A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Simulated Annealing and Genetic Algorithms for the Facility Layout Problem: A Survey

The facility layout problem (FLP) has many practical applications and is known to be NP-hard. During recent decades exact and heuristic approaches have been proposed in the literature to solve FLPs. In this paper we review the most recent developments regardingsimulated annealing and genetic algorithms for solvingfacility layout problems approximately.     

STaTS: A Slicing Tree and Tabu Search based heuristic for the unequal area facility layout problem

In this paper, a slicing tree based tabu search heuristic for the rectangular, continual plane facility layout problem (FLP) is presented. In addition to the incorporation of facilities with unequal areas we also integrate the possibility to specify various requirements regarding (rectangular) shape and dimensions of each individual facility by using bounding curves. Therefore, it is possible to solve problems containing facilities of fixed and facilities of flexible shapes at the same time. We present a procedure that calculates the layout corresponding to a given slicing tree on the basis of bounding curves. These layouts are slicing structures which are able to contain empty spaces to guarantee that stringent shape restrictions of facilities are kept. Due to these features this approach is better suited for practical use than so far existing ones. The effectiveness of our approach in terms of objective function value is shown by comparing our results to those found in the literature. Even a large problem instance comprised of 62 facilities has been solved.     

Solving the failure-to-fit problem for plant layout: By changing department shapes and sizes

The problem of assigning positions to unequal fixed size and shape departments within a given plant area has prompted numerous research efforts in recent years. Most of the published papers discuss techniques that place departments in such a way that the local cost of material flow between departments is minimized. Many of these techniques assume that there exists at least one-feasible layout. When no such layout exists, a failure-to-fit problem arises. In this paper, we present a new method, which combines the existing layout techniques with heuristic search techniques to construct a solution to the failure-to-fit problem by changing the size and/or shape of the departments in a systematic manner without the help of humans. Experimental results show that the proposed technique is able to produce high quality and practical solutions.     

A decomposition heuristics for the container ship stowage problem

In this paper we face the problem of stowing a containership, referred to as the Master Bay Plan Problem (MBPP); this problem is difficult to solve due to its combinatorial nature and the constraints related to both the ship and the containers. We present a decomposition approach that allows us to assign a priori the bays of a containership to the set of containers to be loaded according to their final destination, such that different portions of the ship are independently considered for the stowage. Then, we find the optimal solution of each subset of bays by using a 0/1 Linear Programming model. Finally, we check the global ship stability of the overall stowage plan and look for its feasibility by using an exchange algorithm which is based on local search techniques. The validation of the proposed approach is performed with some real life test cases. This work has been developed within the research area: “The harbour as a logistic node” of the Italian Centre of Excellence on Integrated Logistics (CIELI) of the University of Genoa, Italy     

A graph-pair representation and MIP-model-based heuristic for the unequal-area facility layout problem

Owing to its theoretical as well as practical significance, the facility layout problem with unequal-area departments has been studied for several decades, with a wide range of heuristic and a few exact solution procedures developed by numerous researchers. In one of the exact procedures, the facility layout problem is formulated as a mixed-integer programming (MIP) model in which binary (0/1) variables are used to prevent departments from overlapping with one another. Obtaining an optimal solution to the MIP model is difficult, and currently only problems with a limited number of departments can be solved to optimality. Motivated by this situation, we developed a heuristic procedure which uses a “graph pair” to determine and manipulate the relative location of the departments in the layout. The graph-pair representation technique essentially eliminates the binary variables in the MIP model, which allows the heuristic to solve a large number of linear programming models to construct and improve the layout in a comparatively short period of time. The search procedure to improve the layout is driven by a simulated annealing algorithm. The effectiveness of the proposed graph-pair heuristic is demonstrated by comparing the results with those reported in recent papers. Possible extensions to the graph-pair representation technique are discussed at the end of the paper.     

基于弹性区带架构的多目标设施布局问题研究

为解决设施面积不等的连续型设施布局问题,建立了基于弹性区带架构布置形式,以物料搬运成本最小、邻近关系最大、距离要求满足度最大的多目标设施布局模型。模型中考虑了区域内的横向、纵向过道,对设施的长宽比进行了限制,使得结果更符合实际情况。为克服传统多目标单一化方法需要人为设置子目标函数权重、主观性过强的缺陷,采用基于带有精英保留策略的非支配排序遗传算法(NSGA Ⅱ)的多目标优化算法求解模型,设计了相应的编码方式、交叉算子、变异算子、罚函数。最后通过某物流园区的实例分析证明了模型与方法的有效性。     

Duality in Fuzzy Linear Programming: Some New Concepts and Results

A class of fuzzy linear programming (FLP) problems based on fuzzy relations is introduced, the concepts of feasible and -efficient solutions are defined. The class of crisp (classical) LP problems and interval LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The previous results are applied to the special case of interval LP and compared to the existing literature.     

A Constraint Programming model for fast optimal stowage of container vessel bays

Container vessel stowage planning is a hard combinatorial optimization problem with both high economic and environmental impact. We have developed an approach that often is able to generate near-optimal plans for large container vessels within a few minutes. It decomposes the problem into a master planning phase that distributes the containers to bay sections and a slot planning phase that assigns containers of each bay section to slots. In this paper, we focus on the slot planning phase of this approach and present a Constraint Programming and Integer Programming model for stowing a set of containers in a single bay section. This so-called slot planning problem is NP-hard and often involves stowing several hundred containers. Using state-of-the-art constraint solvers and modeling techniques, however, we were able to solve 90% of 236 real instances from our industrial collaborator to optimality within 1 second. Thus, somewhat to our surprise, it is possible to solve most of these problems optimally within the time required for practical application.     

Heuristics for the dynamic facility layout problem with unequal-area departments

The dynamic facility layout problem (DFLP) is the problem of finding positions of departments on the plant floor for multiple periods (material flows between departments change during the planning horizon) such that departments do not overlap, and the sum of the material handling and rearrangement costs is minimized. In this paper, the departments may have unequal-areas and free orientations, and the layout for each period is generated on the continuous plant floor. Because of the complexity of the problem, only small-size problems can be solved in reasonable time using exact techniques. As a result, a boundary search (construction) technique, which places departments along the boundaries of already placed departments, is developed for the DFLP. The solution is improved using a tabu search heuristic. The heuristics were tested on some instances from the DFLP and static facility layout problem (SFLP) literature. The results obtained demonstrate the effectiveness of the heuristics.     

Layout design for flexible manufacturing systems considering single-loop directional flow patterns

To achieve high productivity in a flexible manufacturing system (FMS), an efficient layout arrangement and material flow path design are important due to the large percentage of product cost that is related to material handling. The layout design problem addressed in this paper has departments with fixed shapes and pick-up/drop-off points. It is an open-field type layout with single-loop directed flow path. A two-step heuristic is proposed to solve the problem. It first solves a traditional block layout with directed-loop flow path to minimize material handling costs by using a combined spacefilling curve and simulated annealing algorithm. The second step of the proposed methodology uses the resulting flow sequence and relative positioning information from the first step as input to solve the detailed FMS layout, which includes the spatial coordinates and orientation of each FMS cell. This detailed FMS layout problem is formulated and solved as a mixed integer program. Empirical illustrations show promising results for the proposed methodology in solving real-world type problems.     

Fuzzy linear programming with vague objective coefficients in an uncertain environment

In this paper, a new fuzzy linear programming (FLP)-based methodology using a specific membership function named modified logistic membership function is proposed. The modified logistic membership function is first formulated and its flexibility established by an analytical approach. This membership function is tested for its useful performance through an illustrative example by employing FLP. The developed methodology of FLP has provided confidence in applying to real-life industrial production planning problem. This approach of solving industrial production planning problem can provide feedback to the decision maker, implementer and analyst. In such cases, this approach can be called interactive FLP. There is a possibility to design the self-organizing of the fuzzy system for the product mix selection problem in order to find a satisfactory solution. The decision maker, analyst and implementer can incorporate their knowledge and experience to obtain the best outcome.     

Enhancing the performance of constraint programming through the introduction of linear programming

Constraint Programming (CP) has been successful in a number of combinatorial search and discrete optimisation problems. Yet other more traditional approaches, such as Integer Programming (IP), can still give a better performance on the same problem types. Central to IP's success is its reliance on a fast Linear Programming (LP) solver providing solutions during the search to the corresponding relaxed problems. These solutions are used to guide the search within IP as well as a means of detecting infeasibility and integrality. This paper shows that there is scope also to include LP within the CP framework, in order to similarly guide the CP search. The problems examined here are one for which CP on its own had proved markedly inferior to IP. Hence a hybrid solver based on the CP search and using an LP solver is configured and run on these problems. The outcome shows that using the LP solver within the CP search is a valuable addition to the available search strategies. An improved performance over the CP-only strategies is obtained and, further, comparable results are obtained to those from IP. Overall, CP+LP can be considered as a more robust approach than either CP or IP on their own on a variety of combinatorial search problems.     

A simple direct cosine simplex algorithm

Linear programming (LP) is the core model of constrained optimization. The Simplex method (Simplex in short) has been proven in practice to perform very well in small- or medium-sized LP problems. A new algorithm called the direct cosine Simplex algorithm (DCA) is presented here to improve upon Simplex and to solve LP problems. The proposed DCA implements a specific cosine criterion to choose the entering variable instead of the traditional most negative rule used in Simplex. Three examples are given to illustrate the implementation of the proposed DCA to improve Simplex and to serve as the optimization tool. The utility of the proposed approach is evident from the extensive computational results on test problems adapted from NETLIB. DCA reduced the number of iterations of Simplex in most cases in our computational experiment. Preliminary results for medium-sized problems are encouraging.     

Solving the short-term electrical generation scheduling problem by an adaptive evolutionary approach

In this paper, we introduce an adaptive evolutionary approach to solve the short-term electrical generation scheduling problem (STEGS). The STEGS is a hard constraint satisfaction optimization problem. The algorithm includes various strategies proposed in the literature to tackle hard problems with constraints such as: the representation used a non-binary coding scheme that drastically reduces the search space compared with the traditional evolutionary approaches. Specialized operators are especially designed for this problem and for this kind of representation, which also includes a local search procedure. Furthermore, the algorithm is guided by an adaptive parameter control strategy. We used some very well known benchmarks for STEGS to evaluate our approach. The results are very encouraging and we have obtained new better values for all the systems tested. Our aim here is to show that evolutionary approaches can be considered as good techniques to be used to solve real-world highly constrained problems.     

A scatter search algorithm for the single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging facilities with given lengths on a line, with the objective of minimizing the weighted sum of the distances between all pairs of facilities. The problem is NP-hard and research has focused on heuristics to solve large instances of the problem. In this paper we present a scatter search algorithm to solve large size SRFLP instances. Our computational experiments show that the scatter search algorithm is an algorithm of choice when solving large size SRFLP instances within limited time.     

The checkpoint ordering problem

We suggest a new variant of a row layout problem: Find an ordering of n departments with given lengths such that the total weighted sum of their distances to a given checkpoint is minimized. The Checkpoint Ordering Problem (COP) is both of theoretical and practical interest. It has several applications and is conceptually related to some well-studied combinatorial optimization problems, namely the Single-Row Facility Layout Problem, the Linear Ordering Problem and a variant of parallel machine scheduling. In this paper we study the complexity of the (COP) and its special cases. The general version of the (COP) with an arbitrary but fixed number of checkpoints is NP-hard in the weak sense. We propose both a dynamic programming algorithm and an integer linear programming approach for the (COP) . Our computational experiments indicate that the (COP) is hard to solve in practice. While the run time of the dynamic programming algorithm strongly depends on the length of the departments, the integer linear programming approach is able to solve instances with up to 25 departments to optimality.     

A novel approach for considering layout problem in cellular manufacturing systems with alternative processing routings and subcontracting approach

In this paper a new integrated approach is presented for designing cellular manufacturing system and its inter- and intra-cell layouts. Various production factors such as part demands, alternative processing routings, operation sequences, processing times, capacity of machines, etc. are incorporated in the problem in order to extend its applicability. To increase the accuracy of the inter- and intra-cell layout design, the material handling cost is calculated in terms of the actual position of machines within the cells and regarding the dimensions of the machines and aisle distances. Also, a subcontracting approach is proposed to determine the production volume of parts within the CF and layout design process regarding the production, material handling and outsourcing costs and under demand and machine capacity constraints. To the best of our knowledge, this is the first study that addresses all these design features simultaneously. As the proposed problem is NP-hard, an efficient GA is employed to solve it. Finally, numerical examples adopted from the literature are used to verify the proposed approach.     

圆形区域分散布局问题研究

针对圆形区域分散布局问题, 文中给出了一个带约束的非线性规划模型. 当布局点数量较少时, 通过将模型转化为无约束优化问题, 利用梯度方法进行求解; 对于布局点数量较多的情况, 提出了一个界为1/2的多项式时间的近似算法, 并进行了相应的算例分析, 进一步来验证算法解的合理性. 研究的结论及方法一定程度上丰富和完善了圆形区域的分散布局理论.