Proposed Mixed Integer Linear Programming Model Research Articles

Matheuristic and learning-oriented multi-objective artificial bee colony algorithm for energy-aware flexible assembly job shop scheduling problem

With the increase of mass customization, flexible job shop scheduling problem considering assembly stage has widely existed in many manufacturing industries, such as die-casting mould factories. This problem is to find a reasonable machine assignment and operation sequence both in fabrication and assembly stages and simultaneously maximize production efficiency. In reality, energy shortages and environmental pollution have given an impetus to the development of energy-aware production scheduling problems. In this study, we address an energy-aware flexible assembly job shop scheduling problem (EFAJSP) with the objectives of minimizing flow time and energy consumption and first develop a mixed-integer linear programming (MILP) model to solve EFAJSP problem. Then, the model-specific characteristics are extracted and applied to a matheuristic decoding method for exploring the Pareto optimal solution. Due to the complexity of EFAJSP problem, a matheuristic and learning-oriented multi-objective artificial bee colony algorithm (MLABC), which combines the advantages of mathematical programming, reinforcement learning and meta-heuristic algorithm, is proposed. In addition, an initialization, destruction/construction operator and population update operator are proposed and work together to improve the exploration and exploitation performance of the proposed MLABC. Finally, numerical experimental results demonstrate the effectiveness of the proposed MILP model and the superiority of the MLABC over other algorithms in the literature.

Modeling and optimization algorithm for energy-efficient distributed assembly hybrid flowshop scheduling problem considering worker resources

Considering increasingly serious environmental issues, sustainable development and green manufacturing have received much attention. Meanwhile, with the development of economic globalization and requirement of customization production, distributed hybrid flowshop scheduling problem (DHFSP) and assembly shop problem (ASP) have widely existed in realistic manufacturing systems. In addition to machine resources, worker resources are a key element affecting production efficiency. However, previous studies have not considered the integration mode of DHFSP, ASP, and worker resources in green manufacturing systems. Therefore, this paper focuses on an energy-efficient distributed assembly hybrid flowshop scheduling problem considering worker resources (EDAHFSPW) for the first time. To solve this problem, a mixed-integer linear programming (MILP) model and a multi-objective memetic algorithm (MOMA) are proposed with minimization the total tardiness (TTD) and total energy consumption (TEC) objectives. In MOMA, a speed-related decoding method is developed to improve the quality of solutions. To generate excellent initial solutions, an initialization strategy is proposed based on problem characteristics. A local search strategy is presented to improve the exploitation capability. An energy-saving strategy is designed to further optimize TEC. Additionally, to validate the proposed MILP model, we implement CPLEX to solve it on 12 small-sized instances. To verify the effectiveness of the proposed MOMA, extensive experiments are conducted to compare with other 5 comparison algorithms on 90 large-sized instances. Experimental results illustrate that MOMA is superior to its competitors.

Optimal strategy to reduce energy waste in an electricity distribution network through direct/indirect bulk load control

In recent decades, the ever-increasing demand for electricity in large cities is leading the electricity distribution network (EDN) to an operational state of fatigue, especially at peak load times. In this context, load congestion, technical losses, and voltage deviations across the EDN can contribute to the high waste of electricity as well as to the worsening of the supply service. To face this problem, this paper proposes a mixed integer linear programming (MILP) model that aims to reshape the demand profile (i.e., load reduction in peak periods without load rebound in the remaining periods of the day) of total consumers while the voltage deviations and technical losses are minimized. In the proposed model, the demand profile reshaping is carried out through direct (DC) and indirect (IC) bulk control of residential, commercial, and industrial loads, while the load factor (LF), i.e., indicator related to the efficient use of electricity, is improved. Due to the scheduling of these loads at certain times of the day, the occurrence of voltage deviations and technical losses can be mitigated by the efficient control of capacitor banks (CBs). To corroborate the applicability of the proposed MILP model, the 33-node IEEE test system was used. The results show the technical and economic gains for the distribution company (DISCO) and the total number of consumers.

A genetic algorithm with critical path-based variable neighborhood search for distributed assembly job shop scheduling problem

Production scheduling in distributed manufacturing systems has become an active research field, where large-sized complicated products, such as airplanes and ships, are taken as the primary focus. This paper investigates a distributed assembly job shop scheduling problem (DAJSP) which consists of two production phases. The first stage processes components in several job shops, and the second stage assembles the processed parts into final products. First, a mixed integer linear programming (MILP) model is established to describe the problem with minimizing maximum completion time and find optimal schedules for small-scale scenarios. Afterwards, a genetic algorithm with variable neighborhood search (GA-VNS) is proposed to address more complex instances, which adopts the genetic algorithm as the main framework and employs the variable neighborhood search for exploration. A problem-specific three-vector encoding scheme is designed to represent three decision-making processes of the DAJSP accordingly. To improve candidate solutions, a disjunctive graph model for DAJSP is formulated and three critical path-based neighborhood structures which directly perform on encoding vectors are designed. Numerical experiments are conducted on four groups of instances with different scales and the experimental results demonstrate the effectiveness of the proposed MILP model and GA-VNS. To sum up, the proposed GA-VNS shows the best performance on 30 instances out of 40 instances, while the superior stability has also been proved by statistical tests. In addition, two complicated DAJSP cases are abstracted from an enterprise for fabricating large complex components to further validate the GA-VNS.

Modeling and optimization of the hybrid flow shop scheduling problem with sequence-dependent setup times

The hybrid flow shop scheduling problem (HFSP) is an extension of the classic flow shop scheduling problem and widely exists in real industrial production systems. In real production, sequence-dependent setup times (SDST) are very important and cannot be neglected. Therefore, this study focuses HFSP with SDST (HFSP-SDST) to minimize the makespan. To solve this problem, a mixed-integer linear programming (MILP) model to obtain the optimal solutions for small-scale instances is proposed. Given the NP-hard characteristics of HFSP-SDST, an improved artificial bee colony (IABC) algorithm is developed to efficiently solve large-sized instances. In IABC, permutation encoding is used and a hybrid representation that combines forward decoding and backward decoding methods is designed. To search for the solution space that is not included in the encoding and decoding, a problem-specific local search strategy is developed to enlarge the solution space. Experiments are conducted to evaluate the effectiveness of the MILP model and IABC. The results indicate that the proposed MILP model can find the optimal solutions for small-scale instances. The proposed IABC performs much better than the existing algorithms and improves 61 current best solutions of benchmark instances.

Electric vehicle charging scheduling with mobile charging stations

Mobile charging station (MCS) is a brand-new technology to electric vehicle charging, which is a supplementary service for addressing the shortcomings associated with fixed charging station (FCS), such as prolonged waiting time of charging, charging congestion of FCS, and user travel anxiety. While MCSs offer convenience to electric vehicle users, the challenge faced in the MCS operation is that MCS equipped with large batteries to provide charging services to users by visiting them one by one not only leads to high operating costs, but also prolongs the waiting time for users to utilize MCS services. To overcome the issues, a novel framework is proposed by optimizing the MCS operation in economic efficiency. In this study, a variant of mixed integer linear programming (MILP) model is developed to maximize the MCS operator's profits with considering the user's perspective, including clustering, and covering user demands, setting temporary charging location for users, dispatching MCSs for charging and discharging, and scheduling EV charging. As the scale of the problem increases, the solution time of using the proposed MILP model is inefficient. Whereas an improved genetic algorithm is developed for solving a large-scale of the proposed model. Solomon's benchmarking instance data is adopted to evaluate the performance and validity of the proposed algorithm, complemented by various sensitivity analyses aimed at providing managerial insights into MCS operations.

Learning From Images: Proactive Caching With Parallel Convolutional Neural Networks

With the continuous trend of data explosion, delivering packets from data servers to end users causes increased stress on both the fronthaul and backhaul traffic of mobile networks. To mitigate this problem, caching popular content closer to the end-users has emerged as an effective method for reducing network congestion and improving user experience. To find the optimal locations for content caching, many conventional approaches construct various Mixed Integer Linear Programming (MILP) models. However, such methods may fail to support online decision making due to the inherent curse of dimensionality. In this paper, a novel framework for proactive caching is proposed. This framework merges model-based optimization with data-driven techniques by transforming an optimization problem into a grayscale image. For parallel training and simple design purposes, the proposed MILP model is first decomposed into a number of sub-problems and, then, Convolutional Neural Networks (CNNs) are trained to predict content caching locations of these sub-problems. Furthermore, since the MILP model decomposition neglects the network resources (such as caching space and link bandwidth) competition among sub-problems, the CNNs' outputs have the risk to be infeasible solutions. Therefore, two algorithms are provided: the first uses predictions from CNNs as an extra constraint to reduce the number of decision variables; the second employs CNNs' outputs to accelerate local search. Numerical results show that the proposed scheme can reduce 71.6% computation time, whose computation time reaches around 28.9 ms, with only 0.8% additional performance cost compared to the MILP solution, which provides high quality decision making in pseudo real-time.

Balancing and scheduling human-robot collaborated assembly lines with layout and objective consideration

The recent Industry 4.0 trend, followed by the technological advancement of collaborative robots, has urged many industries to shift towards new types of assembly lines with human-robot collaboration (HRC). This type of manufacturing line, in which human skill is supported by robot agility, demands an integrated balancing and scheduling of tasks and operators among the stations. This study attempts to deal with these joint problems in the straight and U-shaped assembly lines while considering different objectives, namely, the number of stations (Type-1), the cycle time (Type-2), and the cost of stations, operators, and robot energy consumption (Type-rw). The latter type often arises in the real world, where multiple types of humans and robots with different skills and energy levels can perform the assembly tasks collaboratively or in parallel at stations. Additionally, practical constraints, namely robot tool changes, zoning, and technological requirements, are considered in Type-rw. Accordingly, different mixed-integer linear programming (MILP) models for straight and U-shaped layouts are proposed with efficient lower and upper bounds for each objective. The computational results validate the efficiency of the proposed MILP model with bounded objectives while addressing an application case and different test problem sizes. In addition, the analysis of results shows that the U-shaped layout offers greater flexibility than the straight line, leading to more efficient solutions for JIT production, particularly in objective Type-2 followed by Type-rw and Type-1. Moreover, the U-shaped lines featuring a high HRC level can further enhance the achievement of desired objectives compared to the straight lines with no or limited HRC.

Load Factor Improvement of the Electricity Grid Considering Distributed Energy Resources Operation and Regulation of Peak Load

In recent decades, in urban centers, several services (such as health, food, security, connectivity, etc.) have become more dependent on electricity. With the population increase, the energy requirements of these services can lead the electricity network (EN) to a state of operational stress. In this scenario, distribution companies (DISCOs) need to develop efficient energy management strategies to avoid wasting electricity in its EN. The load factor (LF) of an EN is an indicator related to efficient electricity usage within a time horizon (e.g., day, month, year, etc.). LF improvement can be achieved through an efficient operational scheme that combines several strategies. Among them, peak load regulation in the demand profile of all the consumers, together with the operation of distributed energy resources (DERs), such as renewable energy sources (RESs) and capacitor banks (CBs), represents a promising alternative. In order to address this efficient scheme, this work proposes a mixed-integer linear programming (MILP) model that aims to increase the LF through the bulk scheduling of residential, commercial, and industrial electrical appliances during the day, considering a baseline demand and an hourly rate. At the same time, voltage fluctuations and technical losses are mitigated by the presence of DERs in the EN. Operational constraints related to the performance of EN, RESs, and CBs are considered. Uncertainties in the habitual demand profiles and solar irradiance during the day are represented through simulation algorithms. In order to evaluate the proposed model, a modified 14-node IEEE EN was used. The results corroborate the applicability of this proposed MILP model, reducing electricity waste and ensuring an efficient EN operation.

Modelling and heuristically solving three-dimensional loading constrained vehicle routing problem with cross-docking

Cross-docking is a logistics strategy widely applied in numerous enterprises, where vehicle scheduling is one of the most important factor in operational level for cost reduction. In sight of the widespread applications of the three-dimensional loading on the vehicle scheduling, this study addresses a vehicle routing problem with cross-docking and three-dimensional loading constraints (3L-VRPCD). A relaxation mixed integer linear programming (MILP) model is proposed for solving the 3L-VRPCD. In order to effectively solve large-scale 3L-VRPCD instances, a hybrid heuristic algorithm is then proposed, which combines adaptive large neighbourhood search (ALNS) and multi-order local search packing (LSP) algorithm. In addition, a storage-pool-based strategy is integrated into the proposed algorithm for efficiency enhancement. A large number of 3L-VRPCD instances with wide-ranging properties are generated to test the feasibility and effectiveness of the proposed MILP model and heuristic method. Experimental results demonstrate remarkable advantages of the proposed heuristic over the MILP-based method with respect to solution quality and computational efficiency, especially for medium to large-scale instances. Meanwhile, the results verify that the proposed storage-pool-based strategy is capable of accelerating the search process and reducing the computational burden of the heuristic. Furthermore, the impacts of different properties (e.g., loading conditions) on the solution of 3L-VRPCD have been analysed and discussed.

Supply chain planning of vaccine and pharmaceutical clusters under uncertainty: The case of COVID-19

As an abrupt epidemic occurs, healthcare systems are shocked by the surge in the number of susceptible patients' demands, and decision-makers mostly rely on their frame of reference for urgent decision-making. Many reports have declared the COVID-19 impediments to trading and global economic growth. This study aims to provide a mathematical model to support pharmaceutical supply chain planning during the COVID-19 epidemic. Additionally, it aims to offer new insights into hospital supply chain problems by unifying cold and non-cold chains and considering a wide range of pharmaceuticals and vaccines. This approach is unprecedented and includes an analysis of various pharmaceutical features such as temperature, shelf life, priority, and clustering. To propose a model for planning the pharmaceutical supply chains, a mixed-integer linear programming (MILP) model is used for a four-echelon supply chain design. This model aims to minimize the costs involved in the pharmaceutical supply chain by maintaining an acceptable service level. Also, this paper considers uncertainty as an intrinsic part of the problem and addresses it through the wait-and-see method. Furthermore, an unexplored unsupervised learning method in the realm of supply chain planning has been used to cluster the pharmaceuticals and the vaccines and its merits and drawbacks are proposed. A case of Tehran hospitals with real data has been used to show the model's capabilities, as well. Based on the obtained results, the proposed approach is able to reach the optimum service level in the COVID conditions while maintaining a reduced cost. The experiment illustrates that the hospitals' adjacency and emergency orders alleviated the service level significantly. The proposed MILP model has proven to be efficient in providing a practical intuition for decision-makers. The clustering technique reduced the size of the problem and the time required to solve the model considerably.

Optimal Phase Balancing in Electricity Distribution Feeders Using Mixed-Integer Linear Programming

A mixed-integer linear programming (MILP) model that includes reductions in neutral current, feeder energy-loss cost, customer interruption cost, and labor cost is developed to derive the optimal phase-swapping strategy to enhance the phase balancing of distribution feeders. The neutral current of the distribution feeder is reduced by the phase-swapping strategy so that the tripping of the low-energy overcurrent relay can be prevented and customer-service interruption costs and the labor cost to execute the phase-swapping works can be justified by the energy-loss reduction obtained. The novelty of the study is its derivation of the phase-swapping strategy using mixed-integer linear programming to solve the problem of the unbalance of the distribution feeders. A Taipower distribution feeder is used to derive the phase-swapping strategy to demonstrate the proposed MILP model for phase balancing. The comparison of the phase currents and neutral current before phase-swapping reveals that the three-phase balance was not only significantly improved, but that the voltage unbalance was also decreased dramatically using the proposed phase-swapping strategy.

Mathematical Modeling and A Novel Heuristic Method for Flexible Job-Shop Batch Scheduling Problem with Incompatible Jobs

This paper investigates a novel flexible job-shop scheduling problem, where the machines have batch-processing capacity, but incompatible jobs cannot be processed in a batch (FJSPBI) simultaneously. This problem has wide applications in discrete manufacturing, especially in chemical and steel casting industries. For the first time, in this study, a 3-indexed mixed-integer linear programming (MILP) model is proposed, which can be efficiently and optimally solved by commercial solvers for small-scale problems. In addition, an improved large neighborhood search (LNS) algorithmic framework with an optimal insertion and tabu-based components (LNSIT) is proposed, which can achieve high-quality solutions for a large-scale FJSPBI in a reasonable time. A perturbation strategy and an optimal insertion strategy are then additionally embedded to improve the exploitation and exploration ability of the algorithm. The proposed model and algorithm are tested on numerous existing benchmark instances without the incompatibility characteristics, and on newly generated instances of the FJSPBI. The experimental results indicate the effectiveness of the proposed MILP model and the algorithm, including the proposed strategies, and the optimal insertion strategy can significantly reduce the computational burden of the LNS algorithm. The comparison results further verify that the proposed LNSIT can directly solve the specific flexible job-shop batch scheduling problem without incompatibility, with better results than existing methods, especially for large-scale instances. Additionally, the impacts of a wide range of characteristics, including batch capacity, incompatibility rate, instance scale, and machine processing rate, on the performance of the LNSIT and the scheduling results are analyzed and presented.

Optimal Feeder Reconfiguration and Placement of Voltage Regulators in Electrical Distribution Networks Using a Linear Mathematical Model

Power distribution systems face continuous challenges from increased demand and lengthening of feeders, resulting in power loss augmentation and unacceptable voltage drops. Thus, to reduce technical losses and improve the voltage profile, common techniques such as reactive compensation, network reconfiguration, and placing of voltage regulators are employed. Distribution network reconfiguration (DNR) consists of modifying the system topology with the aim of minimizing power losses, enhancing voltage profile, and improving network reliability. Optimal placement of voltage regulators (OPVRs) improves the voltage profile and helps to reduce power losses. DNR and OPVRs are challenging optimization problems involving both integer and continuous decision variables. In this paper, a mixed-integer linear programming (MILP) model is presented to simultaneously solve the problems of DNR and OPVRs in radial distribution networks. The combined optimal DNR and OPVRs aim at both the minimization of power losses and the improvement of the voltage profile. This approach has not been reported in the specialized literature. The proposed MILP model may be solved through commercially available software, obtaining global optimal solutions with lower computational effort than metaheuristic techniques applied for the same purpose. Several tests were conducted on three benchmark distribution test systems to demonstrate the efficacy and applicability of the proposed approach.

On the MILP Modeling of Remote-Controlled Switch and Field Circuit Breaker Malfunctions in Distribution System Switch Placement

Installing sectionalizing switches and field circuit breakers (FCBs) is vital for the fast restoration of customer electricity supply in distribution systems. However, the high capital costs of these protection devices, especially remote-controlled switches (RCSs) and FCBs, necessitate finding a trade-off between their costs and financial benefits. In this study, we propose a mixed-integer linear programming (MILP) model for optimizing switch planning in distribution systems. The proposed model determines the optimal allocation of manual switches, RCSs, and FCBs to minimize the costs of switches and the reliability-oriented expenses. While the former includes the costs of installing and operating the switches, the latter consists of the distribution company’s lost revenue due to the undelivered energy and the regulatory incentives (or penalties) associated with service reliability indices. Two penalty-reward mechanisms are used to account for the financial benefits of increasing the service reliability through reducing the duration and frequency of interruptions. Proposing a novel reliability assessment model, we consider the possibility of malfunctions in both RCSs and FCBs in a highly efficient manner, which is a major contribution of this work. The proposed MILP model is applied to three test networks to validate its applicability and efficacy. The results show the importance of considering the possibility of switch malfunctions in distribution networks.

Mixed integer linear programming formulations for the permutation flow shop scheduling problem with preventive maintenance activities

Integrating the maintenance activities into the production process allows for keeping/restoring the good functional state of machines and improving the system productivity. According to the maintenance strategy adopted by a company, maintenance activities can be carried out in fixed positions/intervals or can be flexible, with an optimised maintenance approach being the key to efficient production maintenance. Consequently, the scheduling problems with maintenance activities (machine availability constraints) have received an increasing attention for both cases of fixed and flexible maintenance activities. This paper investigates the flow shop scheduling problem in the non-resumable case using an arbitrary number of preventive maintenance activities on each machine, with the optimised criterion being the makespan. Three mixed integer linear programming (MILP) models are proposed for three different problem types with respect to the considered maintenance policies within fixed-time windows, after a fixed number of finished jobs, and within an optimal preventive maintenance period. The computational results based on the randomly generated test problems demonstrate the high performance of the proposed MILP models in solving a wide range of instances, particularly, in small- and medium-scale problems.

Signal timing at an isolated intersection under mixed traffic environment with self‐organizing connected and automated vehicles

AbstractThis study provides a signal timing model for isolated intersections under the mixed traffic environment consisting of connected and human‐driven vehicles (CHVs) and connected and automated vehicles (CAVs). Different from existing studies, CAVs are not controlled by traffic controllers and conduct trajectory planning themselves, which are called self‐organizing CAVs (SOCAVs). The specific trajectory planning strategies of SOCAVs are not accessible to traffic controllers either. The signal optimization problem is formulated as a mixed integer linear programming (MILP) model for total vehicle delay minimization. The states of SOCAVs and CHVs passing the stop bar are predicted without prior information of the trajectory planning strategies of SOCAVs. SOCAVs can lead approaching platoons to pass the intersection effectively, and such “leading effects” of SOCAVs are utilized. Phase sequence and duration are optimized with the “structure‐free” phasing scheme. A parallel particle swarm optimization algorithm with a grouping strategy is designed to solve the optimization model at a reduced scale for computational efficiency. Numerical studies validate that (1) the proposed algorithm significantly outperforms the benchmark method, which directly solves the proposed MILP model using the solver Gurobi 9.0, under medium and high traffic demand; and (2) the proposed model significantly outperforms fixed‐time and vehicle‐actuated signal control in terms of vehicle delay and throughput. Sensitivity analysis shows that the SOCAV penetration rate of 30% is sufficient to guarantee satisfactory performance of the proposed signal timing model.

A new MILP model and fast heuristics for the variable-sized bin packing problem with time windows

Variable-Sized Bin Packing Problem with Time Windows (VSBPPTW) is a variant of the classical bin packing problem in which several types of bins with different sizes and costs are available, and the total cost of bins used to pack all items is minimized. Additionally, a time window is defined for each item, and the set of items packed into each bin must be time-compatible, i.e., the items in a bin must have at least one common time point within their time windows. We propose a new mixed-integer linear programming (MILP) model for VSBPPTW that (i) ensures time-compatibility by filling each bin with items from one maximal time-compatible set only and (ii) uses a compressed network flow formulation to represent the packing of items into bins under each maximal time-compatible set. New fast heuristics for VSBPPTW are also proposed to warm-start the MILP solver. Computational experiments are carried out on two sets of benchmark problem instances from the literature. We show that the new fast heuristics obtain on average much better solutions than the previously proposed fast heuristic, and the new MILP model greatly outperforms the previously proposed MILP model. While the previously proposed MILP model solves to optimality only 13% of the problem instances in the first set and none in the second, the new MILP model solves to optimality over 70% problem instances in both sets.

Collaborative optimization of last-train timetables for metro network to increase service time for passengers

Last-train timetables are significant in metro systems and directly influence transportation organization efficiency and passenger service levels. This study focuses on the collaborative optimization of last-train timetables for a large-scale metro network. Different from most studies, which often focus on improving the ability of passengers to transfer between different metro lines (transfer accessibility) during the last-train period, or the ability of passengers to reach their destinations after boarding the last train service from the origin station (origin-destination (OD) accessibility), this work aims to optimize the latest time for passengers (LTP) to reach their destinations using the metro services. A mixed-integer linear programming (MILP) model is established to optimize the last-train timetables with maximizing LTPs. For comparison, two MILP models respectively aim at maximizing transfer accessibilities and OD accessibilities, are adopted as benchmarks. An improved genetic algorithm based on Q-learning (QGA) is developed to solve the proposed MILP models for optimizing last-train timetables for a large-scale metro network. The proposed method is validated by optimizing the last-train timetable of the metro network of Chengdu, China. The results indicate that compared with optimizing the transfer and OD accessibilities, optimizing LTPs can consider both single-line and multiple-line passenger benefits, and directly increase their accessibilities and feasible times to use the metro service.

Unit commitment with frequency‐arresting adequacy constraint: Modelling and pricing aspects

Frequency-arresting (FA) adequacy has received an increasing attention in operation of power systems with intermittent, electronically connected wind and sfolar generation. Presence of high penetration of renewable energy leads to reduced amount of inertia in the power system and the need for more sustainable market design. This paper addresses the FA adequacy by integrating Nadir and rate of change of frequency constraints in unit commitment formulation. Here, a mixed-integer linear program (MILP) is proposed for addressing the FA adequacy. The proposed MILP model of FA adequacy is non-convex and accordingly, FA adequacy pricing requires non-convex pricing techniques. The explicit and implicit uplift payment techniques are proposed and implemented for FA adequacy pricing using the authors' proposed MILP model. The primal–dual formulation is employed for implicit uplift technique. The proposed MILP model of FA adequacy along with its pricing techniques is carefully studied through an illustrative example system. For further discussions, the Nordic 44-node test network is studied.