraint. Hence, a forward procedure is employed to eliminate the error. The simulation results on four test systems show that TSMILP makes satisfactory performances, in comparison with the existing methods.

The moments produced by the aerodynamic control effectors on aircraft are typically non-linear functions of displacement with a region of linearity about zero displacement. In this paper we briefly review a novel approach that was developed at AFRL for solving the control allocation problem where the control moments are piecewise linear functions of the control displacement. The advantages of this approach are that it avoids cumbersome, high-order polynomial curve fits of the aerodynamic data and thus avoids solution of a non-linear programming problem. Stability and control derivatives are almost always stored in multi-dimensional look-up tables where it is assumed that the data is piecewise linear. The approach that is presented utilizes this piecewise linear assumption for the control effector moment data and accounts for non-linearities in the moment-effector relationships and position constraints on the effectors as well. The control allocation problem using piecewise linear functions is posed as a mixed-integer linear program (MILP) and solved using a freely available GNU-licensed mixed-integer linear programming code. We demonstrate two applications of the piecewise linear control allocation (PLCA) approach. The first application to be considered is to use the PLCA approach in the inner-loop control law of a re-usable launch vehicle on approach and landing. The second application is constraint estimation for trajectory reshaping and re-targeting. In order to successfully re-target a trajectory when an aircraft has experienced degraded performance due to a failure or damage to the vehicle, the effects of the failure or damage on the lift, drag, and trimmability of the vehicle must be known a priori over the entire flight envelope

This paper is devoted to designing a mixed-integer linear programming to solve the simultaneous line planning and timetabling problem. A directed graph is first constructed to represent the combinational travel network for both candidate trains and passengers. Through this graph, it is easy to simultaneously describe trains’ operation choices, passengers’ travel choices, and their restrictive relations. Based on it, a mixed-integer linear programming model with many binary and non-negative integer decision variables is built to simultaneously optimize the line plan, timetable, and passenger travel choices, which can be efficiently solved using commercial solver CPLEX for small-scale network, while a decomposition strategy is required to combine with CPLEX for large-scale network. A set of numerical experiments are carried out to analyse the impacts of flow sub-units, number of candidate trains, and weight factors. Finally, we conduct two case studies for testing the practicability of our proposed method.

A novel MILP approach for simultaneous optimization of water and energy: Application to a Canadian softwood Kraft pulping mill

Plithogenic cubic sets extend the cubic set framework into new mathematical territory that enables superior handling of decision-making and pattern recognition when dealing with uncertainty. Through Plithogenic logic researchers could develop advanced methods to represent complex patterns by supporting inconsistent and self-contradictory or combined features. This research paper explores the schematic explanation of Plithogenic cubic sets and their use in pattern recognition applications. The strong nature of pattern recognition emerge from Plithogenic logic's dimensionality flexibility when combined with a two-member structure in cubic sets while resolving imprecision and vagueness and the complexity of information aspects. Through theoretical foundations and illustrative examples the author presented their capability to employ Plithogenic cubic sets successfully in pattern recognition and classification applications. The schematic method delivers a complete grasp of Plithogenic cubic sets as an enabling instrument for pattern recognition research.

To attain the highest performance of energy supply systems, it is necessary to rationally determine design specifications in consideration of operational strategies corresponding to energy demands. Mixed-integer linear programming (MILP) approaches have been applied widely to such optimal design problems. A MILP method utilizing the hierarchical relationship between design and operation variables have been proposed to solve them efficiently. However, it cannot necessarily be effective to multi-objective optimal design problems because of the existence of a large number of competing design candidates. In this paper, the hierarchical MILP method is revised from the viewpoint of computation efficiency so that it can be applied practically to multi-objective optimal design problems. At the lower level, the order of the optimal operation problems to be solved is changed based on incumbents obtained previously to increase a lower bound for the optimal value of the combined objective function and reduce the number of the optimal operation problems to be solved. At the upper level, a lower bound for the optimal value of the combined objective function is incorporated into the solution method to reduce the number of the design candidates to be generated. This revised hierarchical MILP method is applied to a multiobjective optimal design of a gas turbine cogeneration plant, and its validity and effectiveness are clarified.

A Video-on-Demand system usually consists of a large number of independent video servers. In order to utilize network resources as efficiently as possible the overall network load should be balanced among the available servers. We consider a problem formulation based on an estimation of the expected number of requests per movie during the period of highest user interest. Apart from load balancing our formulation also deals with the minimization of reorganization costs associated with a newly obtained solution. We present two approaches to solve this problem: an exact formulation as a mixed-integer linear program (MIP) and a metaheuristic hybrid based on variable neighborhood search (VNS). Among others the VNS features two special large neighborhood structures searched using the MIP approach and by efficiently calculating cyclic exchanges, respectively. While the MIP approach alone is only able to obtain good solutions for instances involving few servers, the hybrid VNS performs well especially also on larger instances.

This paper presents the mixed-integer linear programming (MILP) based model to approach the admission control in flow shop scheduling problem with queue time constraints, where there are various upper bounds limit in each queue. The scheduling proposed in this paper iteratively retrieves the real-time status of a production system such as machine failures and recoveries, and job arrivals in each step and generate the most updated scheduling result at each decision time. Our objective function is to minimize the occurrence of queue time violation. We solve the MILP using combinatorial Benders’ cut (CBC), where the MILP model is decomposed into two independent parts: the binary variables as a master problem and the continuous variables as a slave problem. We compare the CBC with the results gained from the CPLEX. The numerical results indicate that the CBC indeed effectively and efficiently reaches the good feasible solution within a reasonable timeframe in the context of timely updating scheduling problem.

With the increasing interdependence of various energy carriers, the operation of power systems is found to correlate closely with the limitations on the other energy infrastructures. This paper presents a mixed-integer linear programming (MILP) model for the microgrid (MG) optimal scheduling considering technical and economic ties between electricity and natural gas (NG) systems. In the proposed methodology, different energy converters and storages, including combined heat and power (CHP) units, electricity/heat storage units, and distributed energy resources (DERs) are considered. The proposed model allows the MG operator to minimize the operation cost of the MG while different operational limitations on the energy hub are satisfied. The model is developed based on AC power flow constraints so as to respect reactive power and voltage security constraints. The efficiency and robustness of the proposed MILP formulation are successfully verified using a large-scale test MG.

A computational comparison of cargo prioritization and terminal allocation problem models

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/ .

Flowshop scheduling via mixed integer linear programming (MILP) has received considerable attention in the past four decades. However, alternate models are limited; most numerical studies have used small problem sizes. A need for good model evaluation methodology exists; and limited work exists on flowshops with no intermediate storage. This paper presents a classification of flowshops and MILP scheduling models, and addresses some of these issues. It develops a host of new MILP formulations for minimizing makespan in a permutation flowshop with no storage and with or without unit setups. It presents some useful insights into model building by employing a variety of new and old binary variables and coupling them creatively. In contrast to previous work, it evaluates a range of new and existing MILP models using many larger test problems with no or unlimited intermediate storage, and presents a reliable procedure to rank various models based on problems with varying data and sizes. It shows that the top models for the two flowshops indeed show slightly different computational performance.

Environmental concerns prompt the world for a transition to renewable energy sources from fossil energy. Reducing the dependency on non-renewable sources of energy is prerequisite for the sustainable world and less environmental pollution. Biogas energy, which is one of the most important renewable energy sources, is produced by burning organic wastes and can be used in many different fields. In this study, a two-stage approach is undertaken to optimize a biogas supply chain design by incorporating of 30 districts in Izmir. In the first stage, the selection of the most suitable biogas plants is considered by the goal programming approach, which is of great importance to decide the optimal location with high energy potential. The most suitable sites for the biogas plants are obtained as Konak and Narlıdere districts. In the second stage, the location problem of the biogas vehicle charging stations (BVS) for biogas vehicles is handled considering the results of the first stage using mixed integer linear programming (MILP) approach. Computational results demonstrate that it would be more appropriate to establish BVS in 12 districts of İzmir. The model and solution approach are pioneering for supply chain problems and an efficient tool for renewable energy plans.

The inherent variability and unpredictability of renewable energy output pose significant challenges to power grid stability. Pumped Storage Power Stations (PSPS) play a pivotal role in mitigating these challenges, enhancing the grid’s reliability and operational efficiency. This study proposes an advanced linear analytical method based on Mixed-Integer Linear Programming (MILP) to optimize the short-term scheduling of PSPSs. The goal is to simultaneously maximize the reduction in equivalent load fluctuations and improve power generation benefits. The model linearizes the objective functions, constraints, and decision variables, applying MILP to efficiently derive optimal dispatch solutions. Using the Heimifeng (HMF) PSPS in Hunan Province as a case study, data from four representative daily load scenarios in 2023 are employed to optimize both power output and pumping processes. The results highlight the following nonlinear, competitive relationship between load fluctuation improvements and power generation benefits: as power benefits increase the rate of improvement in load fluctuations tends to decrease. The optimal solutions demonstrate significant outcomes, with improvements exceeding 11.5% in equivalent load fluctuations across all scenarios and daily power benefits surpassing $41,100, reaching a peak of $55,700. This study introduces a robust linear analytical framework capable of simultaneously enhancing power benefits and stabilizing load fluctuations, thereby offering valuable technical support for decision-makers.