Fuzzy Chance-constrained Programming Model Research Articles

Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

Abstract Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

Solving a new bi-objective mathematical model for a hybrid flow shop scheduling problem with robots and fuzzy maintenance time

This paper presents a bi-objective mathematical model of a hybrid flow shop scheduling problem (HFSSP) with robots and fuzzy maintenance time. The proposed model minimizes the mean completion time of jobs and the total cost of the processes based on maintenance costs and incurred transportation costs via robots. A fuzzy chance-constrained programming (FCCP) model is used to present its deterministic-equivalent problem. It is then solved by two multi-objective decision-making approaches, namely LP-metric and goal attainment (GA). A sensitivity analysis of the weights of the objective functions is conducted and the behavior of the model is studied for each instance. Also, a TOPSIS (the technique for order of preference by similarity to ideal solution) method is used to rank the efficiency of the proposed solution approaches. Future research areas are also suggested.

Fuzzy Demand Vehicle Routing Problem with Soft Time Windows

Considering the vehicle routing problem with fuzzy demand and fuzzy time windows, a vehicle routing optimization method is proposed considering both soft time windows and uncertain customer demand. First, a fuzzy chance-constrained programming model is established based on credibility theory, minimizing the total logistics cost. At the same time, a random simulation algorithm is designed to calculate the penalty cost of delivery failures caused by demand that cannot be satisfied. In order to overcome the shortcomings of GA, which easily falls into the local optimum in the process of searching, and the slow convergence speed of SA when the population is too large, a hybrid simulated annealing–genetic algorithm is adopted to improve the solution quality and efficiency. Finally, the Solomon standard example is used to verify the effectiveness of the algorithm, and the influence of decision-makers’ subjective cost preference is analyzed.

Inexact [Formula: see text] fuzzy chance-constrained programming of booster chlorination for water distribution system under uncertainty.

To sustain water quality in water distribution system (WDS), disinfectant generally chlorine is boosted to water distribution system. However, the concentration of chlorine should be limited to acceptable scope. The upper bound of the scope is set for preventing the occurrence of disinfectant byproduct, which is harmful to human health. The lower bound of the scope is set for controlling the growth of microorganism as well as reducing the cost. As such, the optimization model was applied to solve the water quality issue in WDS. However, in WDS, chlorine decays and varies with time and space, affected by pipe material, temperature, pH value, and chlorine injection. Therefore, in this paper, an inexact [Formula: see text] fuzzy chance-constrained programming (IMFCCP) model was proposed to optimize the chlorine injection to maintain chlorine in WDS at an acceptable level with consideration of uncertainty in water quality simulation. The results indicated that the upper bounds, the lower bounds, and intervals of the injection mass increased with preference parameter λ, which means that the results are more unreliable with higher preference parameter λ. However, the effect of reliability level ζ on the injection mass is determined by the relationship between the preference parameter λ and reliability level ζ. In case of [Formula: see text], the effect is not more significant than the case of [Formula: see text]. The results can help managers determine the injection strategy under uncertainty.

Integrated production-inventory-routing problem for multi-perishable products under uncertainty by meta-heuristic algorithms

ABSTRACT The present study aims to introduce an integrated production-inventory-routing problem (PIRP) with a mixed-integer linear programming model, remarking a multi-perishable product, multi-period, and heterogeneous fleets with time windows in a distribution network. The objective of the proposed model is to maximise the total profit, which equals the selling revenue subtract the aggregation of the holding, production, transportation, and utility preference costs. At the production level, a multi-period production system with production capacity constraints is considered, in which the inventory at each stage of production is intended to compute the related holding costs and schedule more appropriate planning. The vehicle routing problem is tackled at the distribution level regarding vehicles with various capacities in a multi-period condition. Consequently, a fuzzy chance-constrained programming model is used to deal with fuzzy parameters. Furthermore, two evolutionary algorithms, namely a hybrid imperialist competitive algorithm (HICA) and self-adaptive differential evolution (SADE), are proposed to solve the given problem. Subsequently, several numerical examples with managerial insights are solved to evaluate the performances of the proposed algorithms and show their effectiveness and efficiency. Computational results demonstrate the superiority of the proposed algorithms for this problem. Finally, the applicability of the proposed algorithms is investigated by a real-case study.

Designing an integrated municipal solid waste management system using a fuzzy chance-constrained programming model considering economic and environmental aspects under uncertainty

In recent decades, municipal solid waste (MSW) management has become one of the major concerns of human societies. Therefore, an efficient MSW system of collecting, recycling, and disposing is required to be designed, maintained and improved continuously. However, designing a proper waste collection management system by controlling the spread of diseases and environmental pollutants is very costly. To provide an efficient MSW management system, this study proposes an integrated location-allocation problem to plan the operations such as collection, recycling, disposal, and transportation under uncertainty. To deal with the problem uncertainty, a triangular fuzzy number approach is employed and finally, a fuzzy chance-constrained programming model is developed. Moreover, the proposed bi-objective mathematical programming model aims to minimize the total costs and the total volume of pollutant emissions simultaneously. To solve and validate the model, a weighted goal programming (WGP) method is employed and solved by CPLEX solver of GAMS software. Finally, a real case study problem was implemented in Qazvin, Iran. The results showed that, in most cases, increasing the confidence level of the case study problem leads to the improvement of objective functions. Managers can take into account the output of this research provided along with a sensitivity analysis under different conditions.

A fuzzy chance-constrained programming model for integrated planning of the wheat supply chain considering wheat quality and sleep period: a case study

Wheat is a staple food in many countries, and as a result, the first most cultivated crop worldwide. Since wheat is a vital product in terms of food security, and its supply chain needs to be studied and planned carefully. This paper proposes a mixed-integer linear programming model for integrated planning of imported and domestically-produced wheat that addresses supplier selection, order planning, transportation, storage, and distribution problems at the same time. Specifically, this model focuses on the wheat quality and wheat sleep period. Moreover, differentiation of long-term and short-term storage facilities and consideration of intra-layer flows between storage facilities are other characteristics of this model. A fuzzy chance-constrained programming approach is employed to cope with the uncertainties associated with domestic supply, demand, and global wheat prices. Applicability and advantages of the developed model are demonstrated using real data from the wheat supply chain of Iran. Results show that the current status of the wheat supply chain of Iran is far from being optimal, and there are many opportunities for improvement.

Fuzzy Chance-Constrained Integer Programming Models for Portfolio Investment Selection and Optimization Under Uncertainty

Portfolio investment optimization is the process of selecting the best portfolio out of the set of all projects being considered. A high financial return is not the only concern since minimization of associated risk is as important. Its objective should be set to maximize the expected return and minimize the risk in the investment as most data need to be justified based on vagueness and future values. Thus, the portfolio investment optimization problem under a fuzzy environment is studied here by incorporating a classical mathematical optimization model with the fuzzy theory. It is solved with the fuzzy chance-constrained integer programming model by linear programming under predetermined conditions and limitations. This study also uses both the credibility index and credibilistic risk index for measuring the investment return and investment risk. A numerical example is illustrated to demonstrate the effectiveness and benefits of the proposed algorithm.

Fuzzy green vehicle routing problem for designing a three echelons supply chain

In this study, a three-echelon fuzzy green vehicle routing problem (3E-FGVRP) is considered for designing a regional agri-food supply chain on a time horizon. To account for the variability associated with the quantities requested by customers, it is assumed that the demands are fuzzy numbers simulated by a time-dependent algorithm. Moreover, the vehicle fleet and distribution centres are considered with a defined capacity. The credibility theory of fuzzy sets is used to implement a multi-objective fuzzy chance-constrained programming model, where the total costs and carbon emissions are minimised. The resolution of the 3E-FGVRP is conducted by using a non-dominated sorting genetic algorithm. The multiple-criteria decision-making ELECTRE III method is applied to select the best solutions belonging to each Pareto front. Finally, the validity of the model is demonstrated by performing an optimisation procedure with three different initial random sets of populations. The application of the model to a case study of the Sicilian agri-food context confirms the robustness of the model, and the optimal configurations of the three-echelon supply chain can be found.

An Optimization Framework for Collaborative Control of Power Loss and Voltage in Distribution Systems With DGs and EVs Using Stochastic Fuzzy Chance Constrained Programming

A stochastic fuzzy chance-constrained programming model with multi-objective optimization for coordinated control of power loss and voltage in distribution systems with renewable energy is presented by taking power output of DGs and charging-discharging power of EVs as random fuzzy variables and load power as random variables. Considering the fuzziness and randomness of active power output of distributed generation systems with wind and solar energy and charging power of electric vehicles, the key parameters of probability density function are determined by fitting incomplete data of uncertainties such as wind speed, sunlight intensity and charging power of electric vehicles. According to the principle of random fuzzy compatibility, the probability density function of the uncertainties is transformed into the probability distribution function of the uncertainties. The NDC(Normal Distribution Crossover)-based non-dominated sorting genetic algorithm is used to solve the optimization problem, and the Pareto solution set of the multi-objective optimization problem is obtained. The feasibility and applicability of the proposed model and algorithm are verified by simulating IEEE-33 and IEEE-118 distribution system.

Solving fuzzy capacitated location routing problem using hybrid variable neighborhood search and evolutionary local search

A fuzzy capacitated location routing problem (FCLRP) is solved by using a heuristic method that combines variable neighborhood search (VNS) and evolutionary local search (ELS). Demands of the customer and travel times between customers and depots are considered as fuzzy and deterministic variables, respectively in FCLRP. Heterogeneous and homogeneous fleet sizes are performed together to reach the least multi-objective cost in a case study. The multi-objective cost consists of transportation cost, additional cost, vehicle waiting cost and delay cost. A fuzzy chance constrained programming model is added by using credibility theory. The proposed method reaches the solution by performing four stages. In the first stage, initial solutions are obtained by using a greedy heuristic method, and then VNS heuristic, which consists of seven different neighborhood structures, is performed to improve the solution quality in the second stage. In the third stage, a perturbation procedure is applied to the improved solution using ELS algorithm, and then VNS heuristic is applied again in the last stage. The combination of VNS and ELS is called VNSxELS algorithm and applied to a case study, which has fifty-seven customers and five distributing points, effectively in a reasonable time.

A credibility-based multi-objective temporary logistics hub location-allocation model for relief supply and distribution under uncertainty

Disaster response operations revolve around uncertainties. While uncertainties arising due to randomness can be avoided for post-disaster location problem, those arising because of impreciseness may persist long after the disaster's occurrence. Despite the uncertainties and lack of sufficient information about the extent of the damage, disaster response facilities must be established quickly after the occurrence of the disaster. Moreover, the decisions of whether to open, where to locate, and when to open disaster response facilities are based on the amount and quality of information available during the decision-making period. To address these issues, we develop a multi-objective location-allocation model for relief supply and distribution that accounts for the imprecise and time-varying nature of different parameters and time-varying coverage, while also accommodating the subjective attributes necessary to enable establishment and operation of the temporary logistics hubs (TLHs). A credibility-based fuzzy chance-constrained programming model is employed to account for the impreciseness inherent in predicting parameter values during disaster response. The results show where, when, and how many TLHs to open and how to allocate relief supplies. Meanwhile, the sensitivity analysis provides a broader understanding of the impact of limiting the number of TLHs as well as the confidence level and the spread of the symmetric triangular fuzzy numbers on the attainment of the model objectives.

A genetic-algorithm-aided fuzzy chance-constrained programming model for municipal solid waste management

ABSTRACTA genetic-algorithm-aided fuzzy chance-constrained programming (GAFCCP) model is developed for supporting municipal solid waste (MSW) management under uncertainty. The proposed model is an innovative combination of the genetic algorithm (GA) and fuzzy chance-constrained programming (FCCP) method and thus makes a unique contribution to enhancing the feasibility and applicability of the optimization model. The GA was capable of tackling the complicated fuzzy membership function and was used to seek optimal solutions by progressively evaluating the performance of the individual solutions; meanwhile, FCCP ensured that the fuzzy constraints were satisfied at specified confidence levels, leading to cost-effective solutions under acceptable risk magnitudes. A long-term regional waste management model of Zhongshan City, China, was formulated to demonstrate the applicability of the proposed GAFCCP model. The comparison results with ongoing treatment schemes demonstrated the superiority of the generated model solutions in the aspects of cost reduction and greenhouse gas emission mitigation.

Application of a Fuzzy Two-Stage Chance Constrained Stochastic Programming Model for Optimization of the Ecological Services Value of the Interconnected River System Network Project in the Western Jilin Province, China

Incremental ecological service value is an important assessment indicator for ecosystem restoration projects, such as the interconnected river system network project in western Jilin Province, China. In this paper, a fuzzy two-stage chance-constrained stochastic programming (FTSCCP) model is proposed for the optimization of regional incremental ecological service value through adjustments to the original water diversions and ecological water supplement schemes for ecosystem restoration under different management requirements and scenarios. The proposed method can effectively address uncertainties such as fuzzy parameters, probability distributions, and random variables in optimizing processes simultaneously. The results reveal that, under the requirement of meeting the original plan, a decrease in restoration scale would mainly appear under low-flow scenarios, and for the other scenarios, the original plan would be fully realized with an increase in flood utilization and a decrease in normal water supplementation. For the requirement of maximizing the incremental ecological service value, the adjustments would mainly be increases in water supplementation for reed wetlands and marsh wetlands and decreases in water surface area, including fish ponds and crab ponds. The ecological service values would be 11,256.79 million CNY (Chinese Yuan), 16,465.11 million CNY and 17,639.59 million CNY under low-, normal- and high-flow scenarios, respectively, with increases of 3.2%, 6.9% and 14.5% compared to the values based on the requirement of meeting the original plan. The results would be helpful for guiding regional ecosystem restoration and water resource management in the western Jilin Province.

Simulation based evolutionary algorithms for fuzzy chance-constrained biogas supply chain design

This study proposes a fuzzy chance-constrained programming model to include uncertainty in the biogas supply chain design problem. Uncertain parameters of the model are available workforce, biomass demand, available biomass and biomass price. A hybrid solution framework consisting of Monte Carlo simulation and evolutionary algorithms (genetic algorithm and differential evolution) is put forward to find the exact and near global optimal solution for the fuzzy chance-constrained model. The results of the test problems show that evolutionary algorithms can effectively solve the mixed integer nonlinear model of biogas location allocation within a reasonable computational time. Also, validation of the hybrid solution framework at different confidence levels is verified. The impacts of uncertainty in available biomass, biomass demand and available workforce on the overall cost of biogas supply chain are studied through sensitivity analysis. A real-world case study with real-life data available from the Province of Khorasan Razavi is performed. This is the first study that designs a biogas supply chain for a province of Iran.

Inexact Fuzzy Chance-Constrained Fractional Programming for Sustainable Management of Electric Power Systems

An inexact fuzzy chance-constrained fractional programming model is developed and applied to the planning of electric power systems management under uncertainty. An electric power system management system involves several processes with socioeconomic and environmental influenced. Due to the multiobjective, multilayer and multiperiod features, associated with these various factors and their interactions extensive uncertainties, may exist in the study system. As an extension of the existing fractional programming approach, the inexact fuzzy chance-constrained fractional programming can explicitly address system uncertainties with complex presentations. The approach can not only deal with multiple uncertainties presented as random variables, fuzzy sets, interval values, and their combinations but also reflect the tradeoff in conflicting objectives between greenhouse gas mitigation and system economic profit. Different from using least-cost models, a more sustainable management approach is to maximize the ratio between clean energy power generation and system cost. Results of the case study indicate that useful solutions for planning electric power systems management practices can be generated.

A Time-Dependent Fuzzy Programming Approach for the Green Multimodal Routing Problem with Rail Service Capacity Uncertainty and Road Traffic Congestion

This study explores an operational-level container routing problem in the road-rail multimodal service network. In response to the demand for an environmentally friendly transportation, we extend the problem into a green version by using both emission charging method and bi-objective optimization to optimize the CO2 emissions in the routing. Two uncertain factors, including capacity uncertainty of rail services and travel time uncertainty of road services, are formulated in order to improve the reliability of the routes. By using the triangular fuzzy numbers and time-dependent travel time to separately model the capacity uncertainty and travel time uncertainty, we establish a fuzzy chance-constrained mixed integer nonlinear programming model. A linearization-based exact solution strategy is designed, so that the problem can be effectively solved by any exact solution algorithm on any mathematical programming software. An empirical case is presented to demonstrate the feasibility of the proposed methods. In the case discussion, sensitivity analysis and bi-objective optimization analysis are used to find that the bi-objective optimization method is more effective than the emission charging method in lowering the CO2 emissions for the given case. Then, we combine sensitivity analysis and fuzzy simulation to identify the best confidence value in the fuzzy chance constraint. All the discussion will help decision makers to better organize the green multimodal transportation.

Solving the green-fuzzy vehicle routing problem using a revised hybrid intelligent algorithm

Green logistics is an emerging area in supply chain management, which has been shown to have tremendous impacts in recent years to face the serious climate changes risks. In this paper, the fuel consumption and fuzzy travel time have been delineated in developing and solving the green-fuzzy vehicle routing problem as an extension of the celebrated VRP in which routes are performed to reduce the total expenditure. Different from the existing solution manners, we transform the original fuzzy chance constrained programming model into an equivalent deterministic model, and then revise the original hybrid intelligent algorithm by replacing the embedded fuzzy simulation with analytical function calculation. Finally, a comparative study with the corresponding literature is performed, which shows that the revised algorithm can not only improve the solution accuracy but also shorten the runtime greatly.

Inexact fuzzy chance-constrained programming for community-scale urban stormwater management

Abstract Due to frequent and serious waterlog and environmental pollution in cities in recent years, urban stormwater management (USM) has become an essential issue of urban sustainable development in China. Low Impact Development (LID) technologies are effective and popular measures of USM, and can be used to reduce urban waterlog and the associated environmental pollution. In this paper, in order to identify an optimal strategy of LID technologies based on economic efficiencies and environmental performances, the construction costs and environmental benefits of the four LID technologies (including grass swales, bioretention cells, green roofs, and permeable pavement) were analyzed. Then, due to the uncertain feature of rainfall, concentration of pollutants and many social-economic factors, an inexact fuzzy chance-constrained programming (IFCCP) model was developed. Multiple uncertainties that can be expressed as interval parameters, fuzzy sets, and stochastic distribution can be addressed effectively and incorporated directly into the modeling process. Based on the analysis of the precipitation probability distribution and land use data, the developed model was applied to a university campus with relatively high building density and plot ratio in Beijing. Then, optimal solutions of multiple precipitation probabilities ( p m ) under varying USM goals were obtained. The four LID technologies were chosen to meet multiple USM goals based on multiple p m values. When p m = 0.1 , 0.05 , 0.02 and 0.015 , the total construction investments would be 1.46 to 4.66, 1.46 to 5.25, 2.31 to 8.54 and 3.95 to 12.7 million yuan, respectively. The relationship between construction costs and p m represented the relationship between economic benefit and system risk. The results indicated that USM was influenced by the rainfall capacity, construction costs would increase with the increase of precipitation, and the risks of constraint violation would decrease as the construction costs increase. When considering the four LID technologies during the planning stage of USM, the preferred selection order would be bioretention cells, permeable pavements, grass swales and green roofs. Therefore, the solutions under each p m level could provide the references for desired USM plans.

How Uncertain Information on Service Capacity Influences the Intermodal Routing Decision: A Fuzzy Programming Perspective

Capacity uncertainty is a common issue in the transportation planning field. However, few studies discuss the intermodal routing problem with service capacity uncertainty. Based on our previous study on the intermodal routing under deterministic capacity consideration, we systematically explore how service capacity uncertainty influences the intermodal routing decision. First of all, we adopt trapezoidal fuzzy numbers to describe the uncertain information of the service capacity, and further transform the deterministic capacity constraint into a fuzzy chance constraint based on fuzzy credibility measure. We then integrate such fuzzy chance constraint into the mixed-integer linear programming (MILP) model proposed in our previous study to develop a fuzzy chance-constrained programming model. To enable the improved model to be effectively programmed in the standard mathematical programming software and solved by exact solution algorithms, a crisp equivalent linear reformulation of the fuzzy chance constraint is generated. Finally, we modify the empirical case presented in our previous study by replacing the deterministic service capacities with trapezoidal fuzzy ones. Using the modified empirical case, we utilize sensitivity analysis and fuzzy simulation to analyze the influence of service capacity uncertainty on the intermodal routing decision, and summarize some interesting insights that are helpful for decision makers.