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.     

Industrial waste recycling strategies optimization problem: mixed integer programming model and heuristics

Manufacturers have a legal accountability to deal with industrial waste generated from their production processes in order to avoid pollution. Along with advances in waste recovery techniques, manufacturers may adopt various recycling strategies in dealing with industrial waste. With reuse strategies and technologies, byproducts or wastes will be returned to production processes in the iron and steel industry, and some waste can be recycled back to base material for reuse in other industries. This article focuses on a recovery strategies optimization problem for a typical class of industrial waste recycling process in order to maximize profit. There are multiple strategies for waste recycling available to generate multiple byproducts; these byproducts are then further transformed into several types of chemical products via different production patterns. A mixed integer programming model is developed to determine which recycling strategy and which production pattern should be selected with what quantity of chemical products corresponding to this strategy and pattern in order to yield maximum marginal profits. The sales profits of chemical products and the set-up costs of these strategies, patterns and operation costs of production are considered. A simulated annealing (SA) based heuristic algorithm is developed to solve the problem. Finally, an experiment is designed to verify the effectiveness and feasibility of the proposed method. By comparing a single strategy to multiple strategies in an example, it is shown that the total sales profit of chemical products can be increased by around 25% through the simultaneous use of multiple strategies. This illustrates the superiority of combinatorial multiple strategies. Furthermore, the effects of the model parameters on profit are discussed to help manufacturers organize their waste recycling network.     

混合整数双层线性规划的全局优化算法

本文讨论了上层决策变量为整数变量、下层决策变量为连续变量的混合整数双层线性规划问题,利用其可行解均落在约束域边界上的性质,提出了一种求解混合整数双层线性规划全局最优解的算法,并举例说明了算法的执行过程。     

A mixed integer programming model for a closed-loop supply-chain network

In this paper, a new mixed integer mathematical model for a closed-loop supply-chain network that includes both forward and reverse flows with multi-periods and multi-parts is proposed. The proposed model guarantees the optimal values of transportation amounts of manufactured and disassembled products in a closed-loop supply chain while determining the location of plants and retailers. Finally, computational results are presented for a number of scenarios to show and validate the applicability of the model.     

A comparison of mixed integer programming formulations of the capacitated lot-sizing problem

We propose a novel mixed integer programming formulation for the capacitated lot-sizing problem with set-up times and set-up carryover. We compare our formulation to two earlier formulations, the Classical and Modified formulations, and a more recent formulation due to Suerie and Stadtler. Extensive computational experiments show that our formulation consistently outperforms the Classical and Modified formulations in terms of CPU time and solution quality. It is competitive with the Suerie–Stadtler (S&S) formulation, but outperforms all other formulations on the most challenging instances, those with low-capacity slack and a dense jobs matrix. We show that some of the differences in the performance of these various formulations arise from their different use of binary variables to represent production or set-up states. We also show that the LP relaxation of our Novel formulation provides a tighter lower bound than that of the Modified formulation. Our experiments demonstrate that, while the S&S formulation provides a much tighter LP bound, the Novel formulation is better able to exploit the intelligence of the CPLEX solution engine.     

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.     

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.     

Dynamic programming and mixed integer programming based algorithms for the online glass cutting problem with defects and production targets

In flat glass manufacturing, glass products of various dimensions are cut from a glass ribbon that runs continuously on a conveyor belt. Placement of glass products on the glass ribbon is restricted by the defects of varying severity located on the ribbon as well as the quality grades of the products to be cut. In addition to cutting products, a common practice is to remove defective parts of the glass ribbon as scrap glass. As the glass ribbon moves continuously, cutting decisions need to be made within seconds, which makes this online problem very challenging. A simplifying assumption is to limit scrap cuts to those made immediately behind a defect (a cut-behind-fault or CBF). We propose an online algorithm for the glass cutting problem that solves a series of static cutting problems over a rolling horizon. We solve the static problem using two methods: a dynamic programming algorithm (DP) that utilises the CBF assumption and a mixed integer programming (MIP) formulation with no CBF restriction. While both methods improve the process yield substantially, the results indicate that MIP significantly outperforms DP, which suggests that the computational benefit of the CBF assumption comes at a cost of inferior solution quality.     

Robust interval linear programming for environmental decision making under uncertainty

In this study, a robust interval linear programming (RILP) method is developed for dealing with uncertainties expressed as intervals with deterministic boundaries. An enhanced two-step method (ETSM) is also advanced to solve the RILP model. The developed RILP improves upon the conventional interval linear programming (ILP) method since it can generate solution intervals within a larger feasible zone. The decision space based on ETSM contains all feasible solutions, such that no useful information is neglected. Moreover, the RILP can guarantee the stability of the optimization model due to no violation for the best-case constraints. The results also suggest that the RILP is effective for practical environmental and engineering problems that involve uncertainties.     

A mixed integer linear programming solution for single hoist multi-degree cyclic scheduling with reentrance

This article considers single hoist multi-degree cyclic scheduling problems with reentrance. Time window constraints are also considered. Firstly, a mixed integer programming model is formulated for multi-degree cyclic hoist scheduling without reentrance, referred to as basic lines in this article. Two valid inequalities corresponding to this problem are also presented. Based on the model for basic lines, an extended mixed integer programming model is proposed for more complicated scheduling problems with reentrance. Phillips and Unger's benchmark instance and randomly generated instances are applied to test the model without reentrance, solved using the commercial software CPLEX. The efficiency of the model is analysed based on computational time. Moreover, an example is given to demonstrate the effectiveness of the model with reentrance.     

A mixed integer linear programming formulation for optimal balancing of mixed-model U-lines

The mixed-model U-line balancing problem was first studied by Sparling and Miltenburg (Sparling, D. and Miltenburg, J., 1998. The mixed-model U-line balancing problem. International Journal of Production Research, 36(2), 485–501) but has not been mathematically formulated to date. This paper presents a mixed integer programming formulation for optimal balancing of mixed-model U-lines. The proposed approach minimises the number of workstations required on the line for a given model sequence. The proposed formulation is illustrated and tested on an example problem and compared with an existing approach. This paper also proposes a new heuristic solution procedure to handle large scale mixed-model U-line balancing problems. A comprehensive experimental analysis is also conducted to evaluate the performance of the proposed heuristic. The results show the validity and usefulness of the proposed integer formulation and effectiveness of the proposed heuristic procedure.     

多核集群任务分配问题的0-1整数规划求解模型

研究了典型多核集群任务分配中的节点内通讯特性。基于0-1整数非线性规划模型和线性松弛技术,给出了一种0-1整数线性规划任务分配问题求解优化模型。由于节点内的通讯量与通讯延迟较大,以最小化计算代价和节点间通讯代价为研究目标的传统求解模型具有严重的局限性,而该求解模型考虑了节点内通讯代价,并采用了线性规划松弛技术,其目标是最小化计算代价、节点间通讯代价和节点内通讯代价。计算结果验证了提出的模型的有效性。     

Fuzzy goal programming for material requirements planning under uncertainty and integrity conditions

In this paper, we formulate the material requirements planning) problem of a first-tier supplier in an automobile supply chain through a fuzzy multi-objective decision model, which considers three conflictive objectives to optimise: minimisation of normal, overtime and subcontracted production costs of finished goods plus the inventory costs of finished goods, raw materials and components; minimisation of idle time; minimisation of backorder quantities. Lack of knowledge or epistemic uncertainty is considered in the demand, available and required capacity data. Integrity conditions for the main decision variables of the problem are also considered. For the solution methodology, we use a fuzzy goal programming approach where the importance of the relations among the goals is considered fuzzy instead of using a crisp definition of goal weights. For illustration purposes, an example based on modifications of real-world industrial problems is used.     

A fuzzy fractional chance-constrained programming model for air quality management under uncertainty

A fuzzy fractional chance-constrained programming model (FFCCPM) was developed for dealing with air quality management under uncertainty. FFCCPM integrates a fractional programming model and a double-sided fuzzy chance-constrained programming model. It considers the ratio between total treated pollutant amounts and system cost in the objective function; the constraints with fuzzy variables can be satisfied under some predetermined confidence levels and reliability scenarios. The air quality management system in Fengrun district, Tangshan City, China, was used to demonstrate the applicability of the proposed method. The obtained results indicated that the proposed model was suitable in describing and providing an overview of a studied management system for decision makers, generating various cost-effective air pollution-abatement alternatives. The strategy with a balance between system economy and reliability was recommended for decision makers. The successful application of FFCCPM in Fengrun district provides a good example of real-world regional air quality management.     

Inexact fuzzy integer chance constraint programming approach for noise control within an urban environment

This article introduces an inexact fuzzy integer chance constraint programming (IFICCP) approach for identifying noise reduction strategy under uncertainty. The IFICCP method integrates the interval programming and fuzzy chance constraint programming approaches into a framework, which is able to deal with uncertainties expressed as intervals and fuzziness. The proposed IFICCP model can be converted into two deterministic submodels corresponding to the optimistic and pessimistic conditions. The modelling approach is applied to a hypothetical control measure selection problem for noise reduction. Results of the case study indicate that useful solutions for noise control practices can be acquired. Three acceptable noise levels for two communities are considered. For each acceptable noise level, several decision alternatives have been obtained and analysed under different fuzzy confidence levels, which reflect the trade-offs between environmental and economic considerations.     

Inexact rough-interval type-2 fuzzy stochastic optimization model supporting municipal solid waste management under uncertainty

In this study, an inexact rough-interval type-2 fuzzy stochastic linear programming (IRIT2FSLP) approach is developed for addressing uncertainties presented as rough-interval, type-2 fuzzy and random variables. The proposed method is applied to the case of a long-term municipal solid waste management system. The IRIT2FSLP approach is an extension of the inexact interval linear programming for handling nonlinear stochastic optimization problems where rough-interval and type-2 fuzzy parameters are integrated into a general framework. The results indicate that IRIT2FSLP normally leads to rough-interval solutions. Comparisons of the proposed model with scenarios without rough-interval and type-2 fuzzy parameters are also conducted. The results indicate the significant impact of dual-uncertain information on the system, which implies the reliability of IRIT2FSLP in handling waste flow allocation.     

Reliable design of a closed loop supply chain network under uncertainty: An interval fuzzy possibilistic chance-constrained model

This article seeks to offer a systematic approach to establishing a reliable network of facilities in closed loop supply chains (CLSCs) under uncertainties. Facilities that are located in this article concurrently satisfy both traditional objective functions and reliability considerations in CLSC network designs. To attack this problem, a novel mathematical model is developed that integrates the network design decisions in both forward and reverse supply chain networks. The model also utilizes an effective reliability approach to find a robust network design. In order to make the results of this article more realistic, a CLSC for a case study in the iron and steel industry has been explored. The considered CLSC is multi-echelon, multi-facility, multi-product and multi-supplier. Furthermore, multiple facilities exist in the reverse logistics network leading to high complexities. Since the collection centres play an important role in this network, the reliability concept of these facilities is taken into consideration. To solve the proposed model, a novel interactive hybrid solution methodology is developed by combining a number of efficient solution approaches from the recent literature. The proposed solution methodology is a bi-objective interval fuzzy possibilistic chance-constraint mixed integer linear programming (BOIFPCCMILP). Finally, computational experiments are provided to demonstrate the applicability and suitability of the proposed model in a supply chain environment and to help decision makers facilitate their analyses.     

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.     

Interval-parameter robust quadratic programming for water quality management under uncertainty

Effective planning of water quality management is important for facilitating sustainable socio-economic development in watershed systems. An interval-parameter robust quadratic programming (IRQP) method is developed by incorporating techniques of robust programming and interval quadratic programming within a general optimization framework. The IRQP improves upon existing quadratic programming methods, and can tackle uncertainties presented as interval numbers and fuzzy sets as well as their combinations. Moreover, it can deal with nonlinearities in the objective function such that economies-of-scale effects can be reflected. The developed method is applied to a case study of a water quality management under uncertainty. A number of decision alternatives are generated based on the interval solutions as well as the projected applicable conditions. They represent multiple decision options with various environmental and economic considerations. Willingness to accept a low economic revenue will guarantee satisfying the water quality requirements. A strong desire to acquire a high benefit will run the risk of violating environmental criteria.     

A 0-1 integer programming model and solving strategies for the slab storage problem

We consider the slab storage problem (SSP) in slab yard operations. A set of slabs enter a slab yard in a specific order. A proper stack needs to be selected for each inbound slab, so that the number of relocations in the subsequent retrieval stage is minimised. We present a 0-1 integer programming model of the SSP that minimises the lower bound of the number of relocations. Four solving strategies are derived from several interesting properties of the mathematical model to speed up the solving process of the model. Making use of randomly generated instances and practical instances, we testify the effectiveness of the solving strategies and study the influence of problem parameters on the computational time of the model. Computational results show that the solving strategies can effectively reduce the computational time of the model and is applicable in medium-sized practical instances.