Stochastic Mixed-integer Linear Programming Model Research Articles

Resilience-oriented planning for active distribution systems: A probabilistic approach considering regional weather profiles

Recent experiences with increasing climatic variability highlighted the importance of enhanced power distribution system resilience. Conventional methods of power distribution system planning usually focus on persistentinvestments associated with predicted faulty events, aiming to enhance the overall system reliability. Improving distribution system resilience in response to extreme weather events involves reduction of impact of the events having varying probability of occurrence. The probability of an extreme event varies significantly by location due to a combination of geographical, climatological, and meteorological factors. Thus, instead of requiring persistent investments, the solutions for resilience-oriented system upgrades should be guided by the potential effects and the probability of occurrence of extreme weather events in a specific area. To achieve this objective, the present paper proposes a two-stage stochastic mixed-integer linear programming (MILP) model based on regional weather profiles for the long-term resilience planning of distribution systems against extreme weather events. In the planning stage, investment decisions are made regarding overhead distribution line hardening and the distributed generation (DG) installation. During restoration process in the operational stage, dynamic microgrid formation is carried out for advanced distribution grid operations to minimize the cost corresponding to loss of load. The inclusion of regional weather profiles into planning objectives provide added flexibility to the network operators for analysing the trade-off between investment cost and load loss cost in resilient distribution system planning strategies.

Optimal Design of Resilient Carbon Capture, Utilization and Storage (CCUS) Supply Chain Networks under Facility Disruption

In recent years, various kinds of carbon dioxide capture, utilization and storage supply chain network design (CCUS SCND) problems have been extensively studied by scholars from the supply chain management community and other fields. The existing works mainly focus on the various deterministic or uncertainty problems; few works consider the CCUS SCND resilience problem in the context of utilization/storage facility disruptions due to unexpected natural disasters or other geological anomaly events. This paper aims to study the CCUS SCND resilience problem under utilization/storage facility capacity disruption risk. We propose a stochastic mixed-integer linear programming model for the considered problem. In the considered problem, the main decisions related to the following areas are taken into account: supply chain design and planning; facility disruption risk handling, including the optimal determination of facility locations and the matching of carbon dioxide emission sources and utilization/storage facilities; carbon dioxide normal transportation planning; and transshipment planning for various disruption scenarios. Finally, an experimental study comprising a case study from China is conducted to validate the effectiveness and performance of our proposed model. The obtained results show that the supply chain networks for the case study obtained by our proposed model are efficient, cost-effective and resilient in mitigating various kinds of utilization/storage facility disruption scenarios, showing the model can be applied to large-scale CCUS projects to help managers effectively deal with disruption risks. Future research should consider multiple disruption events and propose multiple effective resilience strategies.

Efficient expansion planning of modern multi-energy distribution networks with electric vehicle charging stations: A stochastic MILP model

This paper presents a stochastic Mixed-Integer Linear Programming (MILP) model developed to address the collaborative expansion planning of modern multi-energy distribution networks, taking into account the integration of Electric Vehicle Charging Stations (EVCSs) in the presence of uncertainties. The model is comprehensively formulated to encompass Distributed Generation (DG) resources, Electric Vehicles (EVs), and Capacitor Banks (CBs), alongside traditional expansion options like substations and circuit construction/reinforcement. The essence of this Distribution System Expansion Planning (DSEP) problem formulation lies in its stochastic approach, featuring chance constraints to effectively manage and address uncertainties. The central objective of the model is to minimize the total investment and operational expenditures associated with the DSEP problem over a designated planning horizon. The rigorous application of optimization techniques ensures the convergence of the MILP model, thereby delivering an efficient and reliable solution. The model's effectiveness is initially assessed using an 18-bus system, followed by comprehensive testing on a larger 123- bus system to gauge its robustness. Thorough performance evaluations are conducted across various case studies, encompassing both the 18- bus and 123- bus systems, culminating in the validation of the model's resilience, scalability, and replicability. These case studies encompass deterministic and stochastic methodologies across six distinct scenarios. The numerical outcomes conclusively demonstrate that the DSEP solution, when utilizing the stochastic approach without considering EVCSs, is associated with elevated costs. However, with a strategic allocation strategy for EVCSs, noteworthy reductions in investment expenditures related to substations, lines, and overall costs are notably achieved. The incorporation of uncertainty management within the DSEP problem underscores the flexibility and adaptability intrinsic to the proposed stochastic MILP model. These attributes render it ideally suited to effectively orchestrating the expansion of multi-energy distribution networks in the presence of EVCSs, thereby enhancing its utility and relevance in addressing modern energy distribution challenges.

Stochastic MILP Model for Merging EV Charging Stations with Active Distribution System Expansion Planning by considering Uncertainties

Radial Power Distribution Networks (PDNs) often suffer from limited reliability, flexibility, and efficiency, leading to service interruptions. Planning for radial PDNs is essential to enhance redundancy resilience, reduce disruptions, and improve overall efficiency. However, traditional PDN planning methods have become obsolete due to the proliferation of Distributed Generation (DG) resources and energy storage systems. Additionally, the rise of Electric Vehicles (EVs) demands sophisticated charging infrastructure planning. This article presents a Mixed-Integer Linear Programming (MILP) model for joint expansion planning of PDN and Electric Vehicle Charging Stations (EVCSs). The model takes into account the construction or reinforcement of substations and circuits, along with the integration of EVs, the installation of DGs, and the placement of capacitor banks, all regarded as traditional conventional expansion options alternatives. To address uncertainties associated with DG generation, conventional loads, and EV demand, our model identifies optimal installation and asset locations. We formulate this as a stochastic scenario-based program with chance constraints for Power Distribution Network Expansion Planning (PDNEP), minimizing investment, operational, and energy loss cost costs over a planning horizon. Through two deterministic and stochastic approaches, encompassing six case studies on an 18-node test system, we evaluate the effectiveness of our model. Results are further validated on a 54-node system, confirming the model’s robustness. Notably, the numerical findings underscore the substantial cost reduction achieved by including EVCSs in the stochastic expansion planning approach, demonstrating its cost-effectiveness. In case study I, where all EVs charge at home during peak hours, it’s the worst case for the PDN. The 54-node system, more complex, demands longer computational time. In the 18-node system, costs improve from 9.97% (case study II) to 3.96% (case study VI) versus the worst-case (case I). In the 54-node system, improvements range from 10.47% (case study II) to 1.40% (case study VI). As a result, In comparative analyses against deterministic and stochastic approaches, our model consistently outperforms in diverse test case studies. The proposed model’s adaptability to address uncertainties underscores its suitability for solving the PDNEP problem in PND.

Industrial energy hubs with electric, thermal and hydrogen demands for resilience enhancement of mobile storage-integrated power systems

In recent decades, climate change has severely increased the concerns over the resilience of power systems. Extreme events may lead to the failure of a large number of system components and result in interruption of supply to lots of consumers. In this research, the main objective is to use and assess the potential of large industrial energy hubs (EHs) in resilience improvement of power systems. Mobile energy storage and demand response programs are also used to decrease involuntary demand shed in system and enhance system resilience. A stochastic mixed-integer linear programming model is developed considering the uncertainties in damaged transmission lines, hurricane time and repair time. Case study is a modified IEEE 24-bus power system with EHs. The studied EH receives gas from a gas network, exchanges power with power system, includes combined heat and power (CHP), generator, electrolyzer, boiler, thermal storage and hydrogen storage system and feeds electric, heat and hydrogen demands. Expected load not supplied (ELNS) is used as resilience metric. The results approve significant impact of EH in improvement of power system resilience. In scenarios in which the bus connected to EH is isolated, EH supplies the power system demand located at that bus and thereby improves system resilience; on the other hand, in other scenarios, EH typically behaves as an electricity consumer of power system. According to the results, mobile energy storage causes a 2.4% improvement in ELNS. The results also show that demand response program improves ELNS by 7.1%.

Towards a responsive-sustainable-resilient tea supply chain network design under uncertainty using big data

Nowadays, issues such as limited natural resources, environmental problems, social matters, and significance of resilience in agricultural supply chain (ASC) have dragged considerable attention worldwide. In this research, a five-level multi-objective stochastic mixed-integer linear programming model is designed for tea supply chain (TSC) in Iran. The objective functions of the suggested network are minimizing total costs of the supply chain (SC), the total water consumption, and non-resilience measures, and maximizing job opportunities of facilities. Literally, considering uncertainty for SC networks is extremely beneficial due to the existence of some variations in different parameters like demand. As a consequence, a robust possibilistic optimization (RPO) is implemented to manage the uncertainty. Due to the nature of the multi-objective optimization problem, the weighted-normalized-extended goal programming (WNEGP) approach is employed to solve the model. In order to credit the model, real data is collected from the tea organization of Iran. It is worth mentioning that parameters are gathered according to three aspects of big data: volume, velocity, and variety. The results validated the functionality of the model regarding planning strategy. In addition, it showed applying more costs on SC triggers an effective sustainable-resilient-responsive network. In terms of managerial insights, this study offers a far-reaching perspective to managers especially in ASC to develop their industries. Finally, some sensitivity analyses are discussed on key parameters such as demand, robustness coefficients, and also the value of the objective functions in various states. It is worth mentioning that sensitivity analyses on different states of the problem show how sustainability and resiliency affect the supply chain efficiency.

Integrating pre and post-disaster activities for designing an equitable humanitarian relief supply chain

The design of an efficient humanitarian relief supply chain (HRSC) for disaster management depends on the integration of pre- and post-disaster decisions, which reduces the suffering of the victims and saves lives. The use of strategies such as quantity flexibility contracts in the pre-disaster phase and evacuation management in the post-disaster phase significantly increase the efficiency of HRSCs for disaster management. Therefore, in this paper, for the first time, a comprehensive two-stage stochastic mixed-integer linear programming model is presented for integrating the pre- and post-disaster activities for possible disasters such as floods and earthquakes by considering quantity flexibility contract and equitable relief goods distribution. In addition, the proposed model determines the location of warehouses and inventory levels in these warehouses in the pre-disaster phase. It also deals with public donations and budget planning, planning vehicles and helicopters, locating hospitals and shelters, and optimizing the inventory flow between echelons in the post-disaster phase. In the proposed model, the quantity flexibility contract is applied to decrease the inventory level in the pre-disaster phase, and reduce the supply risk in the post-disaster phase. The efficiency and performance of the proposed model are evaluated using the data related to a possible earthquake in Karaj (one of the metropolises of Iran).

A Multiple Objective Genetic Algorithm Approach for Stochastic Open Pit Production Scheduling Optimisation

ABSTRACT The conventional approach to mine planning is to use a single estimated orebody model as the basis for production scheduling. This approach, however, does not consider grade uncertainties associated with grade estimation. These uncertainties have a significant impact on the net present value (NPV) and can only be accounted for when modelled as part of the production scheduling optimisation problem. In this research, a set of equally probable simulated orebodies generated through Sequential Gaussian Simulation is used as input to a stochastic optimisation model solved with genetic algorithm (GA). Grade variability is considered as part of the stochastic model. The problem definition and resource constraints are formulated and optimised using a specially designed mining-specific GA. This GA is employed to handle partial block processing through a specialised chromosome encoding technique resulting in near-optimal solutions. Two case studies are presented which compare results from the stochastic model solved with GA (SGA) and a Stochastic Mixed Integer Linear Programming (SMILP) model solved with CPLEX. For the second case study, while the SMILP model was at an optimality gap of 101% after 28 days, the SGA model generated an NPV of $10,045 M at 10.16% optimality gap after 1.5 h.

Resilience enhancement of distribution network under typhoon disaster based on two-stage stochastic programming

The reliability of power supply in distribution network is vulnerable to extreme weather events such as typhoon. Pre-event preparation can effectively mitigate the deterioration of system resilience. Therefore, we propose a distribution network resilience enhancement decision-making framework which is formulated as a two-stage stochastic mixed-integer linear programming (SMILP) model. The first stage invests coordinately in four strategies, including hardening lines, installing distributed generators (DG), allocating mobile emergency generators (MEG), and deploying switches, etc. And the goal at the first stage is to minimize the investment cost of resilience enhancement strategies. The objective at the second stage is to ensure the minimum expected recourse operation cost of the comprehensive strategies for all typical scenarios. The proposed model can combine distribution network long-term resilience planning with short-term post-event recovery. Furthermore, in order to address the uncertain problems of wind field, line damage and load fluctuations under typhoon disaster, this paper proposes the following solutions. For wind field uncertainty, a detailed wind field model considering time transition, sea-land transition and extreme value distribution is established to deal with wind speed prediction. For line damage uncertainty, an improved stress-strength interference model is set up. And for the load fluctuation uncertainty, scenario generation using load random multipliers is applied to address load uncertainty. The proposed framework is tested in IEEE 33-bus distribution system using historical data from 2018 super typhoon “Mangkhut” in China and demonstrates the SMILP model can significantly reduce the post-event expected recourse operation cost and meanwhile improve the distribution network resilience.

Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems

Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.

Multi-Objective Bus Timetable Coordination Considering Travel Time Uncertainty

This paper proposes a timetable coordination method for transfer problems in a bus transit system. With a given bus network, a stochastic mixed-integer linear programming (MILP) model has been formulated to obtain coordinated bus timetables with the objective of minimizing a weighted sum of the average value of total waiting time and its average absolute deviation value, allowing for random bus travel time. The vital decision variable is the terminal departure offset time of each target bus trip within a certain off-peak period. The robust MILP model can also be used to solve the first-bus transfer problem with the introduction of several new linear constraints. A solution method based on the Monte Carlo simulation has been developed to solve the MILP model. Numerical experiments have been conducted for different scenarios. The results indicate that bus timetables coordinated by the developed model are capable of substantially reducing waiting time for transfer and non-transfer passengers. In addition, the feasibility of simplifying a common sub-route into a single transfer stop in a timetable coordination problem has been explored based on numerical experiments.

A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

<p style='text-indent:20px;'>Providing new models or designing sustainable networks in recent studies represents a growing trend. However, there is still a gap in the simultaneous modeling of the three dimensions of sustainability in the electronic medical device supply chain (SC). In this paper, a novel hybrid chance-constrained programming and cost function model is presented for a green and sustainable closed-loop medical ventilator SC network design. To bring the problem closer to reality, a wide range of parameters including all cost parameters, demands, the upper bound of the released <inline-formula><tex-math id="M1">\begin{document}$ co_2 $\end{document}</tex-math></inline-formula>, and the minimum percentage of the units of product to be disposed and collected from a customer and to be dismantled and shipped from DCs are modeled as uncertain along with the normal probability distribution. The problem was first formulated into the framework of a bi-objective stochastic mixed-integer linear programming (MILP) model; then, it was reformulated into a tri-objective deterministic mixed-integer nonlinear programming (MINLP) one. In order to model the environmental sustainability dimension, in addition to handling the total greenhouse gas emissions, the total waste products were also controlled. The efficiency and applicability of the proposed model were tested in an Iranian medical ventilator production and distribution network. For sensitivity analyses, the effect of some critical parameters on the values of the objective functions was carefully examined. Finally, valuable managerial insights into the challenges of companies during the COVID-19 pandemic were presented. Numerical results showed that with the increase in the number of customers in the COVID-19 crisis, social responsibility could improve cost mean by up to 8%.</p>

Optimizing the aluminum supply chain network subject to the uncertainty of carbon emissions trading market

Aluminum production is characterized by high electricity consumption; hence its decarbonization hinges on upstream low-carbon electricity generation. As a result, the supply chain of aluminum product manufacturing can be strongly affected by the uncertainty of carbon emissions price via electricity generation. This paper investigates these impacts on the strategic design and the tactic operations of the aluminum supply chain by constructing a stochastic mixed-integer linear programming (SMILP) model, in which carbon price uncertainty and demand uncertainty are both accounted for. The supply chain case in the study covers a full process from upstream power generation to end-use consumption. A variety of stochastic carbon price scenarios are designed and assessed. Sensitivity analysis with respect to demand fluctuation is also performed. The results show that carbon price uncertainty significantly affects the configuration of the whole system. The main driving forces are the supplier selection and configuration alteration triggered by the carbon price variation, whereas other metrics such as carbon emissions costs and the responses to demand fluctuations display complicated patterns of change. This study provides a useful analytical framework that can be applied to the planning problem for other industries facing the same challenge.

A chance-constraint optimization model for a multi-echelon multi-product closed-loop supply chain considering brand diversity: An accelerated Benders decomposition algorithm

The closed-loop supply chain (CLSC) network design has become one of the most critical issues due to the importance of resource optimization. Moreover, increasing competition in commercial markets leads to the diversity of a brand’s product portfolio to meet the customer’s demand. Hence, this paper develops a multi-period, multi-brand stochastic mixed-integer linear programming (MILP) model with direct and indirect distribution for the proposed CLSC network. Because of the uncertain nature of demand, the uncertainties for new and second-hand product demand are considered. Besides, a chance-constraint optimization (CCO) approach is applied to deal with uncertainty. Moreover, the accelerated Benders decomposition (BD) algorithm is designed to solve the proposed model. Several test problems are created and used to solve the accelerated BD compared with the conventional BD algorithm to investigate this model. Finally, the results are compared and described analytically, and some future research is suggested.

A strategy to enhance the distribution systems recoverability via the simultaneous coordination of planning actions and operational resources

This paper presents a planning and operational strategy to improve the recoverability of distribution systems (DSs) to deal with a set of possible line fault scenarios. The strategy simultaneously optimizes the allocation of dispatchable distributed generation (DG) units while coordinating a dynamic restoration process based on a radial topology reconfiguration, an islanding operation, a demand response program, and the pre-positioning and dispatch of mobile emergency storage units. The uncertainty and variability associated with solar irradiation and demand are captured via a multi-period formulation based on a stochastic mixed-integer linear programming model. The objective function of this model minimizes the investment cost of new dispatchable DG units and the amount of energy shedding within the system. Simulations are performed on adapted 33-node and 53-node test systems to validate the proposed strategy under four different test conditions, numerical results reveal the advantages of simultaneously solving the planning and operational stages to improve the recoverability of the system.

Optimisation models for the day-ahead energy and reserve self-scheduling of a hybrid wind–battery virtual power plant

This work presents a suite of two optimisation models for the short-term self-scheduling and redispatch of a virtual power plant (VPP) composed of a wind farm and a Li-ion battery, that participates in the day-ahead energy and secondary regulation reserve markets of the Spanish power system. First, a two-stage stochastic mixed-integer linear programming model is used to obtain the VPP’s generation and reserve schedule and the opportunity cost of the energy stored in the battery. The model has an hourly resolution and a look-ahead period beyond the markets’ scheduling horizon and considers the hourly battery degradation costs as a function of both the depth of discharge and the discharge rate. Different strategies are evaluated to forecast the real-time use of the committed secondary regulation reserves. Second, a deterministic MILP model is used to determine the redispatch of the VPP using as input the generation and reserve schedule and the VPP’s storage opportunity cost provided by the former model and is executed on an hourly rolling basis. The results obtained show that the proposed models are effective for the short-term scheduling and redispatch of the VPP used with a low computational time, making them tractable and practical for their daily use.

Impact of Dynamic Thermal Rating on optimal siting and sizing of energy storage systems under Renewable Portfolio Standards requirements

Renewable Portfolio Standards (RPS), enforced as part of environmental regulations, are taking precedence worldwide to promote investments in renewable energy sources (RES). From the grid perspective, one of the bottle-neck factors that limits large-scale RES integration is the transmission network congestion. Dynamic Thermal Rating (DTR) is a smart grid technology that utilizes line condition and real-time weather conditions to alleviate network congestion by exploiting the transmission network inherent flexibility. Due to the intermittency of the RES, energy storage systems (ESS) have emerged as an essential component of power system that stores the excess generated energy by RES and pumps it back into the network at peak hours. However, despite the benefits of these technologies, the coordination of these two technologies with RPS regulations are not studied together. This paper presents a co-optimized model to evaluate the role of DTR in optimal siting and sizing of ESS units in connection with RPS requirements. The modeling framework co-optimizes bulk ESS expansion and DTR investment planning subject to constraints on RPS regulations and is formulated as a two-stage stochastic mixed integer linear programming (MILP) optimization problem. The model utilizes DTR to overcome the problem of network congestion and ESS to mitigate RES curtailment by saving the extra generated energy in non-peak hours. The efficacy of the proposed model in realizing impact of the DTR on ESS (location, energy capacity and power rating) in achieving high RPS targets is investigated on the modified IEEE 24-Bus system. Results demonstrate that DTR complements ESS under different RPS constraints by reducing the total system cost and ESS size significantly. • A stochastic MILP model is presented to model DTR implementation and ESS sizing simultaneously. • The model incorporates RPS policies and demand side management with various limits. • The impact of the DTR on investment cost, location and size of ESS units are discussed. • We find that the co-optimized model is noteworthy to study the power systems with high RPS targets.

A Benders decomposition approach for solving a two-stage local energy market problem under uncertainty

New consumer-centric electricity market schemes have emerged as an alternative to manage the energy balancing locally in distribution networks with high distributed energy resources (DERs) penetration. Peer-to-peer (P2P) energy trading has become an attractive scheme for reducing user and distribution system operators (DSO) costs through the energy surplus exchange between the energy community users, where the uncertainty of DERs and the consumption are challenging issues that require suitable managing for a proper market operation. This article presents a novel two-stage stochastic mixed-integer linear programming model to address the scheduling day-ahead problem of an energy community operating under a P2P energy trading scheme. The formulation proposed minimizes the expected community cost considering the network limitations, ensuring fair trading in the local energy market (LEM), and allowing the prosumers to act as buyers or sellers depending on their load consumption and self-generation. In order to reduce the shared information by consumers/prosumers that trade energy, the proposed model is decomposed into a master problem (MP) and subproblems (SPs) to manage the network constraints and the market requirements, respectively. With this purpose, the Benders decomposition approach is implemented using the recently introduced Strengthened Benders cuts to address the binary variables related to the market and battery operation in the SPs. The model and the algorithm are tested in the 69-bus radial distribution system, considering from 3 to 39 agents trading energy to measure the model scalability. In this analysis, the algorithm convergence shows that the proposed methodology reduces the LEM’s shared information without increasing the energy community cost.

Optimal location and operation of waste-to-energy plants when future waste composition is uncertain

In many countries, waste management is increasingly geared towards a circular economy, aiming for a sustainable society with less waste generation, fewer landfills, and a higher rate of recycling. Waste-to-Energy (WtE) plants, which convert waste into heat and energy, can contribute to the circular economy by utilizing types of waste that cannot be recycled. Due to the varying quality of sorting and socio-economic conditions in individual regions, the waste composition differs between regions and has an uncertain future development. Waste composition significantly affects the operation of WtE plants due to differences in energy potential. This paper supports strategic capacity planning for waste energy recovery by introducing a two-stage stochastic mixed-integer linear programming model that captures waste composition uncertainty through scenarios of possible future development. The results of the model provide insights into the economics of operation and identify important factors in the sustainability of the waste handling system. The model is demonstrated on an instance with six scenarios for waste management in the Czech Republic for the year 2030. The solution of the proposed model is to build 14 new WtE plants with a total capacity of 1970 kt in addition to the four existing plants with a capacity of 831 kt. The annual energy recovery capacity is expected to increase almost four times to satisfy EU directives that restrict waste landfilling.

A stochastic programming model for the aircraft sequencing and scheduling problem considering flight duration uncertainties

AbstractThis study presents a stochastic mixed-integer linear programming model for the aircraft sequencing and scheduling problem. The proposed model aims to minimise the average fuel consumption per aircraft in the Terminal Manoeuvring Area while considering uncertain flight durations for each flight. The tabu search algorithm was selected to solve the problem. The stochastic solution and deterministic solution results were compared to show the benefits of the stochastic solution. The average sample approximation technique was applied to this problem, and enhancement rates of the average fuel consumption per aircraft were 8.78% and 9.11% comparing the deterministic approach