I-VFRP: An interval-valued fuzzy robust programming approach for municipal waste-management planning under uncertainty

In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorporated into the modeling formulation and solution process. The results also suggest that the proposed methodology can be applied to practical problems that are associated with complex and uncertain information.     

A fuzzy linear programming approach for municipal solid-waste management under uncertainty

In this study, a fuzzy linear programming (FLP) method is developed for dealing with uncertainties expressed as fuzzy sets that exist in the constraints’ left-hand and right-hand sides and the objective function. A direct transforming algorithm is advanced for solving the FLP model that improves upon the existing method through provision of a quantitative expression for uncertain relationships among a large number of fuzzy sets. The proposed solution method can greatly reduce computational requirements, which is particularly meaningful for the application of FLP to large-scale practical problems with many fuzzy sets. The developed FLP method is applied to a case of long-term waste-management planning. The results indicate that reasonable solutions have been obtained. They can be used for generating decision alternatives and to help managers identify desired policies for waste management under uncertainty. Compared with the conventional interval-parameter linear programming approach, FLP can provide more information for solutions, containing not only the lower and upper bounds but also the most possible value for decision variables and objective function.     

Task assignment under uncertainty: stochastic programming and robust optimisation approaches

The assignment of tasks to teams is a challenging combinatorial optimisation problem. The uncertainty in the tasks’ execution processes further complicates the assignment decisions. This study investigates a variant of the typical assignment problem, in which each task can be divided into two parts, one is deterministic and the other is uncertain with respect to their workloads. From the stochastic perspective, this paper proposes both a stochastic programming model that can cope with arbitrary probability distributions of tasks’ random workload requirements, and a robust optimisation model that is applicable to situations in which limited information about probability distributions is available. An example of its application in the software project management is given. Some numerical experiments are also performed to validate the effectiveness of the proposed models and the relationships between the two models.     

A hybrid interval-parameter fuzzy robust programming method and its application to filter management strategy in fluid power systems

An interval-parameter fuzzy robust programming (IFRP) method is developed for the assessment of filter allocation and replacement strategies in a fluid power system (FPS) under uncertainty. The developed IFRP can effectively handle the uncertainties expressed as fuzzy sets, interval values, and their combinations, which exist in contaminant ingression/generation of the system and contaminant-holding capacity of filter without making assumptions on their probabilistic distributions. The fuzzy decision space can be delimited into a more robust one with the uncertainties being specified through dimensional enlargement of the original fuzzy constraints, leading to enhanced robustness for the optimization process. Results indicate that the developed IFRP can not only help decision-maker to identify optimal filter allocation and replacement strategies to control the contamination level of FPS with a minimized system-cost and system-failure risk under multiple uncertainties, but also mitigate uncertainties through abating interval widths of the replacement periods and service life under different contamination ingression/generation rates.     

A generalized interval fuzzy mixed integer programming model for a multimodal transportation problem under uncertainty

A generalized interval fuzzy mixed integer programming model is proposed for the multimodal freight transportation problem under uncertainty, in which the optimal mode of transport and the optimal amount of each type of freight transported through each path need to be decided. For practical purposes, three mathematical methods, i.e. the interval ranking method, fuzzy linear programming method and linear weighted summation method, are applied to obtain equivalents of constraints and parameters, and then a fuzzy expected value model is presented. A heuristic algorithm based on a greedy criterion and the linear relaxation algorithm are designed to solve the model.     

Part-period balancing with uncertainty: a fuzzy sets theory approach

The part-period balancing lot-sizing algorithm is modified to use fuzzy data for the single-stage lot-sizing problem. Triangular fuzzy numbers are used to represent uncertainty in the master production schedule. This paper shows that uncertain demand can be easily incorporated into the part-period balancing lot-sizing algorithm and that a fuzzy master production schedule can be used to determine production lot sizes. A detailed example is presented to illustrate the technique.     

Lending decisions under uncertainty: a DEA approach

Credit markets have experienced phenomenal growth in the last years, and there is no evidence to expect that this trend will invert in the next decade. Given that, virtually every segment of the industry is called to improve its skills in credit risk management with the threefold aim of managing concentration risk, meeting regulatory requirements and enhancing revenues. Clearly, for financial institutions, exposure to credit risk continues to be the leading source of problems and the challenging issue of managing credit risk is considered a crucial topic. This paper demonstrates the use of a stochastic DEA model for credit scoring which is one of the prevailing analytical technique to evaluate credit risk. Although the principles contained in this paper are most clearly applicable to the business of lending, they can be applied to all activities where credit risk is present.     

基于遗传算法解决河流环境需水量优化调度的模糊多目标规划方法的研究

为实现城市受污染河流的环境需水量的合理调配,以各蓄水构筑物的调水量为决策变量,研究了在保证河流生态环境需求的基础上,实现整个流域的污染负荷的环境容量最大化、水量最大节约和调水成本最小化的方法,建立了环境需水调度的模糊多目标规划模型.模型中采用整合的非线性梯度隶属度函数作为目标函数,以遗传算法作为求解此模型的工具,并将该模型用于大庆市黎明河流域的环境水量调配,取得较好的效果.     

A fuzzy chance-constrained programming model with type 1 and type 2 fuzzy sets for solid waste management under uncertainty

Effective management of municipal solid waste (MSW) is critical for urban planning and development. This study aims to develop an integrated type 1 and type 2 fuzzy sets chance-constrained programming (ITFCCP) model for tackling regional MSW management problem under a fuzzy environment, where waste generation amounts are supposed to be type 2 fuzzy variables and treated capacities of facilities are assumed to be type 1 fuzzy variables. The evaluation and expression of uncertainty overcome the drawbacks in describing fuzzy possibility distributions as oversimplified forms. The fuzzy constraints are converted to their crisp equivalents through chance-constrained programming under the same or different confidence levels. Regional waste management of the City of Dalian, China, was used as a case study for demonstration. The solutions under various confidence levels reflect the trade-off between system economy and reliability. It is concluded that the ITFCCP model is capable of helping decision makers to generate reasonable waste-allocation alternatives under uncertainties.     

IFQP: A hybrid optimization method for filter management in fluid power systems under uncertainty

An interval-fuzzy quadratic programming (IFQP) method is developed for the assessment of filter allocation and replacement strategies in fluid power systems (FPS) under uncertainty. It can directly handle uncertainties expressed as interval values and/or fuzzy sets that exist in the left-hand and right-hand sides of constraints, as well as in the objective function. Multiple control variables are used to tackle independent uncertainties in the model's right-hand sides and thus optimize the overall satisfaction of the system performance. The IFQP method is applied to a case of planning filter allocation and replacement strategies under uncertainty for an FPS with a single circuit. A piecewise linearization approach is firstly employed to convert the nonlinear FPS problem into a linear one. The generated decision alternatives can help decision makers to identify desired policies for contamination control under various total costs, satisfaction degrees, and system-failure risks under different contaminant-ingression/generation rates.     

A two-stage robust programming approach to demand-driven disassembly planning for a closed-loop supply chain system

The closed-loop supply chain system, which integrates forward and reverse logistics, is a desirable policy for retaining recoverable resources and extending the life cycles of products. In this study, we propose a methodology to contend with a demand-driven disassembly planning problem under a closed-loop supply chain system. A two-stage robust programming model is developed correspondingly, such that multiple products with a hierarchical product's structure are disassembled to satisfy uncertain demands in multiple periods. The objective of the model is to determine a robust decision for recycle volume and timing of each type of end-of-life (EOL) product, as well as recovery strategies. The results provide two-stage decisions by considering future scenarios of periodic demands at the beginning of a planning horizon. The first-stage decision is to determine a compromise solution that is close to the optimal solution for every scenario while retaining a certain level of infeasibility of constraints, such as unsatisfied demand. Afterward, when the outcome of a scenario has been realised, the second-stage decision, such as, inventory volume, is conducted to become a buffer for mitigating uncertain impacts. Furthermore, the computational results confirm the trade-off relationship between solution robustness and model robustness, which are core results of the robust model apart from expected profit. The different types of decision makers’ preferences toward risk can be accounted for to determine a compromise robust solution.     

Forest harvesting planning under uncertainty: a cardinality-constrained approach

Harvesting planning (HP) is a key tactical decision in lumber supply chains. Harvesting areas in the forests are divided into different blocks with different types and quantities of raw materials (logs). Predicting the availability of raw materials in each block along with log demand is impossible in this industry. Hence, incorporating uncertainty into the HP problem is essential in order to obtain robust plans that do not drastically fluctuate in the presence of future perturbations in the forest and log market. In this paper, we propose a robust harvesting planning model formulated based on cardinality-constrained method. The latter provides some insights into the adjustment of the level of robustness of the harvesting plan over the planning horizon and protection against uncertainty. An extensive set of experiments based on Monte-Carlo simulation is also conducted in order to better validate the proposed robust optimisation approach.     

Multi-site integrated production-distribution planning with trans-shipment: a fuzzy goal programming approach

In multi-site (parallel) manufacturing systems, each selling region (SR) is usually allocated to a specific manufacturing site (MS) in order to minimise the transportation costs, simplify the planning process, etc. However, such manufacturing systems usually encounter great drawbacks in terms of over-achievement or under-achievement of the forecasted demand due to isolation of the MSs. To cope with this drawback, this paper proposes a novel framework for preparing an optimal aggregate production plan by interconnecting the MSs through lateral trans-shipment. In the presence of fluctuating and dynamic demands, we apply some classic strategies including the inventory holding, back-ordering, and additional capacity options (i.e. overtime and changing workforce level) as well as lateral trans-shipment of products among MSs as the feasible strategies undertaken. Maximising the total profit and minimising the manufacturing lead time are considered as two managerial goals. Due to ambiguousness of some critical parameters as well as vagueness of objectives’ target values, a fuzzy goal programming (FGP) approach with imprecise goal hierarchy is developed for modelling the aggregate production-distribution planning (APDP) problem with trans-shipment. The proposed FGP model is then converted to an equivalent crisp one by combining the two recently developed fuzzy programming approaches. An illustrative example inspired by a real case study is provided to show the usefulness and applicability of the proposed model.     

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.     

Duality in fuzzy linear programming: a survey

The concepts of both duality and fuzzy uncertainty in linear programming have been theoretically analyzed and comprehensively and practically applied in an abundance of cases. Consequently, their joint application is highly appealing for both scholars and practitioners. However, the literature contributions on duality in fuzzy linear programming (FLP) are neither complete nor consistent. For example, there are no consistent concepts of weak duality and strong duality. The contributions of this survey are (1) to provide the first comprehensive overview of literature results on duality in FLP, (2) to analyze these results in terms of research gaps in FLP duality theory, and (3) to show avenues for further research. We systematically analyze duality in fuzzy linear programming along potential fuzzifications of linear programs (fuzzy classes) and along fuzzy order operators. Our results show that research on FLP duality is fragmented along both dimensions; more specifically, duality approaches and related results vary in terms of homogeneity, completeness, consistency with crisp duality, and complexity. Fuzzy linear programming is still far away from a unifying theory as we know it from crisp linear programming. We suggest further research directions, including the suggestion of comprehensive duality theories for specific fuzzy classes while dispensing with restrictive mathematical assumptions, the development of consistent duality theories for specific fuzzy order operators, and the proposition of a unifying fuzzy duality theory.     

Maintenance supplier selection considering life cycle costs and risks: a fuzzy goal programming approach

In this study, we address a new variant of supplier selection problem named maintenance supplier selection problem faced by a manufacturer. The production system consists of different multi-component equipments whose maintenance activities require several components (parts) each of which could be provided by multiple suppliers. A multi-objective mathematical model is developed to decide about the supply base of each part as well as the purchasing quantity of each part from each selected supplier. The model accounts for the total life cycle costs of purchased parts and various risks threatening the candidate suppliers. A fuzzy/soft lexicographic goal programming approach with soft priorities between objectives is proposed to enable the decision-maker to make preferred trade-offs between objectives by which the effects of various risks in each phase of life cycle of procured parts are investigated. The capability and effectiveness of the proposed model is validated through a case study. Some sensitivity analyses are also carried out for investigating the impact of cost, risk and objectives’ priorities on the final preferred compromise solution. Finally, some managerial insights and concluding remarks are provided.     

Horsetail matching: a flexible approach to optimization under uncertainty*

It is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated.     

A possibilistic programming approach for the location problem of multiple cross-docks and vehicle routing scheduling under uncertainty

This article considers the design of cross-docking systems under uncertainty in a model that consists of two phases: (1) a strategic-based decision-making process for selecting the location of cross-docks to operate, and (2) an operational-based decision-making process for vehicle routing scheduling with multiple cross-docks. This logistic system contains three echelons, namely suppliers, cross-docks and retailers, in an uncertain environment. In the first phase, a new multi-period cross-dock location model is introduced to determine the minimum number of cross-docks among a set of location sites so that each retailer demand should be met. Then, in the second phase, a new vehicle routing scheduling model with multiple cross-docks is formulated in which each vehicle is able to pickup from or deliver to more than one supplier or retailer, and the pickup and delivery routes start and end at the corresponding cross-dock. This article is the first attempt to introduce an integrated model for cross-docking systems design under a fuzzy environment. To solve the presented two-phase mixed-integer programming (MIP) model, a new fuzzy mathematical programming-based possibilistic approach is used. Furthermore, experimental tests are carried out to demonstrate the effectiveness of the presented model. The computational results reveal the applicability and suitability of the developed fuzzy possibilistic two-phase model in a variety of problems in the domain of cross-docking systems.     

Operation allocation and materials-handling system selection in a flexible manufacturing system: A sequential modelling approach

Materials-handling systems are an integrating component of manufacturing operations and as such must be considered within a common framework in manufacturing systems design. This work proposes a first approach to the simultaneous consideration of the operation allocation and the materials-handling system selection problems in a flexible manufacturing system. The objective of the operation allocation model is to select a group of machines where the operations of the part types will be performed and then to assign those operations to the selected machines. The operation allocation model interfaces with the materials-handling system selection model by providing input data in the form of the manufacturing operations to be performed at each machining centre. The selection of the materials-handling system is centred on the matching of the parts visiting a machining centre to perform a manufacturing operation and the abilities of the handling equipment to perform the required materials handling functions of those part types. The objective is to select an optimal group of materials-handling equipment to be assigned to a cell.     

Asset replacement under improving operating and capital costs: a practical approach

We analyse the serial asset replacement problem under incomplete data about technological change that that affects the capital cost, operating cost and salvage value of newer assets. We construct an efficient discrete-time algorithm for this problem in the case where only partial information is available about future costs of new assets. The algorithm is based on introducing a corrected annual capital recovery factor into the classic Economic Life method. It produces the same lifetime of the first asset as the benchmark infinite-horizon cost minimisation when the operating cost, capital cost and salvage value of new assets decrease proportionally. The algorithm has the same complexity as the Economic Life method but performs much better under improving technology. Numeric simulation demonstrates a superior efficiency of the suggested algorithm vs. existing methods in practical situations when only few discrete measurements of technological change are available.