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 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.     

Fuzzy scheduling of a build-to-order supply chain

The overwhelming majority of the literature in the area of supply chain planning and scheduling considers the traditional make-to-stock (MTS) environment. However, manufacturers of assembled products such as cars, computers, furniture, etc. adopt the build-to-order supply chain (BOSC) to become agile in a mass customization process in order to meet diversified customer requirements. In this paper we propose an integrated production–distribution planning model for a multi-echelon, multi-plant and multi-product supply chain operating in a build-to-order (BTO) environment. The uncertainties associated with estimation of the various operational cost parameters are represented by fuzzy numbers. The BOSC scheduling model is thus constructed as a mixed-integer fuzzy programming (MIFP) problem with the goal of reducing the overall operating costs related to component fabrication, procurement, assembling, inspection, logistics and inventory, while improving customer satisfaction by allowing product customization and meeting delivery promise dates at each market outlet. An efficient compromise solution approach by transforming the problem into an auxiliary multi-objective linear programming model is also suggested.     

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.     

Global supplier selection: a fuzzy-AHP approach

Global supplier selection has a critical effect on the competitiveness of the entire supply chain network. Research results indicate that the supplier selection process appears to be the most significant variable in deciding the success of the supply chain. It helps in achieving high quality products at lower cost with higher customer satisfaction. Apart from the common criteria such as cost and quality, this paper also discusses some of the important decision variables which can play a critical role in case of the international sourcing. The importance of the political-economic situation, geographical location, infrastructure, financial background, performance history, risk factors, etc., have also been pointed out in particularly in the case of global supplier selection. Supplier selection problem related to the global sourcing is more complex than the general domestic sourcing and as a result it needs more critical analysis, which could not be found properly in past available literatures. This paper discusses the fuzzy based Analytic Hierarchy Process (fuzzy-AHP) to efficiently tackle both quantitative and qualitative decision factors involved in selection of global supplier in current business scenario. The fuzzy-AHP is an efficient tool to tackle the fuzziness of the data involved in deciding the preferences of the different decision variables involved in the process of global supplier selection. The triangular fuzzy numbers are used to transform the linguistic comparison of the different decision criteria, sub-criteria and performance of the alternative suppliers. The pairwise comparison matrices help in deciding the synthetic extent value of each comparison and finally, the priority weights of one alternative over another are decided in this paper. An example from a manufacturing industry searching for the global supplier for a critical component is used to demonstrate the effective implementation procedure of proposed fuzzy-AHP technique. The proposed model can provide the guidelines and directions for the decision makers to effectively select their global suppliers in the current competitive business scenario.     

Fuzzy QFD for supply chain management with reliability consideration

Although many products are made through several tiers of supply chains, a systematic way of handling reliability issues in a various product planning stage has drawn attention, only recently, in the context of supply chain management (SCM). The main objective of this paper is to develop a fuzzy quality function deployment (QFD) model in order to convey fuzzy relationship between customers needs and design specification for reliability in the context of SCM. A fuzzy multi criteria decision-making procedure is proposed and is applied to find a set of optimal solution with respect to the performance of the reliability test needed in CRT design. It is expected that the proposed approach can make significant contributions on the following areas: effectively communicating with technical personnel and users; developing relatively error-free reliability review system; and creating consistent and complete documentation for design for reliability.     

An application of cost-effective fuzzy inventory controller to counteract demand fluctuation caused by bullwhip effect

This paper develops a fuzzy inventory model to counteract the demand fluctuation in supply demand networks, which combines fuzzy logic controller with (s,?S) policy based on economic order quantity (EOQ) model. Following a literature review and a discussion of counteractions to the bullwhip effect and the obstruction of general counteraction in supply demand networks, a multi-echelon fuzzy inventory model in supply demand networks is proposed. A simulation model with one- and two-echelon supply demand network is built and tested for (s, S) policy based on the classical EOQ model and the proposed fuzzy inventory model. Based on the simulation, results of the relevance performance are presented and discussed, which show that the proposed multi-echelon fuzzy inventory model provides not only a cost-effective management of inventory (e.g. lower inventory levels and cost) in market uncertainty, but also another effective alternative for counteracting demand fluctuation. In particular, the proposed multi-echelon fuzzy inventory model shows benefit in counteracting demand fluctuation in multi-echelon supply demand networks. Finally, some conclusions and suggestions for further research works are presented.     

Solving large-scale fixed cost integer linear programming models for grid-based location problems with heuristic techniques

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.     

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.     

Fuzzy approach to the robust facility layout in uncertain production environments

The proposed method approaches the problem of the optimal facility layout using fuzzy theory. The optimal layout is a robust layout that minimizes the total material handling cost, when the product market demands are uncertain variables, which are defined as fuzzy numbers. Since each department has a limited production capacity, not all possible combinations, deriving from each product's market demand, are taken into account because some combination could exceed the overall department's productivity. Therefore, the optimal solution results by solving a 'constrained' fuzzy optimization problem, in which the fuzzy material handling costs corresponding to the layouts are evaluated, and a ranking method, which considers the grade of pessimism of the decision maker, is established to determine the optimal layout.     

Fuzzy hybrid decision model for supplier evaluation and selection

This paper proposes a structured, integrated decision model for evaluating suppliers by combining the fuzzy analytical hierarchy process (FAHP) and grey relational analysis (GRA). The qualitative and partially-known information is incorporated in this decision model using the fuzzy set theory. In this proposed methodology, the weights of the evaluation criteria are calculated by using FAHP, then the ranking of the suppliers is determined by using GRA. Finally to show the robustness of the model, a sensitivity analysis is also performed. In this study, the supplier selection problem of an electroplating industry in the southern part of India was investigated, demonstrating the effectiveness of this developed integrated model. This model can help in solving the complex decision in supplier selection practice. The results generated from the model are properly validated and finally a systematic solution with decision support is provided for decision makers. This model can be integrated with other decision support systems of similar kinds of industries.     

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 stochastic dynamic programming model for stream water quality management

This paper deals with development of a seasonal fraction-removal policy model for waste load allocation in streams addressing uncertainties due to randomness and fuzziness. A stochastic dynamic programming (SDP) model is developed to arrive at the steady-state seasonal fraction-removal policy. A fuzzy decision model (FDM) developed by us in an earlier study is used to compute the system performance measure required in the SDP model. The state of the system in a season is defined by streamflows at the headwaters during the season and the initial DO deficit at some pre-specified checkpoints. The random variation of streamflows is included in the SDP model through seasonal transitional probabilities. The decision vector consists of seasonal fraction-removal levels for the effluent dischargers. Uncertainty due to imprecision (fuzziness) associated with water quality goals is addressed using the concept of fuzzy decision. Responses of pollution control agencies to the resulting end-of-season DO deficit vector and that of dischargers to the fraction-removal levels are treated as fuzzy, and modelled with appropriate membership functions. Application of the model is illustrated with a case study of the Tungabhadra river in India.     

Developing a fuzzy proportional–derivative controller optimization engine for engineering design optimization problems

In real world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as purely mathematical. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all. In this article, the optimization process is analogous to a closed-loop control system, and a fuzzy proportional–derivative (PD) controller optimization engine is developed for engineering design optimization problems with monotonicity and implicit constraints. Monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. In this research, monotonicity of the design variables and activities of the constraints determined by the theory of monotonicity analysis are modelled in the fuzzy PD controller optimization engine using generic fuzzy rules. The designer only needs to define the initial values and move limits of the design variables to determine the parameters in the fuzzy PD controller optimization engine. In the optimization process using the fuzzy PD controller optimization engine, the function value of each constraint is evaluated once in each iteration. No sensitivity information is required. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested.     

Sustainable Supplier Selection Model in Supply Chains During the COVID-19 Pandemic

As global supply chains become more developed and complicated, supplier quality has become increasingly influential on the competitiveness of businesses during the Covid-19 pandemic. Consequently, supplier selection is an increasingly important process for any business around the globe. Choosing a supplier is a complex decision that can result in lower procurement costs and increased profits without increasing the cost or lowering the quality of the product. However, these decision-making problems can be complicated in cases with multiple potential suppliers. Vietnam's textile and garment industry, for example, has made rapid progress in recent years but is still facing great difficulties as the supply of raw materials and machinery depends heavily on foreign countries. Therefore, it is extremely important for textile and garment manufacturing companies in Vietnam to implement an effective supplier evaluation and selection process. While multicriteria decision-making models are frequently employed to assist with supplier evaluation and selection problems, few of these models consider the problem under the condition of a fuzzy decision-making environment. The aim of this paper is to create a hybrid MCDM model using the Fuzzy Analytical Hierarchy Process (FAHP) model and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to assist the supplier selection process in the garment industry in a fuzzy decision-making environment. In this study, the FAHP method is used to evaluate the performance and the weight of each criterion. TOPSIS is then used to rank all potential suppliers. The proposed model is then applied to a real-world case study to demonstrate both the process of calculation as well as its real-world applicability. The results from the case study provide empirical evidence that the model is feasible. The proposed approach can also be used in combination with other MCDM models to better support decision makers and can be modified to be applied in similar supplier selection processes for different industries.     

Supplier selection using fuzzy association rules mining approach

Owing to ill-structured, dynamic environments and the presence of multiple decision-makers with conflicting viewpoints, comprehension, analysis and support of the supplier evaluation process becomes more and more difficult. Moreover, with the complexities of issues such as the role of leadership, the influence of group formation, and analysis of disagreements, it cannot be predictable that there will ever exist a solution to cope with all imprecise, multi-criteria/multi-actor situations. A fuzzy association rules-based approach may be suited for the judgement of human subjects. In this paper, we develop an approach based on Fuzzy Association Rule Mining to support the decision makers by enhancing the flexibility in making decisions for evaluating suppliers with both tangibles and intangibles attributes. Also, by checking the fuzzy classification rules, the goal of knowledge acquisition can be achieved in a framework in which assessments could be established without constraints, and consequently checked and compared in several details. The efficacy and intricacy of the proposed model for finding fuzzy association rules from the database for supplier assessment is demonstrated with the help of numerical examples.     

Using fuzzy analytic hierarchy process and particle swarm optimisation for balanced and defective supply chain problems considering WEEE/RoHS directives

This research explored problems concerning production and delivery in a green supply chain, and constructed an optimal mathematical model to provide solutions. This model incorporates WEEE and RoHS in EU directives for the selection of green partners when establishing a supply chain. The weight of each component is calculated by fuzzy analytic hierarchy process (fuzzy AHP). Previous studies suggested that a supply chain is a balanced system, however, in actual practice, there may be processing damages or delivering losses. Thus, such a supply chain with production loss is known as a ‘defective supply chain’. This research analysed the defective supply chain system to discuss its supplier selection, production, and distribution. It developed an optimal mathematical model for both balanced and defective models, and adopted particle swarm optimisation (PSO) to obtain solutions for both models. Finally, case studies for both models with quality solutions were discussed to confirm the efficiency and effectiveness of the proposed approach.     

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.     

Fuzzy bi-objective formulation for a parallel machine scheduling problem with machine eligibility restrictions and sequence-dependent setup times

In this paper, a fuzzy bi-objective mixed-integer linear programming (FBOMILP) model is presented. FBOMILP encompasses the minimisation workload imbalance and total tardiness simultaneously as a bi-objective formulation for an unrelated parallel machine scheduling problem. To make the proposed model more practical, sequence-dependent setup times, machine eligibility restrictions and release dates are also considered. Moreover, the inherent uncertainty of processing times, release dates, setup times and due dates are taken into account and modelled by fuzzy numbers. In order to solve the model for small-scale problems, a two-stage fuzzy approach is proposed. Nevertheless, since the problem belongs to the class of NP-hard problems, the proposed model is solved by two meta-heuristic algorithms, namely fuzzy multi-objective particle swarm optimisation (FMOPSO) and fuzzy non-dominated sorting genetic algorithm (FNSGA-II) for solving large-scale instances. Subsequently, through setting up various numerical examples, the performances of the two mentioned algorithms are compared. When α?=?0.5 (α is a level of risk-taking and when it increases the decision-maker’s risk-taking decreases), FNSGA-II is fairly more effective than FMOPSO and has better performance especially in solving large-sized problems. However, when α rises, it can be stated that FMOPSO moderately becomes more appropriate. Finally, directions for future studies are suggested and conclusion remarks are drawn.     

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.