Integer Nonlinear Programming Research Articles

A bay design problem in less-than-unit-load production warehouse

In this paper, we consider a bay design problem in a less-than-unit-load production warehouse, which is motivated by a real-world problem in a semiconductor company. The objective is to maximize the utilization of the vertical space in bays by considering several practical storage requirements. To solve the problem, a non-linear integer programming model is first formulated. Since the problem is similar to a two-stage cutting stock problem (CSP), a column-and-row generation (CRG) method is developed, in which the original problem is decomposed into a restricted master problem and three subproblems, including two classical column generation subproblems and a row generation subproblem. The two former subproblems are solved as unbounded knapsack problems and for the latter, a two-stage approach is applied. The results of computational experiments on randomly generated instances show that the proposed CRG method is more efficient than the classic column-generation-based method, solving the non-linear model directly and solving a cut model in the literature directly. The results of a case study show that our strategy can improve the utilization of the existing warehouse storage space significantly by about 24%. The CRG method is also tested on basic two-stage two-dimensional CSP benchmarks and its performance is compared to those of other pattern-based methods. The results show its potential for effectively solving the basic two-stage two-dimensional CSPs.

Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling

The cross-line operation (CO) of trains in urban rail transit is an effective method to efficiently satisfy transfer passenger travel demand as well as relieve the pressure of transfer stations. The primary problem of CO is designing train services to satisfy travel demand with an uneven spatial distribution of passengers. This study constructs a nonlinear integer programming model with a novel train operation scheme, i.e., virtual coupling (VC) technology, which allows the coupling/decoupling of trains on different lines at both ends of each operation zone. This scheme makes the train capacity equitably distributed in each operation zone, thereby balancing train capacity utilization over the whole CO system. Regarding the nonlinear characteristics of the proposed model, an adaptive simulated annealing genetic algorithm (ASA-GA) was designed to quickly generate high-quality solutions. Based on real-world data from the Beijing Changping Line and Line 13, the effectiveness of the proposed model and algorithm were verified. The computation results show that in comparison to a single grouping train composition scheme without CO, a VC scheme with CO would reduce operation costs by 46.8%, with 80.6% savings of train capacity equity. Furthermore, the average passenger residence time would be reduced by 25.9%.

Game theoretic approach for the synthesis and optimization of economically optimal coalitions in integrated renewable energy and polygeneration networks

This study presents the generation of an integrated resource network that encompasses a bioenergy supply chain and a polygeneration hub. The resource network is designed using a superstructure that comprises 3 layers. The first layer is a supply chain network of seasonally available renewable and non-renewable energy sources and is connected to the second layer by a transport system comprised of rail, road, and pipeline. The second layer, which is the polygeneration hub, comprises a boiler to produce high-pressure steam, steam turbines for generating power, a multi-effect evaporation system for desalinating salt-water and an absorption refrigeration system to produce chilled water. The option of connecting the boiler to a solar-thermal and heat storage subnetwork for feedwater pre-heating is included in the second layer. Piping and electrical cables connect the polygeneration layer to the third layer, which comprises industrial heat demand, represented by heat exchanger network, and electrical power demand. The model is applied to a case study in which the subnetworks in the overall integrated superstructure layers are considered as individual players in an industrial symbioses network with the possibility to be in a cooperative game. The objective function of the model, which is represented as a mixed integer nonlinear programming model, involves the minimization of the summation of the players' marginal contribution, with equal weighting applied for each player, to the coalition's total annual cost. The result obtained involves preference for a coalition among participating subnetworks that include combined heat and power, multi-effect evaporation, and heat exchanger network.

A distributed economic model predictive control-based FPPT scheme for large-scale solar farm

Increasing solar-photovoltaic-power penetration necessitates the implementation of flexible power point tracking (FPPT) in solar farms. However, coordinating multiple solar photovoltaic (PV) power generation systems (SPVPGSs) poses a significant challenge due to their inherent intermittency and varying dynamic characteristics, complicating FPPT implementation. To overcome this challenge, this paper proposes a distributed economic model predictive control (DEMPC) scheme to achieve FPPT, while simultaneously enhancing overall economic performance in solar farms. By integrating solar farm control and local control into one optimal control framework, this scheme eliminates the need for power allocation, PV voltage reference calculation, and pulse width modulation. Leveraging the SPVPGS model and soft power constraint, DEMPC controllers are designed to achieve the economic targets of solar farms, which cooperate through a communication network to realize FPPT and economic optimization. Additionally, the strong nonlinearity of SPVPGS causes a non-convex mixed integer nonlinear programming (MINLP) problem, solved by a MINLP algorithm using finite converter switching states. Simulations under step-changed irradiance and power reference, as well as urgent maintenance conditions, demonstrate that the DEMPC-based FPPT scheme significantly outperforms the traditional hierarchical model predictive control-based FPPT scheme, presenting both superior dynamic response and enhanced economic performance in FPPT implementation.

Bi-Objective Mixed Integer Nonlinear Programming Model for Low Carbon Location-Inventory-Routing Problem with Time Windows and Customer Satisfaction

In order to support a low-carbon economy and manage market competition, location–inventory–routing logistics management must play a crucial role to minimize carbon emissions while maximizing customer satisfaction. This paper proposes a bi-objective mixed-integer nonlinear programming model with time window constraints that satisfies the normal distribution of stochastic customer demand. The proposed model aims to find Pareto optimal solutions for total cost minimization and customer satisfaction maximization. An improved non-dominated sorting genetic algorithm II (IMNSGA-II) with an elite strategy is developed to solve the model. The model considers cost factors, ensuring that out-of-stock inventory is not allowed. Factors such as a carbon trading mechanism and random variables to address customer needs are also included. An entropy weight method is used to derive the total cost, which is comprised of fixed costs, transportation costs, inventory costs, punishment costs, and the weight of carbon emissions costs. The IMNSGA-II produces the Pareto optimal solution set, and an entropy–TOPSIS method is used to generate an objective ranking of the solution set for decision-makers. Additionally, a sensitivity analysis is performed to evaluate the influence of carbon pricing on carbon emissions and customer satisfaction.

Synthesis of refinery hydrogen networks based on compressor performance models

Synthesis of refinery hydrogen network contributes to reductions in hydrogen consumption, production cost, and emissions. Hydrogen compressors are drivers for hydrogen distribution, which has a direct effect on the performance of hydrogen networks. The existing works focus on the hydrogen network synthesis and often use simple compressor models for easy modelling and solving, which may lead to overly ideal results. This work proposes a novel mathematical method for refinery hydrogen network synthesis considering compressor performances, incorporating both centrifugal and reciprocating compressors. The compression power vs. volumetric flowrate curves and the compression ratio vs. volumetric flowrate curves are considered for a centrifugal compressor, while the efficiency of a reciprocating compressor is correlated with the compression ratio. A mixed integer nonlinear programming model (MINLP) is formulated to minimize the total annualized cost. The developed MINLP model is examined based on two hydrogen network cases reported in literature. The results show that the minimum total annualized cost is a co-optimization of compressor type, operating points, compressor placement, and hydrogen network structure. In the two cases, compression efficiencies of centrifugal compressors range from 0.760 to 0.795 and that of reciprocating compressors are between 0.539 and 0.807. The consideration of centrifugal compressors rather than just reciprocating compressors not only improves the compression efficiencies but also simplify the hydrogen network structures. Moreover, it also brings reductions in the compression power by 2.23% and 1.97%, and decreases in the compressor investment by 7.12% and 6.87%.

Access strategies in mmWave cell-free network: A matching and auction theory based approach

The mmWave cell-free network is viewed as an energy-efficient and emerging architecture for beyond 5G systems. However, many practical issues should be solved for this system before the benefits are enjoyed, whereof feasible implementation of access strategy is one of the most vital. To this end, the access strategy including user scheduling and bandwidth allocation is investigated in this paper. Firstly, to maximize the defined utility function, the joint optimization problem of user scheduling and bandwidth allocation is formulated as a mixed integer nonlinear programming problem, which is intractable to search for an optimal solution in polynomial time. Secondly, a two-stage matching and sealed-auction-based access strategy is proposed to obtain a sub-optimal solution, where UEs with high-quality channel conditions are scheduled in the first-stage matching as much as possible, and the rest unmatched UEs are scheduled in the second-stage matching with the sealed-auction based bandwidth allocation. Thirdly, a simplified full-matching based access strategy with lower complexity is proposed for the high-dynamic scenario caused by the high-speed movement of UEs, and the complexities and convergence of the two proposed strategies are quantitatively analyzed. Finally, the performance of proposed strategies is evaluated through abundant simulations in different scenarios, and the superiorities of the proposed strategies are demonstrated through the comparison with reference strategies.

An Iterative Procurement Combinatorial Auction Mechanism for the Multi-Item, Multi-Sourcing Supplier-Selection and Order-Allocation Problem under a Flexible Bidding Language and Price-Sensitive Demand

This study addresses the multi-item, multi-sourcing supplier-selection and order-allocation problem. We propose an iterative procurement combinatorial auction mechanism that aims to reveal the suppliers’ minimum acceptable selling prices and assign orders optimally. Suppliers use a flexible bidding language to submit procurement bids. The buyer solves a Mixed Integer Non-linear Programming (MINLP) model to determine the winning bids for the current auction iteration. We introduce a buyer’s profit-improvement factor that constrains the suppliers to reduce their selling prices in subsequent bids. Moreover, this factor enables the buyer to strike a balance between computational effort and optimality gap. We develop a separate MINLP model for updating the suppliers’ bids while satisfying the buyer’s profit-improvement constraint. If none of the suppliers can find a feasible solution, the buyer reduces the profit-improvement factor until a pre-determined threshold is reached. A randomly generated numerical example is used to illustrate the proposed mechanism. In this example, the buyer’s profit improved by as much as 118% compared to a single-round auction. The experimental results show that the proposed mechanism is most effective in competitive environments with several suppliers and comparable costs. These results reinforce the importance of fostering competition and diversification in a supply chain.

Distributed reinforcement learning-based optimization of resource scheduling for telematics

As a reliable computational model in telematics, Mobile Edge Computing (MEC) is utilized to lighten the load on individual autonomous vehicles. It enables computationally expensive and latency-sensitive jobs to be offloaded from cars with limited processing power to servers at edge nodes. This paper presents the joint optimization problem as a mixed nonlinear integer programming problem. The scheduling of vehicle tasks and the distribution of communication resources are then rationalized using a resource allocation algorithm based on Deep Deterministic Policy Gradient (DDPG) reinforcement learning to arrive at a suboptimal solution based on the resource allocation policies of single-intelligent DDPG and multi-intelligent DDPG. The numerical simulation demonstrates the superior service experience of the proposed joint optimization strategy of distributed task offloading and resource allocation based on multi-intelligent DDPG over other association and channel selection strategies and joint resource allocation based on single-intelligent DDPG.

A flexible combinatorial auction bidding language for supplier selection and order allocation in a supply chain with price sensitive demand

This study addresses the multi-item multi-sourcing supplier selection and order allocation problem in a Combinatorial Auction (CA) framework. First, we propose a new flexible procurement CA bidding language, then prove that it can convey the same information as existing bidding languages while allowing more efficient auction outcomes and reduced computational complexity. Suppliers submit combinatorial bids to a single manufacturer. Each bid carries information about the supplier’s cost structure and potential discounts. The manufacturer purchases the items and assembles one or more product types that experience price-sensitive demand rates by end consumers. We use the realistic logit function to represent the price-sensitive demand rates of finished products. A Mixed Integer Non-Linear Programming (MINLP) model is developed to help the manufacturer make optimal supplier selection, order allocation, and pricing decisions under the proposed bidding language. The MINLP model considers purchasing, transportation, ordering, administrative, holding, and manufacturing costs. These considerations make the results of this study realistic with the potential to be used in practical applications. Also, we derive a set of necessary conditions that must exist in at least one optimal solution to the problem. These conditions can be utilized to design efficient solution mechanisms for this difficult problem in the future. Finally, an extensive numerical analysis is performed to illustrate the bidding language and the key results.

Task execution latency minimization for energy-sensitive IoTs in wireless powered mobile edge computing: A DRL-based method

Wireless powered mobile edge computing (MEC) has become a vital component of future 6G networks, offering efficient computational capabilities to internet of things (IoT) devices and ensuring a stable energy supply. In this paper, we propose an innovative design for a wireless powered MEC-assisted energy-sensitive IoT systems. The system integrates wireless power technology (WPT), allowing IoT devices to efficiently perform tasks under wireless power supply without consuming their battery energy. Therefore, we formulate a dynamic computational task execution latency minimization (DCTELM) problem to tackle the impacts of unpredictable energy requirements and potential communication latency on resource allocation. We also analyze the differences between time division multiple access and non-orthogonal multiple access communication protocols. Given the complexity and dynamic nature of this problem, we define it as a mixed integer nonlinear programming problem. To address this problem, we introduced an enhanced deep reinforcement learning (DRL) algorithm. First, the DCTELM problem is transformed into a Markov decision process (MDP), which simplifies the structure of the problem. Then, after determining the offloading decision, we reveal the convex relationship between resource allocation and task latency. Furthermore, the action space of MDP is further reduced, and a multi-head proximal policy optimization (PPO) algorithm is proposed to deal with the simplified MDP problem. This process reduces the complexity of problem solving, decreases the consumption of computational resources. Simulation results demonstrate that our method outperforms other baselines in terms of computation latency, particularly with the introduction of an exploration strategy during the training phase, which significantly reduces task latency.

Synthesis of heat-integrated water networks considering detailed heat exchanger design

This paper introduces a mathematical programming approach for synthesising heat-integrated water networks (HIWNs) with a detailed heat exchanger design. A common assumption of constant heat capacities of water streams is relaxed. The specific heat capacity of water is now considered as a nonlinear function of temperature to avoid errors in calculating energy balances. In addition, most previous works have assumed constant individual heat transfer coefficients for water streams and utilities when estimating heat transfer areas. This assumption can lead to significant over- or under-estimation of heat transfer areas. In this research, individual and overall heat transfer coefficients for heat exchangers are calculated based on the detailed design of plate heat exchangers for heat recovery. The objective function of the proposed mixed integer nonlinear programming (MINLP) model is to minimise the total annualised cost (TAC). The TAC includes operating costs for freshwater and utilities, pumping costs for overcoming pressure drops in heat exchangers, and investment costs for heat exchangers. Three examples are solved in this work to verify the proposed model.

An intelligent resource allocation strategy with slicing and auction for private edge cloud systems

The convergence of transformative technologies, including the Internet of Things (IoT), Big Data, and Artificial Intelligence (AI), has driven private edge cloud systems to the forefront of research efforts. The access to massive terminals and the emergence of personalized services pose serious challenges for efficient resource management in power private edge cloud systems. To address the challenge of inequitable resource allocation in the private edge cloud, this work proposes an intelligent resource allocation strategy with a slicing and auction approach. By formalizing the resource allocation problem as a Mixed Integer Nonlinear Programming (MINLP) puzzle, the method transforms it into a hierarchical allocation challenge for Mobile Network Operators (MNOs), Mobile Virtual Network Operators (MVNOs), and power terminals. The proposed Multi-hop Progressive Auction Algorithm (MPAA) addresses the sliced resource allocation problem between MNOs and MVNOs. Furthermore, a Terminal Resource Allocation Strategy (TRAS) based on improved particle swarm optimization is proposed to solve the spectrum resource allocation problem between MVNOs and power terminals. Extensive simulation results show that the bidding overhead of MPAA is reduced by 6.12% and the average terminal satisfaction of TRAS is improved by about 1.3% compared to conventional methods, thus improving the utilization of wireless resources within the power AIoT.

Fleet Repositioning, Flag Switching, Transportation Scheduling, and Speed Optimization for Tanker Shipping Firms

In response to the European Union (EU)’s sanctions on Russian oil products, tanker shipping firms may adopt two strategies to reoptimize their shipping networks. The first strategy is to switch the flag states of tankers that are not eligible to operate on certain routes. The second strategy is to reposition tankers based on their flag states, i.e., moving those tankers that are eligible from other groups to specified routes. To help tanker shipping firms minimize the total operating cost during the planning horizon in the context of EU oil sanctions, including costs of fleet repositioning, flag switching, and fuel, this study investigates an integrated problem of fleet repositioning, flag switching, transportation scheduling, and speed optimization considering the dynamic relationships among fuel consumption, speed, and load. By formulating the problem as a nonlinear integer programming model and applying various linearization techniques to convert the nonlinear model into a linear optimization model solvable by off-the-shelf linear optimization solvers, this study demonstrates the practical application potential of the proposed model, with the longest solution time of less than two hours for a numerical instance with seven routes. Furthermore, through sensitivity analyses on important factors including unit fuel prices, crude oil transportation demand, and the tanker repositioning cost, this study provides managerial insights into the operations management of tanker shipping firms.

Multi-objective design optimization of hermetically sealed core-type distribution transformer considering current harmonics of power grid using NSGA III

In this paper, the design optimization of hermetically sealed oil-immersed distribution transformers is studied. For the first time in the design of this type of transformer, harmonics of the power grid affecting the transformer design, along with all design variables, are included, and their boundaries suitable for the production line are determined. The design probability (search space) for a total of 21 design variables is 1.32×1025. The Brute Force method is not a time-efficient approach to finding the optimal result in this search space. Due to the mixed integer nonlinear programming (MINLP) nature of the transformer design problem, heuristic optimization algorithms are selected because of their shorter solution time and near-optimal results. Considering the principle of the “No Free Lunch” theorem, extensive evaluations were conducted on various heuristic algorithms to identify the most effective ones for addressing the transformer design problem. The most successful algorithms for solving the problem were identified after thorough evaluation, considering three distinct scenarios within a multi-objective optimization framework. These algorithms are namely MOEA/D, MOGWO, MOWOA, and NSGA III. In this study, an optimization tool is developed for the design optimization of hermetically sealed distribution transformers, covering a range from 50 kVA to 3150 kVA and encompassing all design variables. The developed optimization program is fully compatible with the production line and can be easily adjusted to meet all national and international standards, such as IEC and IEEE, as well as customer specifications.

Hybrid Genetic Algorithm and Tabu Search for Solving Preventive Maintenance Scheduling Problem for Cogeneration Plants

Preventive Maintenance (PM) is a periodic maintenance strategy that has great results for devices in extending their lives, increasing productivity, and, most importantly, helping to avoid unexpected breakdowns and their costly consequences. Preventive maintenance scheduling (PMS) is determining the time for carrying out PM, and it represents a sensitive issue in terms of impact on production if the time for the PM process is not optimally distributed. This study employs hybrid heuristic methods, integrating Genetic Algorithm (GA) and Tabu Search (TS), to address the PMS problem. Notably, the search for an optimal solution remained elusive with GA alone until the inclusion of TS. The resultant optimal solution is achieved swiftly, surpassing the time benchmarks set by conventional methods like integer programming and nonlinear integer programming. A comparison with a published article that used metaheuristics was also applied in order to evaluate the effectiveness of the proposed hybrid approach in terms of solution quality and convergence speed. Moreover, sensitivity analysis underscores the robustness and efficacy of the hybrid approach, consistently yielding optimal solutions across diverse scenarios. The schedule created exceeds standards set by waterworks experts, yielding significant water and electricity surpluses—16.6% and 12.1%, respectively—while simultaneously matching or surpassing total production levels. This method can be used for power plants in private or public sectors to generate an optimal PMS, save money, and avoid water or electricity cuts. In summary, this hybrid approach offers an efficient and effective solution for optimizing PMS, presenting opportunities for enhancement across various industries.

Modeling a Multimodal Routing Problem with Flexible Time Window in a Multi-Uncertainty Environment

In this study, we extend the research on the multimodal routing problem by considering flexible time window and multi-uncertainty environment. A multi-uncertainty environment includes uncertainty regarding the demand for goods, the travel speed of the transportation mode, and the transfer time between different transportation modes. This environment further results in uncertainty regarding the delivery time of goods at their destination and the earliness and lateness caused by time window violations. This study adopts triangular fuzzy numbers to model the uncertain parameters and the resulting uncertain variables. Then, a fuzzy mixed integer nonlinear programming model is established to formulate the specific problem, including both fuzzy parameters and fuzzy variables. To make the problem easily solvable, this study employs chance-constrained programming and linearization to process the proposed model to obtain an equivalent credibilistic chance-constrained linear programming reformulation with an attainable global optimum solution. A numerical case study based on a commonly used multimodal network structure is presented to demonstrate the feasibility of the proposed method. Compared to hard and soft time windows, the numerical case analysis reveals the advantages of the flexible time window in reducing the total costs, avoiding low reliability regarding timeliness, and providing confidence level-sensitive route schemes to achieve flexible routing decision-making under uncertainty. Furthermore, the numerical case analysis verifies that it is necessary to model the multi-uncertainty environment to satisfy the improved customer requirements for timeliness and enhance the flexibility of the routing, and multimodal transportation is better than unimodal transportation when routing goods in an uncertain environment. The sensitivity analysis in the numerical case study shows the conflicting relationship between the economic objective and the reliability regarding the timeliness of the routing, and the result provides a reference for the customer to find a balance between them.

Demand Response Power-Gas Interconnection Energy System and Power Metering Based on SOGA

Under the double background of global energy crisis and environmental pollution, China vigorously develops renewable energy while accelerating the construction of energy Internet. Taking demand response into account, this paper proposes the optimal design of power-gas interconnection energy system and power metering, and establishes the optimization research model of IEGES (integrated electricity-gas energy system) with demand response. Firstly, the structure model of the electric-gas interconnection energy system with CHP (Cogeneration, combined heat and power) as the core is constructed, and the energy conversion relationship and different energy flow directions of the coupling equipment are expounded from three aspects. The natural gas source point, pipeline equation, power side branch equation, voltage and current equation are modeled and sorted out, and the square term in the equation is linearized by second-order cone programming method, and the mixed integer nonlinear programming problem is transformed into mixed integer linear programming problem. A single objective genetic algorithm with “elite strategy” was selected to solve the equipment capacity optimization problem of IEGES system with system economy as the optimization objective. After a long time of parameter combination attempts, the current population size is 30, the number of iterations is 600, the crossover rate is 0.8, and the heritability is 0.3. The above parameters can obtain better convergence results on the basis of considering the operation time. Finally, a stochastic optimization method of energy Internet considering integrated demand response and uncertainty of wind power is proposed, which aims to meet the energy demand of end users while minimizing the operating cost of the system. The comprehensive demand response strategy including internal and external demand response is considered in the model. Internal demand response is realized by adjusting the internal operation mode of EH, while external demand response is implemented by the end user’s active response, and the load is time-shifted or interrupted under the guidance of external signals.

Lyapunov-guided Deep Reinforcement Learning for service caching and task offloading in Mobile Edge Computing

With the development of Internet of Things (IoT) networks and Mobile Edge Computing (MEC), many computing-intensive applications have been developed in large quantities. Due to the heterogeneity of tasks, different application services are required to perform each task. Caching application services and related data in edge servers is challenging. Hence, we study the service cache placement and task offloading problem in IoT networks. Since IoT devices and edge servers with limited storage resources can only cache a few services at the same time, we formulate the service cache placement and task offloading of IoT devices problem to minimize task service delay with long-term energy constraint of IoT devices, which is a mixed integer nonlinear programming problem. To solve this problem, an online Deep Reinforcement Learning guided by the Lyapunov optimization framework algorithm (LYADRL) is proposed. We first build a virtual queue model to decouple the problem by Lyapunov optimization technique to transform the problem into a single time slot optimization problem. Then, we use Deep Reinforcement Learning techniques to find the optimal edge service caching and task offloading policies for each time slot. Simulation results show that our algorithm can reduce the service delay compared with other benchmark algorithms.

A new hybrid approach for cold load restoration using deep learning

After an extended outage in a power system, restoration of feeders with high penetration levels of thermostatically controlled loads (TCLs) will be a challenging problem. By restoring these feeders, the demand will increase several times the normal conditions which is called the cold load pick-up (CLPU) phenomenon. This can lead to overload of equipment such as power transformers in transmission substations. In order to increase the reliability of network load supply, in transmission substations, usually several transformers are used which can be operated in parallel by closing circuit breakers on the secondary side called bus section (BS). When the restoration process is long and the loads are cold, more loads can be restored by closing the BSs and paralleling the transformers. On the other hand, it is not possible to close all the BSs simultaneously because paralleling all the transformers at the same time increases the short circuit level on the secondary side. Furthermore, in power systems, some loads are voltage-dependent and by reducing the voltage, their demand can be decreased and therefore the restoration process is expedited. Due to the increase in demand caused by CLPU, additional stress is applied to the power system, which can lead to operational constraints violation and even long-term voltage instability. Therefore, at the time of restoration of each feeder, the settings of voltage control devices are updated to avoid the violation of operational constraints and the occurrence of wider events. In this paper, in order to reduce cold load restoration time, closing the bus sections and voltage reduction are used simultaneously. Also, short circuit and operational constraints are considered. The cold load restoration problem is formulated as a mixed integer nonlinear programming and is solved using the pattern search algorithm. Since there are many decision variables and solving the problem is time-consuming, a combination of deep learning networks, pattern search algorithm and parallel computing is used to accelerate the computations.