Unit commitment ? a fuzzy mixed integer Linear Programming solution

Unit commitment (UC) of a large system is a complex puzzle with integer/continuous variables and numerous inter-temporal constraints. After deregulation, price offers submitted by GenCos are predominantly in the form of linear price quantity (PQ) pairs. A fuzzy UC formulation that uses price offers modeled as PQ pairs. This fuzzy linear optimisation formulation of UC is solved using a mixed integer linear programming (MILP) routine. In this formulation, start up cost is modelled using linear variables. The fuzzy formulation provides modeling flexibility, relaxation in constraint enforcement and allows the method to seek a practical solution. The use of MILP technique makes the proposed solution method rigorous and fast. The method is tested on a 24 h, 104-generator system demonstrating its speed and robustness gained by using the LP technique. A five-generator system is additionally used to create a see-through example demonstrating advantages of using the fuzzy optimisation model.     

Optimal vehicle-dispatch decisions for cement silos using a fuzzy multi-objective approach

In real dispatch problems for cement-silo trucks, input data or parameters, such as forecasting demand, resources, costs, and the objective function, are often imprecise or fuzzy because some information is incomplete, unavailable, or unobtainable. This work presents a novel fuzzy multi-objective linear programming (FMOLP) model that solves the cement-silo vehicle-dispatch problem in a fuzzy environment. This model is applied to solve multi-source, multi-product, multi-vehicle, and multi-ready-mixed-concrete (RMC) plant vehicle-dispatch problems with imprecise goals, input data, and parameters. This work elucidates the relationship between dispatch planning and RMC plants with a focus on the allocation of cement-silo trucks. This work uses a real cement study case to demonstrate the feasibility of the proposed model. The main contribution of this work is its fuzzy mathematical programming methodology for solving the cement-silo vehicle-dispatch problem in a fuzzy environment. The analytical results can help dispatchers analyse systematically the cost-effectiveness of vehicle-dispatch planning in practical applications.     

多产品供应商选择的模糊多目标整数规划模型

以成本、质量、交货为目标,考虑供应商供应能力、采购数量、供应数量的柔性、评价等级等约束,建立了多产品供应商选择的模糊多目标规划模型.采用降半梯形分布的隶属度函数将模糊多目标规划模型转化为单目标线性规划并求解,应用算例证明了模型的有效性和可行性.     

A rolling horizon approach for material requirement planning under fuzzy lead times

This paper proposes a fuzzy multi-objective integer linear programming (FMOILP) approach to model a material requirement planning (MRP) problem with fuzzy lead times. The objective functions minimise the total costs, back-order quantities and idle times of productive resources. Capacity constraints are included by considering overtime resources. Into the crisp MRP multi-objective model, we incorporate the possibility of occurrence of each uncertain lead time using fuzzy numbers. Then FMOILP is transformed into an auxiliary crisp mixed-integer linear programming model by a fuzzy goal programming approach for each fuzzy lead time combination. In order to defuzzify the set of solutions associated with each fuzzy lead time combination, a solution method based on the centre of gravity concept is addressed. Model validation with a numerical example is carried out by a novel rolling horizon procedure where uncertain lead times are updated during each planning period according to the centre of gravity obtained. For illustration purposes, the proposed solution approach is satisfactorily compared to a rolling horizon approach in which lead times are allocated when the possibility of occurrence is established at one.     

Mixed model disassembly line balancing problem with fuzzy goals

The collection of used products is the driving force of remanufacturing systems and enterprises can gain significant economic, technical and social benefits from recycling. All products are disassembled up to some level in remanufacturing systems. The best way to disassemble returned products is valid by a well-balanced disassembly line. In this paper, a mixed integer programming (MIP) model is proposed for a mixed model disassembly line balancing (MMDLB) problem with multiple conflicting objectives: (1) minimising the cycle time, (2) minimising the number of disassembly workstations and (3) providing balanced workload per workstation. In most real world MMDLB problems, the targeted goals of decision makers are frequently imprecise or fuzzy because some information may be incomplete and/or unavailable over the planning horizon. This study is the first in the literature to offer the binary fuzzy goal programming (BFGP) and the fuzzy multi-objective programming (FMOP) approaches for the MMDLB problem in order to take into account the vague aspirations of decision makers. An illustrative example based on two industrial products is presented to demonstrate the validity of the proposed models and to compare the performances of the BFGP and the FMOP approaches.     

Lexicographic fuzzy multi-objective model for minimisation of exceptional and void elements in manufacturing cell formation

This research proposes a lexicographic fuzzy multi-objective model based on perfect grouping for concurrent solving the part-family and machine-cell formation problems in a cellular manufacturing system. New simplified mathematical expressions of exceptional and void elements are proposed, opposing conventional quadratic and absolute functions. The main objectives of the proposed solution model, that is, the minimisation of both the number of exceptional elements and the number of void elements is defined by fuzzy goals as pre-emptive ordering. A lexicographic fuzzy goal model is developed to enhance cell performance and machine utilisation simultaneously. A satisfactory efficient solution can easily be obtained, and alternative solutions can also be generated by capturing flexibility of the proposed fuzzy multi-objective programming model. The formulated model can be solved by existing integer programming solvers. Finally, the evaluation of cell formation problems is briefly discussed to show the performance of the proposed model.     

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.     

A mixed integer goal programming model for discrete values of design requirements in QFD

A main feature of quality function deployment (QFD) planning process is to determine target values for the design requirements (DRs) of a product, with a view to achieving a higher level of overall customer satisfaction. However, in real world applications, values of DRs are often discrete instead of continuous. Therefore, a mixed integer linear programming (MILP) model considering discrete data is suggested. As opposed to the existing literature, the fulfilment levels of DRs are assumed to have a piece-wise linear relationship with cost; because, constraints of technology and resource rarely provides a linear relationship in manufacturing systems. In the proposed MILP model, we considered customer satisfaction as the only goal. But, QFD process may be necessary to optimise cost and technical difficulty goals as well as customer satisfaction. Therefore, by developing the MILP model with multi-objective decision making (MODM) approach, a novel mixed integer goal programming (MIGP) model is proposed to optimise these goals simultaneously. Finally, MILP model solution turns out to be a more realistic approach to real applications because piece-wise linear relationship is taken into account. The solution of MIGP model provided different alternative results to decision makers according to usage of the lexicographic goal programming (LGP) approach. The applicability of the proposed models in practice is demonstrated with a washing machine development problem.     

Multi-site production planning integrating procurement and distribution plans in multi-echelon supply chains: an interactive fuzzy goal programming approach

Motivated by a real case of an automobile company, this study proposes a multi-objective, multi-site production planning model integrating procurement and distribution plans in a multi-echelon supply chain network with multiple suppliers, multiple manufacturing plants and multiple distribution centres. The model incorporates four important conflicting objectives simultaneously: minimisation of the total cost of logistics, maximisation of the total value of purchasing, minimisation of defective items and minimisation of late deliveries subject to some realistic constraints. Due to the imprecise/fuzzy nature of the objectives’ aspiration levels and some critical data, an interactive fuzzy goal programming formulation is first developed. Then, a novel fuzzy approach is proposed to convert the FGP model into an auxiliary crisp formulation to find an efficient compromise solution. The proposed model and solution method are validated through some numerical tests. Computational results indicate the practicality and tractability of the proposed model and also the superiority of the proposed auxiliary crisp formulation in contrast to the current alternative fuzzy approaches.     

A multi-objective optimisation model to integrating flexible process planning and scheduling based on hybrid multi-objective simulated annealing

Process planning and production scheduling play important roles in manufacturing systems. In this paper we present a mixed integer linear programming (MILP) scheduling model, that is to say a slot-based multi-objective multi-product, that readily accounts for sequence-dependent preparation times (transition and set up times or machine changeover time). The proposed scheduling model becomes computationally expensive to solve for long time horizons. The aim is to find a set of high-quality trade-off solutions. This is a combinatorial optimisation problem with substantially large solution space, suggesting that it is highly difficult to find the best solutions with the exact search method. To account for this, the hybrid multi-objective simulated annealing algorithm (MOHSA) is proposed by fully utilising the capability of the exploration search and fast convergence. Two numerical experiments have been performed to demonstrate the effectiveness and robustness of the proposed algorithm.     

Redundancy allocation for multi-state systems using physical programming and genetic algorithms

This paper proposes a multi-objective optimization model for redundancy allocation for multi-state series–parallel systems. This model seeks to maximize system performance utility while minimizing system cost and system weight simultaneously. We use physical programming as an effective approach to optimize the system structure within this multi-objective optimization framework. The physical programming approach offers a flexible and effective way to address the conflicting nature of these different objectives. Genetic algorithm (GA) is used to solve the proposed physical programming-based optimization model due to the following three reasons: (1) the design variables, the number of components of each subsystems, are integer variables; (2) the objective functions in the physical programming-based optimization model do not have nice mathematical properties, and thus traditional optimization approaches are not suitable in this case; (3) GA has good global optimization performance. An example is used to illustrate the flexibility and effectiveness of the proposed physical programming approach over the single-objective method and the fuzzy optimization method.     

New approach for solving intuitionistic fuzzy multi-objective transportation problem

Multi-objective transportation problem (MOTP) under intuitionistic fuzzy (IF) environment is analysed in this paper. Due to the fluctuation of market scenario, we assume that the transportation cost, the supply and the demand parameters are not always precise. Hence, the parameters are imprecise, i.e., they are IF numbers. Considering the specific cut interval, the IF transportation cost matrix is converted to interval cost matrix in our proposed problem. Again, using the same concept, the IF supply and the IF demand of the MOTP are reduced to the interval form. Then the proposed MOTP is changed into the deterministic MOTP, which includes interval form of the objective functions. Two approaches, namely intuitionistic fuzzy programming and goal programming, are used to derive the optimal solutions of our proposed problem, and then the optimal solutions are compared. A numerical example is included to illustrate the feasibility and the applicability of the proposed problem. Finally, we present the conclusions with the future scopes of our study.     

Integrating production-transportation planning decision with fuzzy multiple goals in supply chains

This work presents a novel fuzzy multi-objective linear programming (f-MOLP) model for solving integrated production-transportation planning decision (PTPD) problems in supply chains in a fuzzy environment. The proposed model attempts to simultaneously minimise total production and transportation costs, total number of rejected items, and total delivery time with reference to available capacities, labor level, quota flexibility, and budget constraints at each source, as well as forecast demand and warehouse space at each destination. An industrial case demonstrates that the proposed f-MOLP model achieves an efficient compromise solution and overall decision maker satisfaction with determined goal values. Additionally, the proposed model provides a systematic framework that facilitates decision makers to interactively modify the fuzzy data and parameters until a satisfactory solution is obtained. Overall, the f-MOLP model offers a practical method for solving PTPD problems with fuzzy multiple goals, and can effectively improve producer–distributor relationships within a supply chain.     

Closed loop supply chain network design and optimisation using fuzzy mixed integer linear programming model

Owing to the revolution in sustainable and green manufacturing the production planning and network design of closed loop supply chain concept has got the attention of researchers and managers. In this paper, a multi-product, multi-facility capacitated closed-loop supply chain framework is proposed in an uncertain environment including reuse, refurbish, recycle and disposal of parts. The uncertainty related to demand, fraction of parts recovered for different product recovery processes, product acquisition cost, purchasing cost, transportation cost, processing, and set-up cost is handled with fuzzy numbers. A fuzzy mixed integer linear programming model is proposed to decide optimally the location and allocation of parts at each facility and number of parts to be purchased from external suppliers in order to maximise the profit of organisation. The proposed solution methodology is able to generate a balanced solution between the feasibility degree and degree of satisfaction of the decision maker. The proposed model has been tested with an illustrative example.     

Designing a sustainable stochastic electricity generation network with hybrid production strategies

This paper aims to design a sustainable stochastic electricity production network where fossil fuels-based, biomass-based, and co-firing-based production strategies are simultaneously considered in order to take advantage of all the three production strategies. A multi-objective stochastic mixed integer linear programming model is proposed to achieve economic feasibility, as well as environmental and social benefits under multiple uncertainties. The model is solved by using the improved augmented epsilon constraint method. A case study is used to illustrate the effectiveness of the proposed model. Pareto optimal analysis is conducted to understand the trade-off between economic, environmental, and social aspects of sustainability.     

Multi-product inventory planning with downward substitution, stochastic demand and setup costs

In this paper we consider a single period multi-product inventory problem with stochastic demand, setup cost for production, and one-way product substitution in the downward direction. We model the problem as a two-stage integer stochastic program with recourse where the first stage variables determine which products to produce and how much to produce, and the second stage variables determine how the products are allocated to satisfy the realized demand. We exploit structural properties of the model and utilize a combination of optimization techniques including network flow, dynamic programming, and simulation-based optimization to develop effective heuristics. Through a computational study, we evaluate the performance of our heuristics by comparison with the corresponding optimal solution obtained from a large scale mixed integer linear program. The computational study indicates that our solution methodology can be very effective (98.8% on average) and can handle industrial-sized problems efficiently. We also provide several new qualitative insights on issues such as the effect of demand variance and cost parameters on the optimal number of products setup, the amount produced or inventoried, and the benefits of allowing substitution.     

Fuzzy regression approach to process modelling and optimization of epoxy dispensing

Epoxy dispensing is one of the popular processes to perform microchip encapsulation for chip-on-board (COB) packages. However, determination of proper process parameters setting for optimal quality of the encapsulation is difficult due to the complex behaviour of the encapsulant during dispensing and the uncertainties caused by fuzziness of epoxy dispensing systems. In conventional regression models, deviations between the observed values and the estimated values are supposed to be in probability distribution. However, when data is irregular, the obtained regression model has an unnaturally wide possibility range. In fact, these deviations in some processes such as epoxy dispensing can be regarded as system fuzziness that can be dealt with properly using fuzzy regression method. In this paper, a fuzzy regression approach with fuzzy intervals to process modelling of epoxy dispensing for microchip encapsulation is described. Two fuzzy regression models relating three process parameters and two quality characteristics respectively for epoxy dispensing were developed. They were then introduced to formulate a fuzzy multi-objective optimization problem. A fuzzy linear programming technique was employed to formulate the optimization model. By solving the model, an optimal setting of process parameters can be obtained. Validation experiments were conducted to evaluate the effectiveness of the proposed approach to process modelling and optimization of epoxy dispensing for microchip encapsulation.     

Impact of the use-based maintenance policy on the performance of cellular manufacturing systems

In this paper, a multi-objective integer programming approach is developed to investigate the impact of the use-based preventive maintenance (UPM) policy on the performance of the cellular manufacturing system (CMS). Under the UPM policy a maintenance schedule is established which provides for the performance of preventive maintenance (PM) only after a predetermined number of operating hours of machine use. This research indicates how PM and failure repair (FR) actions affect the effective availability of the machines and accordingly the machine and inter/intra-cell material handling costs under the UPM policy. The objective is to minimise the machine cost, inter- and intra-cell material handling and PM/FR costs. The proposed model is solved by an interactive fuzzy programming (IFP) approach to determine the best compromise solution from the decision maker point of view. IFP assumes that each objective function has a fuzzy goal and focuses on minimising the worst upper bound to obtain an efficient solution which is close to the best lower bound of each objective function. Compromise solutions are prioritised by two efficiency criteria, i.e. grouping efficiency and system availability. The performance of the proposed model is verified by a comprehensive numerical example.     

Fuzzy multi-objective optimisation for master planning in a ceramic supply chain

In this paper, we consider the master planning problem for a centralised replenishment, production and distribution ceramic tile supply chain. A fuzzy multi-objective linear programming (FMOLP) approach is presented which considers the maximisation of the fuzzy gross margin, the minimisation of the fuzzy idle time and the minimisation of the fuzzy backorder quantities. By using an interactive solution methodology to convert this FMOLP model into an auxiliary crisp single-objective linear model, a preferred compromise solution is obtained. For illustration purposes, an example based on modifications of real-world industrial problems is used.     

A fuzzy goal programming model for the simple U-line balancing problem with multiple objectives

A fuzzy goal programming model for the simple U-line balancing (SULB) problem with multiple objectives is presented. In real life applications of the SULB problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore a fuzzy goal programming model is developed for this purpose. The proposed model is the first fuzzy multi-objective decision-making approach to the SULB problem with multiple objectives which aims at simultaneously optimizing several conflicting goals. The proposed model is illustrated using an example. A computational study is conducted by solving a large number of test problems to investigate the relationship between the fuzzy goals and to compare them with the goal programming model proposed by Gökçen and A?pak (Gökçen, H. and A?pak, K., European Journal of Operational Research, 171, 577–585, 2006). The results of the computational study show that the proposed model is more realistic than the existing models for the SULB problem with multiple objectives and also provides increased flexibility for the decision-maker(s) to determine different alternatives.