The mixed and multi model line balancing problem: a comparison

The balancing problem deals with the assignment of tasks to work stations. We can distinguish two approaches in the literature on the mixed model line balancing problem, that both transform this problem into a single model line balancing problem. These approaches use combined precedence diagrams and adjusted task processing times, respectively.An experiment was carried out to compare several heuristics based on the combined precedence diagram. A new optimisation method has been developed. The results indicate that the position of common tasks in the precedence diagram of the different models has a significant effect on both the CPU time and the unequal distribution of the total work content of single models among work stations. Moreover, good solutions with respect to the number of required stations go together with long CPU times. For several instances, we decreased the CPU times considerably without deteriorating the performance of the methods, by using a reversed combined precedence diagram.     

Multiple-source learning precedence graph concept for the automotive industry

In modern production systems, customized mass production of complex products, such as automotive or white goods, is often realized at assembly lines with a high degree of manual labor. For firms that apply assembly systems, the assembly line balancing problem (ALBP) arises, which is to assign optimally tasks to stations or workers with respect to some constraints and objectives. Although the literature provides a number of relevant models and efficient solution methods for ALBP, firms, in most cases, do not use this knowledge to balance their lines. Instead, the planning is mostly performed manually by numerous planners responsible for small sub-problems. This is because of the lack of data, like the precedence relations between the tasks to be performed. Such data is hard to collect and to maintain updated.     

Robust balancing of straight assembly lines with interval task times

This paper addresses the balancing problem for straight assembly lines where task times are not known exactly but given by intervals of their possible values. The objective is to assign the tasks to workstations minimizing the number of workstations while respecting precedence and cycle-time constraints. An adaptable robust optimization model is proposed to hedge against the worst-case scenario for task times. To find the optimal solution(s), a breadth-first search procedure is developed and evaluated on benchmark instances. The results obtained are analysed and some practical recommendations are given.     

A station-oriented enumerative algorithm for two-sided assembly line balancing

In this paper, a station-oriented enumerative algorithm for two-sided assembly lines balancing (TALB) is proposed. First, the time transfer function is defined and combined with the precedence relation to compute the earliest and the latest start time of tasks. With the direction and cycle time constraints, a station-oriented procedure based on the start time is designed to assign tasks, starting from the left station to the rightstation of the position. Some unsuitable position assignments would be finally removed by checking the precedence constraints among the assigned tasks. The proposed algorithm is integrated with the Hoffmann heuristic to develop a system for solving TALB problems. The test is performed on the well-known benchmark set of problem instances. Experimental results demonstrate that the proposed procedure is efficient.     

A two-process implicit enumeration algorithm for the simple assembly line balancing problem

We present an implicit-enumeration algorithm for the simple assembly line balancing problem, with the objective of minimizing the number of stations along the line for a fixed cycle time. The algorithm traverses two branch-and-bound trees, one assigning tasks that appear earlier in the precedence network at lower levels of the tree, the other assigning later tasks at lower levels. The two processes alternately expand nodes of their respective trees, and share information about each other's progress in the form of global bounds. Computational results are reported on several well-known problems from the literature.     

On a learning precedence graph concept for the automotive industry

Assembly line balancing problems (ALBP) consist in assigning the total workload for manufacturing a product to stations of an assembly line as typically applied in automotive industry. The assignment of tasks to stations is due to restrictions which can be expressed in a precedence graph. However, (automotive) manufacturers usually do not have sufficient information on their precedence graphs. As a consequence, the elaborate solution procedures for different versions of ALBP developed by more than 50 years of intensive research are often not applicable in practice.Unfortunately, the known approaches for precedence graph generation are not suitable for the conditions in the automotive industry. Therefore, we describe a new graph generation approach that is based on learning from past feasible production sequences and forms a sufficient precedence graph that guarantees feasible line balances. Computational experiments indicate that the proposed procedure is able to approximate the real precedence graph sufficiently well to detect optimal or nearly optimal solutions for a well-known benchmark data set. Even for additional large instances with up to 1,000 tasks, considerable improvements of line balances are possible. Thus, the new approach seems to be a major step to close the gap between theoretical line balancing research and practice of assembly line planning.     

Optimal task sequencing with precedence constraints

A given finite set of tasks, having known nonnegligible failure probabilities and known costs (or rewards) for their performance, can be performed sequentially until either one of the tasks fails or all tasks have been executed. The allowable task performance sequences are constrained only by certain precedence requirements, which specify that certain tasks must be performed before certain other tasks. Given the individual task failure probabilities and task costs, along with the intertask precedence requirements, the problem is to determine an optimal task performance sequence having minimal expected cost (or maximal expected reward). A number of potential applications of such “task ordering” problems are described, including R&D project organization, design of screening procedures, and determining testing points for sequential manufacturing processes.The main results of this paper are a number of reduction theorems which lead to a very efficient optimization algorithm for a large class of task ordering problems. Though these theorems are not quite sufficient for us to give a fast optimization algorithm, we do show how their use can improve upon exhaustive search techniques.     

Scheduling groups of tasks with precedence constraints on three dedicated processors

We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a specified set of processors. Up to three precedence constraints are considered among groups of tasks requiring the same set of processors. The objective of the problem is to find a nonpreemptive schedule which minimizes the maximum completion time (makespan). This scheduling problem is equivalent to the problem of finding an extension of the constraint graph (i.e. the graph which represents the conflicts between tasks and the precedence constraints) to a comparability graph with minimum (over all the extensions) maximum clique weight. The problem is NP-hard in the strong sense. A normal schedule is such that all the tasks requiring the same set of processors are scheduled consecutively. With a normal schedule the problem reduces to the quotient graph of the constraint graph. In this paper we obtain tight approximation results for the minimum makespan of a normal schedule through tight results on the minimum increase of the maximum clique weight when the (partially oriented) quotient graph is extended to a comparability graph.     

Balancing of simple assembly lines under variations of task processing times

One of the simple assembly line balancing problems (SALBPs), known as SALBP-E, is considered. It consists in assigning a given set V={1,2,??,n} of elementary tasks to linearly ordered workstations with respect to precedence and capacity restrictions while minimizing the following product: number of used workstations × working time on the most loaded one. The stability of feasible and optimal solutions for this problem with regard to possible variations of the processing time of certain tasks is investigated. Two heuristic procedures finding a compromise between the efficiency and the considered stability measure of studied solutions are suggested and evaluated on known benchmarks.     

Shortest-route formulation of mixed-model assembly line balancing problem

A shortest-route formulation of the mixed-model assembly line balancing problem is presented. Common tasks across models are assumed to exist and these tasks are performed in the same stations. The formulation is based on an algorithm which solves the single-model version of the problem. The mixed-model system is transformed into a single-model system with a combined precedence diagram. The model is capable of considering any constraint that can be expressed as a function of task assignments.     

Scheduling tasks and communications on a virtual distributed system

A set of tasks has to be scheduled on identical parallel processors subject to precedence constraints and small communication delays. A polynomial algorithm is known to exist if task duplication is allowed and the number of available processors is not limited. However the problem of communications scheduling is not taken into account. In this paper, we prove that this algorithm also never saturates communication channels and always delivers messages on time, if slightly stronger constraints are imposed on the tasks.     

A minmax regret approach to the critical path method with task interval times

The execution of a given project, with a number of interrelated tasks due to precedence constraints, represents a challenge when one must to control the available resources and the compromised due dates. In this paper, we analyse this problem under uncertain individual task completing times, specifically, we will assume that a given range, for the admissible values of each individual completing time, is available. Taking into account that the precedence relations between tasks must be preserved, each realization of the admissible execution times for the set of tasks will define a new scenario determining the ending time for the project and the subset of critical tasks.     

Periodic schedules for linear precedence constraints

We consider the computation of periodic cyclic schedules for linear precedence constraints graphs: a linear precedence constraint is defined between two tasks and induces an infinite set of usual precedence constraints between their executions such that the difference of iterations is a linear function. The objective function is the minimization of the maximal period of a task.We recall first that this problem may be modelled using linear programming. A polynomial algorithm is then developed to solve it for a particular class of linear precedence graphs called unitary graphs. We also show that a periodic schedule may not exist for unitary graphs. In the general case, a decomposition of the linear precedence graph into unitary components is computed and we assume that a periodic schedule exists for each of these components. Lower bounds on the periods are exhibited and we show that an optimal periodic schedule may not achieve them. The notion of quasi-periodic schedule is then introduced and we prove that this new class of schedules always reaches these bounds.     

Metaheuristic algorithms for balancing robotic assembly lines with sequence-dependent robot setup times

Industries are incorporating robots into assembly lines due to their greater flexibility and reduced costs. Most of the reported studies did not consider scheduling of tasks or the sequence-dependent setup times in an assembly line, which cannot be neglected in a real-world scenario. This paper presents a study on robotic assembly line balancing, with the aim of minimizing cycle time by considering sequence-dependent setup times. A mathematical model for the problem is formulated and CPLEX solver is utilized to solve small-sized problems. A recently developed metaheuristic Migrating Birds Optimization (MBO) algorithm and set of metaheuristics have been implemented to solve the problem. Three different scenarios are tested (with no setup time, and low and high setup times). The comparative experimental study demonstrates that the performance of the MBO algorithm is superior for the tested datasets. The outcomes of this study can help production managers improve their production system in order to perform the assembly tasks with high levels of efficiency and quality.     

The stochastic U-line balancing problem: A heuristic procedure

Many heuristics have been proposed for the assembly line balancing problem due to its computational complexity and difficulty in identifying an optimal solution. Still, the basic line balancing model fails to consider a number of realistic elements. The implementation of a Just-In-Time manufacturing system generally entails the replacement of traditional straight assembly lines with U-shaped lines. An important issue in the U-line balancing problem is the consideration of task time variability due to human factors or various disruptions. In this paper, we consider the stochastic U-line balancing problem. A hybrid heuristic is presented consisting of an initial feasible solution module and a solution improvement module. To gain insight into its performance, we analyze the heuristic under different scenarios of task time variability. Computational results clearly demonstrate the efficiency and robustness of our algorithm.     

Balancing modular transfer lines with serial-parallel activation of spindle heads at stations

The paper deals with an as yet unexplored combinatorial optimization problem concerning balancing complex transfer lines in the machining/process environment. In contrast to similar problems for assembly lines, in transfer line balancing, tasks are grouped into blocks. All tasks of each block are executed simultaneously (in parallel) by one piece of equipment (spindle head). For the transfer lines considered in this paper, spindle heads at each station are activated in serial-parallel order. The set of all available spindle heads is known beforehand. Precedence, cycle time, compatibility, and parallelism constraints for the blocks and tasks are given. The line investment cost is estimated by the sum of block and station costs. The problem is to assign all tasks (using the available blocks) such that all constraints are respected and line investment cost is at a minimum. This paper focuses on solving the problem via a branch-and-bound algorithm. An approach for obtaining an efficient lower bound is offered, based on a reduction of the initial problem to a set partitioning problem. Computational experiments reveal that the proposed approach is efficient mathematically and can be used to solve practical transfer line design problems of a medium size.     

Scheduling UET task systems with concurrency on two parallel identical processors

Problems with unit execution time tasks and two identical parallel processors have received a great deal of attention in scheduling theory. In contrast to the conventional models, where each task requires only one processor, we consider a situation when a task may require both processors simultaneously. For problems without precedence constraints we present several polynomial time algorithms which complement recent results of Lee and Cai. We also show that the introduction of precedence constraints leads to NP-hardness results for maximum lateness and mean flow time objective functions. For the maximum lateness problem, a family of algorithms, based upon the idea of modified due dates, is considered. The worst case behaviour of these algorithms is analysed, and it is shown that the same upper bound is tight for each algorithm of this family.     

A branch and bound algorithm for determining locations of long-term care facilities

This paper focuses on the problem of determining locations for long-term care facilities with the objective of balancing the numbers of patients assigned to the facilities. We present a branch and bound algorithm by developing dominance properties, a lower bounding scheme and a heuristic algorithm for obtaining an upper bound for the problem. For evaluation of the suggested branch and bound algorithm, computational experiments are performed on a number of test problems. Results of the experiments show that the suggested algorithm gives optimal solutions of problems of practical sizes in a reasonable amount of computation time.     

A new diversity induction mechanism for a multi-objective ant colony algorithm to solve a real-world time and space assembly line balancing problem

Time and space assembly line balancing considers realistic multi-objective versions of the classical assembly line balancing industrial problems. It involves the joint optimisation of conflicting criteria such as the cycle time, the number of stations, and/or the area of these stations. The different problems included in this area also inherit the precedence constraints and the cycle time limitations from assembly line balancing problems. The presence of these hard constraints and their multi-criteria nature make these problems very hard to solve. Multi-objective constructive metaheuristics (in particular, multi-objective ant colony optimisation) have demonstrated to be suitable approaches to solve time and space assembly line balancing problems. The aim of this contribution is to present a new mechanism to induce diversity in an existing multi-objective ant colony optimisation algorithm for the 1/3 variant of the time and space assembly line balancing problem. This variant is quite realistic in the automative industry as it involves the joint minimisation of the number and the area of the stations given a fixed cycle time limit. The performance of our proposal is validated considering ten real-like problem instances. Moreover, the diversity induction mechanism is also tested on a real-world instance from the Nissan plant in Barcelona (Spain).     

The complexity of a cyclic scheduling problem with identical machines and precedence constraints

We consider a set T of tasks with unit processing times. Each of them must be executed infinitely often. A uniform constraint is defined between two tasks and induces a set of precedence constraints on their successive executions. We limit our study to a subset of uniform constraints corresponding to two hypotheses often verified in practice: Each execution of T must end by a special task f, and uniform constraints between executions from different iterations start from f. We have a fixed number of identical machines. The problem is to find a periodic schedule of T which maximizes the throughput. We prove that this problem is NP-hard and show that it is polynomial for two machines. We also present another nontrivial polynomial subcase which is a restriction of uniform precedence constraints.