Modelling and optimisation of energy-efficient U-shaped robotic assembly line balancing problems

Within U-shaped assembly lines, the increase of labour costs and subsequent utilisation of robots has led to growing energy consumption, which is the current main expense of auto and electronics industries. However, there are limited researches concerning both energy consumption reduction and productivity improvement on U-shaped robotic assembly lines. This paper first develops a nonlinear multi-objective mixed-integer programming model, reformulates it into a linear form by linearising the multiplication of two binary variables, and then refines the weight of multiple objectives so as to achieve a better approximation of true Pareto frontiers. In addition, Pareto artificial bee colony algorithm (PABC) is extended to tackle this new complex problem. This algorithm stores all the non-dominated solutions into a permanent archive set to keep all the good genes, and selects one solution from this set to overcome the strong local minima. Comparative experiments based on a set of newly generated benchmarks verify the superiority of the proposed PABC over four multi-objective algorithms in terms of generation distance, maximum spread, hypervolume ratio and the ratio of non-dominated solution.     

A realistic multi-manned five-sided mixed-model assembly line balancing and scheduling problem with moving workers and limited workspace

The assembly line balancing problem can completely vary from one production line to the other. This paper deals with a realistic assembly line for the automotive industry inspired by Fiat Chrysler Automotive in North America and Parskhodro in Iran (both large-scale automotive companies). This problem includes some specific requirements that have not been studied in the literature. For example, the assembly line is five-sided, and workers can move along these sides. Due to the limited workspace, all the sides cannot work simultaneously at one station. First, a mixed integer linear programming model is proposed for the problem. Then, the model is improved to have a tighter linear relaxation. Moreover, an effective logic-based Benders’ decomposition algorithm is developed. After careful analysis of problem’s structure, three propositions are introduced. The master problem is well restricted by eight valid inequalities. Two different sub-problem types are defined to extract more information from the master problem’s solution. In this case, the algorithm adds effective cuts that reduce the solution space to the extent possible at each iteration. Thus, the number of iterations is significantly cut down. The performance of the model and algorithm, as well as improvement made on both, is evaluated.     

A Petri net-based heuristic for mixed-model assembly line balancing problem of Type-E

To effectively respond to the changing market demands, a manufacturer should produce variety of products with small lots. Thus, multiple products (models) are assembled simultaneously on a same line. However, it is very challenging to balance such an assembly line. This paper conducts a study on balancing a mixed-model assembly line of Type E. To solve this problem, a coloured-timed Petri net model is developed to describe the task precedence relationship. Also, the optimisation problem is formulated as a mathematical programming model. Then, with the models, a two-stage heuristic algorithm is proposed to solve the problem. At the first stage, based on the Petri net model, a P-invariant algorithm (PA) is presented to minimise the number of workstations. At the second stage, a heuristic is proposed to further minimise the cycle time by combining the PA with a binary search algorithm (BSA). Performance of the proposed method is evaluated by an illustrative example and numerical experiments. It is shown that it works well in terms of both solution accuracy and computational efficiency for large size problems.     

Supply chain optimisation with U-type assembly line balancing

In supply chain optimisation problems, determining the location and number of facilities is considered at a strategic level, while mid-term and short-term decisions such as those concerning assembly policy, inventory levels, lot sizes and scheduling are handled at the tactical and operational levels. However, considering these decisions simultaneously is almost always ignored during the optimisation of distribution/production processes. The aim of the study is to optimise the supply chain network (strategic level), including manufacturers, assemblers and customers, while simultaneously balancing the U-type assembly lines (tactical level) in assemblers. A nonlinear mixed integer mathematical model is proposed to minimise the total costs and the number of assembly stations while minimising the total fixed costs of stations. To show the validity and usefulness of the proposed model, a numerical example with different scenarios and sensitivity analyses is given and discussed.     

Supply chain optimisation with assembly line balancing

Supply chain management operates at three levels, strategic, tactical and operational. While the strategic approach generally pertains to the optimisation of network resources such as designing networks, location and determination of the number of facilities, etc., tactical decisions deal with the mid-term, including production levels at all plants, assembly policy, inventory levels and lot sizes, and operational decisions are related to how to make the tactical decisions happen in the short term, such as production planning and scheduling. This paper mainly discusses and explores how to realise the optimisation of strategic and tactical decisions together in the supply chain. Thus, a supply chain network (SCN) design problem is considered as a strategic decision and the assembly line balancing problem is handled as a tactical decision. The aim of this study is to optimise and design the SCN, including manufacturers, assemblers and customers, that minimises the transportation costs for determined periods while balancing the assembly lines in assemblers, which minimises the total fixed costs of stations, simultaneously. A nonlinear mixed-integer model is developed to minimise the total costs and the number of assembly stations while minimising the total fixed costs. For illustrative purposes, a numerical example is given, the results and the scenarios that are obtained under various conditions are discussed, and a sensitivity analysis is performed based on performance measures of the system, such as total cost, number of stations, cycle times and distribution amounts.     

Stochastic two-sided U-type assembly line balancing: a genetic algorithm approach

In this paper, a novel stochastic two-sided U-type assembly line balancing (STUALB) procedure, an algorithm based on the genetic algorithm and a heuristic priority rule-based procedure to solve STUALB problem are proposed. With this new proposed assembly line design, all advantages of both two-sided assembly lines and U-type assembly lines are combined. Due to the variability of the real-life conditions, stochastic task times are also considered in the study. The proposed approach aims to minimise the number of positions (i.e. the U-type assembly line length) as the primary objective and to minimise the number of stations (i.e. the number of operators) as a secondary objective for a given cycle time. An example problem is solved to illustrate the proposed approach. In order to evaluate the efficiency of the proposed algorithm, test problems taken from the literature are used. The experimental results show that the proposed approach performs well.     

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 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.     

Flexible assembly line design problem with fixed number of workstations

In the paper, we study a flexible assembly line design problem with equipment decisions. We assume the task times and equipment costs are correlated in the sense that for all tasks the cheaper equipment gives no smaller task time. Given the cycle time and number of workstations we aim to find the assignment of tasks and equipment to the workstations so as to minimise the total equipment cost. We develop a branch and bound algorithm that uses powerful lower bounds and reduction mechanisms. Our computational experiments have revealed that our algorithm can solve large-sized problem instances in reasonable solution times.     

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.     

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.     

Extending assembly line balancing problem by incorporating learning effects

In the ramp-up phase, or time to volume of new products, pronounced learning effects are observed. They are present especially on assembly lines producing mass-goods because of a high number of repetitions of the tasks. Shortening the ramp-up phase and reaching the steady-state production as soon as possible generates main advantages for firms that introduce new products. Moreover, a careful planning of the ramp-up stage is getting even more important in view of shorter product life cycles and a growing importance of the ‘time to payback’ financial indicators. Former studies on incorporation of learning effects into assembly line balancing have limited applicability, because they rely on unrealistic assumptions. We model learning effects, based on general and realistic assumptions, as an extension of the Simple Assembly Line Balancing Problem. We propose exact and heuristic solution procedures and perform extensive computational tests. We found that for instances similar to the problems, which arise in firms, the duration of the learning stage can be reduced by up to 10% if our specialised methods are applied.     

Bi criteria flexible assembly line design problem with equipment decisions

In this paper, we address the assembly line balancing and design problem of assigning tasks and equipment to work stations where there are several equipment alternatives for each task. We consider minimizing the total equipment cost and the number of work stations criteria. We aim to generate efficient solutions with respect to these criteria and propose a branch and bound algorithm whose efficiency is enhanced with powerful reduction and bounding mechanisms. We find that our algorithm is capable of solving problem instances with up to 25 tasks and five pieces of equipment.     

Analysis of the type II robotic mixed-model assembly line balancing problem

In recent years, there has been an increasing trend towards using robots in production systems. Robots are used in different areas such as packaging, transportation, loading/unloading and especially assembly lines. One important step in taking advantage of robots on the assembly line is considering them while balancing the line. On the other hand, market conditions have increased the importance of mixed-model assembly lines. Therefore, in this article, the robotic mixed-model assembly line balancing problem is studied. The aim of this study is to develop a new efficient heuristic algorithm based on beam search in order to minimize the sum of cycle times over all models. In addition, mathematical models of the problem are presented for comparison. The proposed heuristic is tested on benchmark problems and compared with the optimal solutions. The results show that the algorithm is very competitive and is a promising tool for further research.     

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.     

The constrained two-dimensional guillotine cutting problem with defects: an ILP formulation,a Benders decomposition and a CP-based algorithm

This paper addresses a variant of two-dimensional cutting problems in which rectangular small pieces are obtained by cutting a rectangular object through guillotine cuts. The characteristics of this variant are (i) the object contains some defects, and the items cut must be defective-free; (ii) there is an upper bound on the number of times an item type may appear in the cutting pattern; (iii) the number of guillotine stages is not restricted. This problem commonly arises in industrial settings that deal with defective materials, e.g. either by intrinsic characteristics of the object as in the cutting of wooden boards with knotholes in the wood industry, or by the manufacturing process as in the production of flat glass in the glass industry. We propose a compact integer linear programming (ILP) model for this problem based on the discretisation of the defective object. As solution methods for the problem, we develop a Benders decomposition algorithm and a constraint-programming (CP) based algorithm. We evaluate these approaches through computational experiments, using benchmark instances from the literature. The results show that the methods are effective on different types of instances and can find optimal solutions even for instances with dimensions close to real-size.     

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.     

A discrete hybrid differential evolution algorithm for solving integer programming problems

Differential evolution (DE) is one of the most prominent new evolutionary algorithms for solving real-valued optimization problems. In this article, a discrete hybrid differential evolution algorithm is developed for solving global numerical optimization problems with discrete variables. Orthogonal crossover is combined with DE crossover to achieve crossover operation, and the simplified quadratic interpolation (SQI) method is employed to improve the algorithm's local search ability. A mixed truncation procedure is incorporated in the operations of DE mutation and SQI to ensure that the integer restriction is satisfied. Numerical experiments on 40 test problems including seventeen large-scale problems with up to 200 variables have demonstrated the applicability and efficiency of the proposed method.     

A corrected formulation for the double row layout problem

This paper presents a corrected formulation to the mixed integer programming model of the double-row layout problem (DRLP), first proposed by Chung and Tanchoco (2010 Chung, J and Tanchoco, J. 2010. The double row layout problem. International Journal of Production Research, 48(3): 709–727. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], The double row layout problem. International Journal of Production Research, 48 (3), 709–727). In the DRLP, machines are placed along two rows of a corridor, where the objective is to minimise the total cost of material handling for products that move between these machines. We highlight the errors in the original formulation, propose corrections to the formulation, and provide an analytical validation of the corrections.     

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.