Fuzzy Mixed Integer Nonlinear Programming Research Articles

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.

Designing a Reverse Logistics Network for End-of-Life Vehicles in an Uncertain Environment

The strategic development of reverse logistics networks is crucial for addressing the common challenge of low recovery rates for end-of-life vehicles (ELVs) in China. To minimize the total cost of the reverse logistics network for ELVs, this paper proposes a mixed-integer linear programming (MILP) model. The model considers the recycling volume of different vehicle types, facility processing capacity, and the proportions of parts and materials. Building on this foundation, a fuzzy mixed-integer nonlinear programming (FMINLP) model is developed to account for the inherent uncertainty associated with recycling volumes and facility processing capacities. The model was solved using Lingo, and its effectiveness was validated using Jiangsu Province of China as a case study, followed by a sensitivity analysis. The results indicate that dismantling and machining centers incur the highest processing costs. Variations in recycling volume and facility handling capacity significantly impact total costs and site selection, with the former having a more pronounced effect. Increasing facility processing capacity effectively increases the recovery rate. Moreover, a higher confidence level corresponds to higher total costs and a greater demand for facilities.

Fuzzy Approaches and Simulation-Based Reliability Modeling to Solve a Road–Rail Intermodal Routing Problem with Soft Delivery Time Windows When Demand and Capacity are Uncertain

In this study, a freight routing problem considering both soft delivery time windows and demand and capacity uncertainty in a road–rail intermodal transportation system is investigated. According to fuzzy set theory, uncertain demands and capacities are formulated as trapezoidal fuzzy numbers. Soft delivery time windows under a fuzzy environment is established, in which fuzzy periods caused by early and late deliveries that lead to penalty are modeled based on maximum functions. To solve the routing problem yielding the above characteristics, this study designs a fuzzy mixed-integer nonlinear programming model whose objective is to minimize the total costs created in the road–rail intermodal transportation activities. After using the fuzzy expected value method to address the fuzzy objective, two fuzzy approaches, i.e., fuzzy chance-constrained programming method and fuzzy ranking method, are separately adopted to undertake the defuzzification of the fuzzy constraints. Improved linear formulations of the model are then produced to make it easier to solve. A simulation-based reliability modeling is developed to quantify the reliability of the optimization results given by different fuzzy approaches under different parameter settings in a simulation environment. Finally, an empirical case is presented to verify the feasibility of the proposed methods. The effects of demand and capacity fuzziness on the routing optimization are revealed, and an optimization procedure that helps decision-makers to select a more suitable fuzzy approach and determine the best parameter setting for a given case is demonstrated. Some insights that are helpful for organizing a reliable transportation are also drawn.

A fuzzy mid-term capacity and production planning model for a manufacturing system with cloud-based capacity

Most of the past cloud manufacturing (CMfg) studies investigated the short-term production planning or job scheduling of a CMfg system, while the mid-term or long-term capacity and production planning of a CMfg system has rarely been addressed. In addition, most existing methods are suitable for CMfg systems comprising three-dimensional (3D) printers, computer numerical control (CNC) machines or robots, but ignore the coordination and transportation required for moving jobs across factories. To fill these gaps, a fuzzy mid-term capacity and production planning model for a manufacturer with cloud-based capacity is proposed in this study. The proposed methodology guides a manufacturer in choosing between non-cloud-based capacity and cloud-based capacity. It can be applied to factories utilizing machines with different degrees of automation including highly automatic equipment (such as 3D printers, CNC machines, and robots) and lowly automatic (legacy) machines, while existing methods assume that orders can be easily transferred between machines that are often highly automatic. In the proposed methodology, first, various types of capacity are unequally prioritized. Then, a fuzzy mixed-integer nonlinear programming model is formulated and optimized to make the mid-term or long-term capacity and production plan of a factory. The fuzzy capacity and production planning model is designed for factories with parallel machines. The proposed methodology has been applied to a case to illustrate its applicability. According to the experimental results, the proposed methodology successfully reduced total costs by up to 8%. The advantage of the proposed methodology over existing practices in fulfilling customers’ demand was also obvious.

Green and Reliable Freight Routing Problem in the Road-Rail Intermodal Transportation Network with Uncertain Parameters: A Fuzzy Goal Programming Approach

In this study, the author focuses on modeling and optimizing a freight routing problem in a road-rail intermodal transportation network that combines the hub-and-spoke and point-to-point structures. The operations of road transportation are time flexible, while rail transportation has fixed departure times. The reliability of the routing is improved by modeling the uncertainty of the road-rail intermodal transportation network. Parameters that are influenced by the real-time status of the network, including capacities, travel times, loading and unloading times, and container trains’ fixed departure times, are considered uncertain in the routing decision-making. Based on fuzzy set theory, triangular fuzzy numbers are employed to formulate the uncertain parameters as well as resulting uncertain variables. Green routing is also discussed by treating the minimization of carbon dioxide emissions as an objective. First of all, a multiobjective fuzzy mixed integer nonlinear programming model is established for the specific reliable and green routing problem. Then, defuzzification, linearization, and weighted sum method are implemented to present a crisp linear model whose global optimum solutions can be effectively obtained by the exact solution algorithm run by mathematical programming software. Finally, a numerical case is given to demonstrate how the proposed methods work. In the case, sensitivity analysis is adopted to reveal the effects of uncertainty on the routing optimization. Fuzzy simulation is then performed to help decision makers to select the best crisp route plan by determining the best confidence level shown in the fuzzy chance constraints.

Fuzzy Programming Approaches for Modeling a Customer-Centred Freight Routing Problem in the Road-Rail Intermodal Hub-and-Spoke Network with Fuzzy Soft Time Windows and Multiple Sources of Time Uncertainty

In this study, we systematically investigate a road-rail intermodal routing problem the optimization of which is oriented on the customer demands on transportation economy, timeliness and reliability. The road-rail intermodal transportation system is modelled as a hub-and-spoke network that contains time-flexible container truck services and scheduled container train services. The transportation timeliness is optimized by using fuzzy soft time windows associated with the service level of the transportation. Reliability is enhanced by considering multiple sources of time uncertainty, including road travel time and loading/unloading time. Such uncertainty is modelled by using fuzzy set theory. Triangular fuzzy numbers are adopted to represent the uncertain time. Under the above consideration, we first establish a fuzzy mixed integer nonlinear programming model with a weighted objective that includes minimizing the costs and maximizing the service level for accomplishing transportation orders. Then we use the fuzzy expected value model and fuzzy chance-constrained programming separately to realize the defuzzification of the fuzzy objective and use fuzzy chance-constrained programming to deal with the fuzzy constraint. After defuzzification and linearization, an equivalent mixed integer linear programming (MILP) model is generated to enable the problem to be solved by mathematical programming software. Finally, a numerical case modified from our previous study is presented to demonstrate the feasibility of the proposed fuzzy programming approaches. Sensitivity analysis and fuzzy simulation are comprehensively utilized to discuss the effects of the fuzzy soft time windows and time uncertainty on the routing optimization and help decision makers to better design a crisp transportation plan that can effectively make tradeoffs among economy, timeliness and reliability.

A Bi-Objective Fuzzy Credibilistic Chance-Constrained Programming Approach for the Hazardous Materials Road-Rail Multimodal Routing Problem under Uncertainty and Sustainability

Hazardous materials transportation involves extensive risk and cannot be avoided in practice. An advanced routing, however, can help to reduce the risk by planning the best transportation routes for hazardous materials that can make effective tradeoffs between the risk objective and the economic objective. In this study, we explore the hazardous materials routing problem in the road-rail multimodal transportation network with a hub-and-spoke structure, in which the risk is measured by the multiplication of population exposure and the associated volume of hazardous materials, and minimizing the total risk of all the transportation orders of hazardous materials is set as the risk objective. It is difficult to estimate the population exposure exactly during the routing decision-making process, which results in its uncertainty. In this study, we formulate the uncertain population exposure from a fuzzy programming perspective by using triangular fuzzy numbers. Moreover, the carbon dioxide emission constraint is formulated to realize the sustainable transportation of hazardous materials. To optimize the problem under the above framework, we first establish a bi-objective fuzzy mixed integer nonlinear programming model, and then develop a three-stage exact solution strategy that the combines fuzzy credibilistic chance constraint, linearization technique, and the normalized weighting method. Finally, a computational experiment is carried out to verify the feasibility of the proposed method in dealing with the problem. The experimental results indicate that tradeoffs between the two conflicting objectives can be effectively made by using the Pareto frontier to the hazardous materials routing problem. Furthermore, the credibility level and carbon dioxide emission cap significantly influence the hazardous materials routing optimization. Their effects on the optimization result are quantified by using sensitivity analysis, which can draw some useful insights to help decision makers to better organize the hazardous materials road-rail multimodal transportation under uncertainty and sustainability.

A Fuzzy Programming Method for Modeling Demand Uncertainty in the Capacitated Road–Rail Multimodal Routing Problem with Time Windows

Demand uncertainty is an important issue that influences the strategic, tactical, and operational-level decision making in the transportation/logistics/supply chain planning. In this study, we explore the effect of demand uncertainty on the operational-level freight routing problem in the capacitated multimodal transportation network that consists of schedule-based rail transportation and time-flexible road transportation. Considering the imprecise characteristic of the demand, we adopt fuzzy set theory to model its uncertainty and use trapezoidal fuzzy numbers to represent the fuzzy demands. We set multiple transportation orders as the optimization object and employ soft time windows to reflect the customer requirement on on-time transportation. Under the above situation, we establish a fuzzy mixed integer nonlinear programming (FMINLP) model to formulate the capacitated road–rail multimodal routing problem with demand uncertainty and time windows. We first use the fuzzy expected value model and credibility measure based fuzzy chance-constrained programming to realize the defuzziness of the model and then adopt linearization technique to reformulate the crisp model to finally generate an equivalent mixed integer linear programming (MILP) model that can be solved by standard mathematical programming software. Finally, a numerical case is presented to demonstrate the feasibility of the proposed method. Sensitivity analysis and fuzzy simulation are combined to quantify the effect of demand uncertainty on the routing problem and also reveal some helpful insights and managerial implications.

Fuzzy resource-constrained project scheduling with multiple routes: A heuristic solution

Abstract The resource-constrained project scheduling problem (RCPSP) with multiple routes by considering flexible activities is one of important subjects in project scheduling problems. The ability to select an appropriate route for implementing the flexible activities is a rational reason for indicating more complexity of the problem relative to common RCPSP that attracted the attention of the researchers in the recent decade. On the other hand, due to lack of access to project crisp information, the needs to consider uncertainty concepts in the RCPSP will be significant. Hence, in this paper, a new fuzzy mixed integer nonlinear programming (MINLP) model is presented under uncertain conditions. A hybrid meta-heuristic approach is also proposed to minimize costs of project completion. In this approach, to generate high quality initial solutions, a heuristic algorithm is designed based on distribution rules. Then, to change and assign an appropriate route from available routes for flexible activities, a meta-heuristic algorithm is presented based on binary particle swarm optimization (PSO). Finally, to generate best solution from routes assigned by the binary PSO, a meta-heuristic based on genetic algorithm (GA) is proposed. To appraise the effectiveness of presented model, different test problems are solved by the proposed approach, and comparisons are provided with results obtained by the GA and PSO.

A fuzzy dynamic multi-objective, multi-item model by considering customer satisfaction in supply chain

Customer satisfaction is an important issue in competitive strategic management of companies. Supply chain logistical and cross-functional drivers have an important role to manage customer satisfaction. Customer satisfaction depends on quality, cost and delivery. In this paper a fuzzy mixed integer nonlinear programming model is proposed for a multi-item multi-period problem in multi-level supply chain. Minimizing costs, manufacturing and transportation time, transportation risks, maximizing quality by minimizing the number of defective products and maximizing customers’ service levels are considered to be objective functions of the model. Furthermore, it is assumed that the demand rates are fuzzy values. An exact -constraint approach is used to solve the problem. The problem is computationally intractable. Therefore, the Non-dominant Sorting Genetic Algorithm (NSGA-II) is developed to solve it. The Taguchi method is utilized to tune the NSGA-II parameters. Finally, some numerical examples are generated and solved to evaluate the performance of the proposed model and solving methods.

Single machine scheduling problems under position-dependent fuzzy learning effect with fuzzy processing times

In this paper, we consider single machine scheduling problems under position-dependent fuzzy learning effect with fuzzy processing times. We study three objectives which are to minimize makespan, total completion time and total weighted completion time. Furthermore, we show that these three problems are polynomially solvable under position-dependent fuzzy learning effects with fuzzy processing times. In order to model the uncertainty of fuzzy model parameters such as processing time and learning effect, we use an approach called likelihood profile that depends on the possibility and necessity measures of fuzzy parameters. For three objective functions, we build Fuzzy Mixed Integer Nonlinear Programming (FMINP) models using dependent chance constrained programming techniques for the same predetermined confidence levels. Furthermore, we present polynomially solvable algorithms for different confidence levels for these problems.

A hybrid multi-objective evolutionary algorithm approach for handling sequence- and machine-dependent set-up times in unrelated parallel machine scheduling problem

This paper addresses a fuzzy mixed-integer non-linear programming (FMINLP) model by considering machine-dependent and job-sequence-dependent set-up times that minimize the total completion time, the number of tardy jobs, the total flow time and the machine load variation in the context of unrelated parallel machine scheduling (UPMS) problem. The above-mentioned multi-objectives were considered based on non-zero ready times, machine- and sequence-dependent set-up times and secondary resource constraints for jobs. The proposed approach considers unrelated parallel machines with inherent uncertainty in processing times and due dates. Since the problem is shown to be NP-hard in nature, it is a challenging task to find the optimal/near-optimal solutions for conflicting objectives simultaneously in a reasonable time. Therefore, we introduced a new multi-objective-based evolutionary artificial immune non-dominated sorting genetic algorithm (AI-NSGA-II) to resolve the above-mentioned complex problem. The performance of the proposed multi-objective AI-NSGA-II algorithm has been compared to that of multi-objective particle swarm optimization (MOPSO) and conventional non-dominated sorting genetic algorithm (CNSGA-II), and it is found that the proposed multi-objective-based hybrid meta-heuristic produces high-quality solutions. Finally, the results obtained from benchmark instances and randomly generated instances as test problems evince the robust performance of the proposed multi-objective algorithm.

RETRACTED: Apply Fuzzy Learning Effect with Fuzzy Processing Times for Single Machine Scheduling Problems

Most common objectives in single-machine scheduling problems are minimize makespan, total completion time and total weighted completion time, respectively. In this paper, we consider single machine scheduling problems under position-dependent fuzzy learning effect with fuzzy processing times. Furthermore, we show that these three problems are polynomially solvable under position dependent fuzzy learning effect with fuzzy processing times. In order to model the uncertainty of fuzzy model parameters such as processing time and learning effect, we use an approach called as likelihood profile that dependent on the possibility and necessity measures of fuzzy parameters. For three objective functions, we build Fuzzy Mixed Integer Nonlinear Programming (FMINP) models those are dependent chance constrained programming technique for the same predetermined confidence levels. Furthermore, we present polynomially solvable algorithms for different confidence levels for these problems.

Fuzzy mixed integer non-linear programming model for the design of an algae-based eco-industrial park with prospective selection of support tenants under product price variability

Eco-industrial parks provide a platform for the application of industrial symbiosis where the synergistic network of companies reuse portions of their by-products to reduce disposed waste, reduce environmental emissions, and improve plant efficiency. However, designing a complex network of material and energy exchanges between companies in an industrial park while satisfying multiple conflicting objectives require a systematic design methodology. In addition, strategic decision-making in an eco-industrial park involves the selection of prospective companies (i.e., support tenants), which complement the existing companies (i.e., anchor tenants). In this study, a fuzzy mixed-integer non-linear programming model is proposed to select prospective support tenants in an eco-industrial park while satisfying the product demand, minimizing the environmental footprint of the eco-industrial park, and also maximizing the annualized profit of each company in the eco-industrial park. A hypothetical but realistic case study involving an algae-based eco-industrial park is used to demonstrate the application of the model. The results demonstrate the selection of the appropriate support tenants for the algae-based eco-industrial park together with the optimal plant configuration. Sensitivity analysis is used to assess the performance of the algae-based eco-industrial park with respect to the changes in prices of the by-products. The developed model thus aid the planners of an eco-industrial park in assessing which among the prospective support tenants would best complements an existing anchor tenant. Furthermore, the model can also identify price negotiation points between tenants for some product streams which may show sensitivity on the plant capacity of each tenant.

Fuzzy optimization of topologically constrained eco-industrial resource conservation networks with incomplete information

It is possible to minimize industrial resource consumption by establishing eco-industrial resource conservation networks (RCN) between different plants. The establishment of these networks requires the satisfaction of quality criteria for material properties deemed significant by an industry. It also necessitates cooperation among the different firms based on the satisfaction of individual cost or resource consumption goals. Furthermore, there may be varying degrees of incomplete information regarding the process data of the participating plants. Eco-industrial RCNs may also be topologically constrained with respect to the number of links connecting different plants. These design aspects are incorporated in the optimization model through fuzzy mixed integer linear programming (FMILP) or fuzzy mixed integer non-linear programming (FMINLP). Case studies from literature involving water integration and hydrogen recovery are used to illustrate the methodology. The model is able to identify the topologically constrained network that achieves the highest level of overall satisfaction among participating plants.

Optimising multi-product multi-chance-constraint inventory control system with stochastic period lengths and total discount under fuzzy purchasing price and holding costs

While the usual assumptions in multi-periodic inventory control problems are that the orders are placed at the beginning of each period (periodic review) or depending on the inventory level they can happen at any time (continuous review), in this article, we relax these assumptions and assume that the periods between two replenishments of the products are independent and identically distributed random variables. Furthermore, assuming that the purchasing price are triangular fuzzy variables, the quantities of the orders are of integer-type and that there are space and service level constraints, total discount are considered to purchase products and a combination of back-order and lost-sales are taken into account for the shortages. We show that the model of this problem is a fuzzy mixed-integer nonlinear programming type and in order to solve it, a hybrid meta-heuristic intelligent algorithm is proposed. At the end, a numerical example is given to demonstrate the applicability of the proposed methodology and to compare its performance with one of the existing algorithms in real world inventory control problems.

Multi-Product Multi-Constraint Inventory Control Systems with Stochastic Replenishment and Discount under Fuzzy Purchasing Price and Holding Costs

While in multi-periodic inventory control problems the usual assumption are that the orders are placed at the beginning of each period (periodic review) or depending on the inventory level they can happen at any time (continuous review), in this research, we relax these assumptions and assume that the periods between two replenishments of the products are independent and identically distributed random variables. Furthermore, assuming the purchasing price are triangular fuzzy variables, the quantities of the orders are of integer-type and that there are space, budget and service level constraints, incremental discount is considered to purchase products and a combination of back-order and lost-sales are taken into account for the shortages. We show that the model of this problem is a fuzzy mixed-integer nonlinear programming type and in order to solve it, a hybrid method of fuzzy simulation and genetic algorithm approach is used. At the end, a numerical example is given to demonstrate the applicability of the proposed methodology in real world inventory control problems.