Optimization of a machining economics model with fuzzy exponents and coefficients

The primary objective of a machining economics model is to determine the optimal cutting parameters that minimize production costs while satisfying some design constraints. This paper develops a solution method that can derive the fuzzy unit production cost of a fuzzy machining economic model when the exponents of decision variables in the objective function, the cost and the constraint coefficients are fuzzy numbers. A pair of two-level machining economics problems is formulated to calculate the upper and lower bounds of the fuzzy unit production cost at possibility level α. Based on the duality theorem and by using a variable substitution technique, the two-level machining economics problem is transformed into the one-level conventional geometric program. Solving the corresponding pair of geometric programs produces the interval of the unit production cost. The examples show that the interval of unit production cost contain more information when the parameters in machining economics problems are fuzzy numbers.     

Simultaneous determination of optimal machining conditions and tool replacement policies in constrained machining economics problems by geometric programming

This paper presents an analytical model for the simultaneous determination of the optimal machining conditions (cutting speed and feed) and the optimal tool replacement policy in constrained machining economics problems by geometric programming. The optimal preventive tool replacement policy is initially determined as a tool life fractile (independent of the underlying tool life distribution) and then it is expressed as actual tool life by utilizing the underlying tool life distribution applicable to the combination of tool material, workpiece properties, and machining conditions. Constraints on the optimal values of cutting speeds, feeds, and/or optimal tool replacement policy based on maximum allowable values and/or surface finish requirements are handled through the optimization of the dual objective function. It is shown that the optimal cost distribution does not depend on the cost coefficients in the objective function. Finally, the model is applied to two-stage systems where the necessary conditions are derived for increasing the synchronization between the two stages.     

Optimal selection of machining rate variables by geometric programming

The selection of economically optimal machining rate variables, i.e. cutting speed and feed rate, is of major importance in the field of metal cutting.

In this paper, apart from the conventional methods used for optimization in machining economics, geometric programming, a relatively new non-linear programming technique, is employed to optimize the constrained unit cost problem in turning. The cutting power available from the machine tool and the permissible CLA value of surface roughness are used as the forced constraints in the primal programme.

Initially, the primal and dual programmes are formulated. Furthermore, a numerical example is presented to illustrate the application of geometric programming which has been proved successful.

It is pointed out that geometric programming may be also used for optimization of unit cost in machining processes other than turning under the assumption that the imposed restrictions, discussed in this paper, are valid.     

Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers

Transportation problem (TP) is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of TP involving interval-valued trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving interval-valued trapezoidal fuzzy numbers transportation problem based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To illustrate the proposed approach an application example is solved. It is demonstrated that study of interval-valued trapezoidal fuzzy numbers transportation problem gives rise to the same expected results as those obtained for TP with trapezoidal fuzzy numbers.     

A mathematical programming approach to fuzzy queues with batch arrivals

This paper proposes a mathematical programming approach for constructing the membership functions of the performance measures in batch-arrival queueing systems with constant batch size and the arrival rate and service rate being fuzzy numbers. The basic idea underlying the proposed method is to transform a fuzzy batch-arrival queue to a family of conventional crisp queues with batch arrivals by applying the α-cut approach. Then the family of crisp batch-arrival queues is described by formulating a pair of parametric nonlinear programs, through which the membership functions of the performance measures can be derived. A numerical example is solved successfully to demonstrate the validity of the proposed approach. Since the performance measures are completely expressed by membership functions rather than by crisp values, more information is provided for designing queueing systems. The successful extension of batch-arrival queues to fuzzy environments permits queueing models to have wider applications in the real world.     

A mathematical programming approach to supply chain models with fuzzy parameters

Supply chain (SC) models play an important role in supply chain management (SCM) for reducing costs and finding better ways to create and deliver value to customers. An approach to deriving the membership function of the fuzzy minimum total cost of the multi-product, multi-echelon, and multi-period SC model with fuzzy parameters is proposed in this article. On the basis of α-cut representation and the extension principle, a pair of mathematical programs are formulated to calculate the lower and upper bounds of the fuzzy minimum total cost at possibility level α. The membership function of the fuzzy minimum total cost is constructed by enumerating different values of α. To demonstrate the validity of the proposed procedure, a four-echelon five-period SC model with fuzzy parameters is solved successfully. Since the objective value is expressed by membership functions rather than by crisp values, they completely conserve the fuzziness of input information when some of the SC data are ambiguous. Thus the proposed approach can represent SCs with fuzzy parameters more accurately, and more information is provided for designing SCs in real-world applications.     

A genetic programming based fuzzy regression approach to modelling manufacturing processes

Fuzzy regression has demonstrated its ability to model manufacturing processes in which the processes have fuzziness and the number of experimental data sets for modelling them is limited. However, previous studies only yield fuzzy linear regression based process models in which variables or higher order terms are not addressed. In fact, it is widely recognised that behaviours of manufacturing processes do often carry interactions among variables or higher order terms. In this paper, a genetic programming based fuzzy regression GP-FR, is proposed for modelling manufacturing processes. The proposed method uses the general outcome of GP to construct models the structure of which is based on a tree representation, which could carry interaction and higher order terms. Then, a fuzzy linear regression algorithm is used to estimate the contributions and the fuzziness of each branch of the tree, so as to determine the fuzzy parameters of the genetic programming based fuzzy regression model. To evaluate the effectiveness of the proposed method for process modelling, it was applied to the modelling of a solder paste dispensing process. Results were compared with those based on statistical regression and fuzzy linear regression. It was found that the proposed method can achieve better goodness-of-fitness than the other two methods. Also the prediction accuracy of the model developed based on GP-FR is better than those based on the other two methods.     

Fuzzy analysis for steady-state availability: a mathematical programming approach

Steady-state availability has been widely applied as a measure to evaluate the reliability characteristics of a repairable system. However, it is generally not realistic to make assumptions concerning failure time and repair time distributions. Thus, this article has developed a procedure to construct the membership function for fuzzy steady-state availability. Based on Zadeh’s extension principle, a pair of mathematical programs is formulated to find α-cuts of fuzzy steady-state availability. An explicit closed-form expression for the membership function is derived by taking the inverse function of the $\alpha$-cut. To illustrate the interpretation and practical value of fuzzy availability in real-world applications, several numerical examples are provided and discussed.     

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.     

Optimizing machining economics models of turning operations using the scatter search approach

Improving performance of metal-cutting operations can lead to considerable savings. In the area of machining optimization, the determination of optimal machining parameters is a substantial problem. The complexity of machining economics problems presents difficulties for some optimization techniques. Scatter search is one of the optimization techniques recently developed in the area of metaheuristics. It is a population-based methodology that shares features with evolutionary methods. It has proved highly successful when solving a diverse array of complex optimization problems. The paper focuses on the application of scatter search to resolve the machining economics models of turning operations. By using several turning models, the computational investigation of scatter search was done by comparison with other optimization methods. The experimental results show the effectiveness of scatter search in the machining economics problems.     

Application of geometric programming to transformer design

This paper considers the transformer design optimization problem. In its most general form, the design problem requires minimizing the total mass (or cost) of the core and wire material while ensuring the satisfaction of the transformer ratings and a number of design constraints. The constraints include appropriate limits on efficiency, voltage regulation, temperature rise, no-load current, and winding fill factor. The design optimization seeks a constrained minimum mass (or cost) solution by optimally setting the transformer geometry parameters and the relevant electrical and magnetic quantities. In cases where the core dimensions are fixed, the optimization problem calls for a constrained maximum volt-ampere or minimum loss solution. This paper shows that the above design problems can be formulated in geometric programming (GP) format. The importance of the GP format stems from two main features. First, GP provides an efficient and reliable solution for the design optimization problem with several variables. Second, it guarantees that the obtained solution is the global optimum. The paper includes a demonstration of the application of the GP technique to transformer design. It also includes a comparative study to emphasize the advantage of including the transformer core dimensions as variables in the design problem.     

The analysis of an inventory control model using posynomial geometric programming

We consider a generalization of the well-known ‘sawtooth’ inventory control model which is used to determine optimal policies for stocking a single item under certain conditions. The case where several items are to be stocked simultaneously and subject to a number of constraints give rise to the generalization that is of interest. We show that the problem of analysing the model may he viewed as a problem in posynomial Geometric Programming. We then point out some of the advantages to be realized from viewing it this way and from using Geometric Programming techniques to solve it. These include, the ability to obtain feasible solutions with known upper bounds on the difference between their associated costs and the minimum cost, a capability for performing an analysis of the sensitivity of the values of the optimized parameters to variations in input parameters, availability of a solution method that is computationally more attractive than methods previously known and, arising out of this last benefit, an ability to deal with much more complicated problems to which the model is applied than previously seemed practical. A simple example problem is solved in detail to illustrate many of these benefits and results are presented for a much more complex case. In addition some possible extensions of the model, which can also be dealt with in this framework, are pointed out.     

Double-sided fuzzy chance-constrained linear fractional programming approach for water resources management

A double-sided fuzzy chance-constrained fractional programming (DFCFP) method is developed for planning water resources management under uncertainty. In DFCFP the system marginal benefit per unit of input under uncertainty can also be balanced. The DFCFP is applied to a real case of water resources management in the Zhangweinan River Basin, China. The results show that the amounts of water allocated to the two cities (Anyang and Handan) would be different under minimum and maximum reliability degrees. It was found that the marginal benefit of the system solved by DFCFP is bigger than the system benefit under the minimum and maximum reliability degrees, which not only improve economic efficiency in the mass, but also remedy water deficiency. Compared with the traditional double-sided fuzzy chance-constrained programming (DFCP) method, the solutions obtained from DFCFP are significantly higher, and the DFCFP has advantages in water conservation.     

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.     

An approach to machining process optimization

Optimization is rightly claimed to be the most significant factor distinguishing the modern approach to machining processes from the orthodox one. Practical results can be obtained from the application of mathematical modelling and optimization techniques.

The paper presents a procedure for solution of machining process optimization problems, with the use of theory of graphs and Bellman's optimum principle.

The finite graph G=(N, A) (where -N denoted a set of nodes and A the set of arcs) here presents the set of feasible solutions. Each are represents a corresponding machining operation and each path the feasible machining process.

A number can be associated with each are : the increment of criterion for optimality. Using the optimum principle we can find the shortest possible path in the graph, thus solving given problems of choice for the optimal machining process.

There is a variety of ways in which the performance criterion can be formulated. In this paper only those performance measures having sufficient economical justification are considered. We may minimize the costs of machining processes, or optimize economical effectiveness of capital engaged.

The method presented can be applied with every additive criterion, as well as with some non-additive ones. A practical example illustrates the field of possible application of this method.     

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.     

Optimum synthesis of a slider-crank mechanism using geometric programming

Geometric programming is a powerful technique and its application to mechanism synthesis has not been fully explored. In this paper, slider-crank mechanism is chosen since the objective function consists of a large number of terms while there are only two design variables. Thus the degree of difficulty is more and does not lend itself to easy application of the geometric programming. Here, modification of the objective function is suggested so that the degree of difficulty reduces to zero even when the working space constraint is imposed. This results in a near optimum solution. Application of signomial geometric programming is illustrated through a numerical example.     

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.     

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.     

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.