Constraint programming for solving various assembly line balancing problems

PurposeAssembly lines are widely employed in manufacturing processes to produce final products in a flow efficiently. The simple assembly line balancing problem is a basic version of the general problem and has still attracted the attention of researchers. The type-I simple assembly line balancing problems (SALBP-I) aim to minimise the number of workstations on an assembly line by keeping the cycle time constant.Design/methodology/approachThis paper focuses on solving multi-objective SALBP-I problems by utilising an artificial bee colony based-hyper heuristic (ABC-HH) algorithm. The algorithm optimises the efficiency and idleness percentage of the assembly line and concurrently minimises the number of workstations. The proposed ABC-HH algorithm is improved by adding new modifications to each phase of the artificial bee colony framework. Parameter control and calibration are also achieved using the irace method. The proposed model has undergone testing on benchmark problems, and the results obtained have been compared with state-of-the-art algorithms.FindingsThe experimental results of the computational study on the benchmark dataset unequivocally establish the superior performance of the ABC-HH algorithm across 61 problem instances, outperforming the state-of-the-art approach.Originality/valueThis research proposes the ABC-HH algorithm with local search to solve the SALBP-I problems more efficiently.

Assembly lines of big-size products such as buses, trucks and helicopters are very different from the lines studied in the literature. These products’ manufacturing processes have a lot of tasks most of which have long task times. Since traditional assembly line models including only one worker in each station (i.e. simple assembly lines) or at most two workers (two-sided assembly lines) may not be suitable for manufacturing these type of products, they need much larger shop floor for a number of stations and long product flow times. In this study, an assembly line balancing problem (ALBP) with parallel multi-manned stations is considered. Following the problem definition, a mixed integer programming formulation is developed. A detailed study of priority rules for simple ALBPs is also presented, and a new efficient constructive heuristic algorithm based on priority rules is proposed. In order to improve solutions found by the constructive heuristic, a genetic algorithm-based solution procedure is also presented. Benchmark instances in the literature are solved by using the proposed mathematical programming formulation. It has been seen that only some of the small-size instances can be solved optimally by this way. So the efficiency of the proposed heuristic method is verified in small-size instances whose optimal solutions are found. For medium- and big-size instances, heuristics’ results and CPU times are demonstrated. A comparative evaluation with a branch and bound algorithm that can be found in the literature is also carried out, and results are presented.

An assembly system consists of work stations where specific tasks are carried in such a way that last station gives the complete product. An assembly line balancing Problem (ALBP) involves optimally assigning tasks among workstations with respect to some performance objective. ALBP problems are of NP hard nature and in literature; many efforts are there to solve the problem efficiently through heuristics. This paper proposes a heuristic algorithm for solving type 2 ALBP for single objective optimization. The proposed algorithm assigns tasks to a fixed no of stations with objective of minimizing cycle time. In literature, type 2 ALBP is mostly solved through Type 1 problem. However, this paper proposes a direct approach to Type 2 ALBP. The effectiveness is tested through application on Gunther problem of 35 tasks with forty five precedence constraints. The task assignment for six stations is computed and it shows competitive performance. The number of fixed stations is varied and corresponding cycle times are computed. The algorithm is also tested on a real industrial problem of 24 tasks. The experimental testing indicates tendency of the proposed algorithm to give effective optimal results.

In this paper a constraint programming (CP) approach for calculating release dates for lots within a supply chain environment is investigated. The lot start times are verified by a simulation model using different dispatching rules focusing on tardiness. To test the presented CP approach a simple fab model is constructed. The fab model consists of parallel batch machines as well as work centers and single machines. The investigated objectives are tardiness, earliness and cycle time. Due to the high complexity decomposition methods for the CP approach are tested. The results from the CP method are lot start dates which are verified by a downstream simulation run. The results show that the presented CP approach could outperform the simulation for all objectives. The content of this paper could be used as a first investigation for new scheduling methods within a supply chain management.

Cell formation deals with grouping of machines and parts in manufacturing systems according to their compatibility. Manufacturing processes are surrounded with an abundance of complex constraints which should be considered carefully and represented clearly for obtaining high efficiency and productivity. Constraint programming is a new approach to combinatorial optimization and provides a rich language to represent complex constraints easily. However, the cell formation problems are well suited to be solved by constraint programming approach since the problem has many constraints such as part-machine requirements, availabilities in the system in terms of capacity, machine and worker abilities. In this study, the cell formation problem is modeled using machine, part processing and worker flexibilities via resource element–based representation. Resource elements define the processing requirements of parts and processing capabilities of machines and workers, which are resource-independent capability units. A total of 12 case problems are generated, and different search phases of constraint programming are defined for the solution procedure. The cell formation problem is modeled in both constraint programming and integer programming, and a comparative analysis of constraint programming and integer programming model solutions is done. The results indicate that both the models are effective and efficient in the solution of the cell formation problem.

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

Assembly line balancing is a hard problem concerned with the assignment of individual work elements or tasks to workstations so as to minimize the assembly cost as much as possible. Decisions associated with line balancing have a direct influence on the costs of production. Basic heuristic algorithms have been widely used to solve the assembly line balancing problem (ALBP). However, to solve hard, large-scale ALBPs, efficient decision tools are essential. Therefore, developing efficient algorithms that can yield optimal or near-optimal solutions is of utmost importance. In this chapter, a hybrid grouping genetic algorithm is developed to address complex problems. The algorithm incorporates basic heuristics, unique genetic operators, and other techniques to enhance the optimization search process. Comparative computational results based on established test problems are presented. The results show that the hybrid grouping genetic algorithm is more efficient and effective than competitive algorithms. The algorithm can be used to assist decision makers in assembly line balancing decisions, with minimal computational requirements.

The peptide-protein docking problem is an important problem in structural biology that facilitates rational and efficient drug design. In this work, we explore modeling and solving this problem with the quantum-amenable quadratic unconstrained binary optimization (QUBO) formalism. Our work extends recent efforts by incorporating the objectives and constraints associated with peptide cyclization and peptide-protein docking in the two-particle model on a tetrahedral lattice. We propose a “resource efficient” QUBO encoding for this problem, and baseline its performance with a novel constraint programming (CP) approach. We implement an end-to-end framework that enables the evaluation of our methods on instances from the Protein Data Bank (PDB). Our results show that the QUBO approach, using a classical simulated annealing solver, is able to find feasible conformations for problems with up to 6 peptide residues and 34 target protein residues (PDB 3WNE, 5LSO), but has trouble scaling beyond this problem size. In contrast, the CP approach can solve the largest instance in our test set, containing 11 peptide residues and 49 target protein residues (PDB 2F58). We conclude that while QUBO can be used to successfully tackle this problem, its scaling limitations and the strong performance of the CP method suggest that it may not be the best choice.

PurposeThe purpose of this paper is to describe the introduction of simple assembly line balancing problems (SALBPs) under four joint combinations of two learning effects and two deterioration effects.Design/methodology/approachIn this paper, effects of learning and deterioration are considered simultaneously. By the effects of learning and deterioration, it is meant that the task time of a task is defined by increasing function of its execution start time and position in the sequence.FindingsIt was shown that polynomial solutions can be obtained for SALBP under four joint combinations of two learning effects and two deterioration effects. Furthermore, this paper also discusses the adaptations of the proposed models by using known ALB problem.Originality/valueIn this paper, SALBPs were introduced under four joint combinations of two learning effects and two deterioration effects. Position dependent learning effect and time dependent learning effect were considered for learning effect when linear deterioration effect and non‐linear deterioration effect were taken up.

The process of optimally assigning the timed tasks required to assemble a product to an ordered sequence of workstations is known as the Assembly Line Balancing (ALB) problem. Typical approaches to ALB assume a strict mathematical posture and mostly treat it as a combinatorial optimization problem with the objective of minimizing the idle time across the workstations, while satisfying precedence constraints. The actual nature of the tasks assigned is seldom taken into consideration. While this approach may yield satisfactory cycle time results on paper, it often leads to inconvenient task assignments in an actual work environment. It has been postulated in the literature that assigning groups of related tasks to the same workstation may lead to assembly lines that exhibit increased robustness in real-world situations at the expense of a slightly increased cycle time. The prototypical example of such an approach, Agrawal’s Largest Set Rule (LSR), utilizes backward work relatedness to assign a set of cohesive tasks to the proper workstation. In this paper, we enhance the performance of the original LSR algorithm through the concept of bi-directional work relatedness, where backward and forward relationships are taken into consideration for task assignments. The proposed concept leads to comparable cycle time and improved work relatedness. Applying this novel concept to a benchmark ALB problem demonstrates the feasibility and applicability of the proposed approach.

This paper presents a VLSI design for a competitive neural network model, known as GENET (Wang and Tsang 1991), for solving Constraint Satisfaction Problems (CSP). The CSP is a mathematical abstraction of the problems in many AI application domains. In essence, a CSP can be defined as a triple (Z, D, C), where Z is a finite set of variables, D is a mapping from every variable to a domain, which is a finite set of arbitrary objects, and C is a set of constraints. Each constraint in C restricts the values that can be simultaneously assigned to a number of variables in Z. If the constraints in C involve up to but no more than n variables it is called an n-ary CSP. The task is to assign one value per variable satisfying all the constraints in C (Mackworth 1977). In addition, associated with the variable assignments might be costs and utilities. This turns CSPs into optimization problems, demanding CSP solvers to find a set of variable assignments that would produce a maximum total utility at a minimal cost. Furthermore, some CSPs might be over-constrained, i.e. not all the constraints in C can be satisfied simultaneously. In this case, the set of assignments to a maximum number of variables without violating any constraints in C, or the set of assignments to all the variables which violates a minimal number of constraints might be sought for.

Enhanced mixed integer programming model for a transfer line design problem

p-Hub location-allocation problem is one of the most interesting subjects in the location theory. Hubs act as switching points to reduce the transportation cost. In this study, two new solution methods, a constraint programming (CP) based model and a hybrid of k-means and genetic algorithm (KGA), are developed to generate exact and approximate solutions, respectively. The proposed CP formulation is more understandable and straightforward in comparison with the MIP model. The experimental results indicate that the CP model uses the memory of the computer (RAM) more efficiently, which enables us to solve the medium size problems. But, in terms of run time, this method cannot be superior to the MIP model. The CP formulation is also extended for the multi allocation p-hub location problem. K-means algorithm, a well-known algorithm for clustering data, is used to generate initial solutions of GA. Furthermore, a new adaptive crossover operator, which is based on the k-means algorithm, is proposed. The experimental results indicate that the KGA algorithm is superior to the GA, regarding time, objective value, and quality of solution measures.

The work describes a scheduling system based on an opportunistic framework. It exploits the potentiality of a predictive module that provides a-priori information about the conflicts that may arise in a schedule, managing a constraint based approach in problem solving. The system SOS has been designed and realized to apply in real productive environments this theoretical approach. Requirements of efficiency and flexibility of use have been particularly considered. GUIDELINES AND SCHEDULING STRATEGIES IN SOS In this paper we will present an approach to scheduling problems which is focused on the requirements of real manufacturing environments. Therefore we considered aspects that are tied to the managerial perspective of the problem. The SOS system is designed both for computing solutions in scheduling problems and for supporting manager in obtaining schedules that reflect the state of shop-floor and organizational problems in production. Two aspects have been considered in these perspective: • it develops a temporal predictive method [7] to support a preventive analysis of activity interactions and to detect bottlenecks. It exploits an a-priori analysis of the effects of organizational constraints on the scheduling problem, evaluating alternatives as quick as possible. Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 628 Artificial Intelligence in Engineering • If some unexpected event happens on the job shop, it works as a support tool for the evaluation of different reaction decisions (reactive scheduling [4]). The contribution of Al-based techniques is focused especially on the analysis of the effects that scheduling variables have on solutions and on the definition of adequate reasoning patterns. We based our approach on opportunistic reasoning [5]. This term characterizes a problem solving process in which the focus of attention is constantly directed to those choices that appear the most promising for reaching a solution. This policy is expected to reduce the search space (and therefore the computational complexity), focusing the reasoning on areas that are considered as the more constrained (island driving). The capability to provide a feasible solution in a sufficiently short time is fundamental for manufacturing environments, but we must remember that the existence of at least one feasible solution is not always guaranteed. The evaluation of effects of constraints becomes fundamental to identify a solution. Production managers want to know how the decision to change or to relaxing the value of some constraints (concerning for example orders due-dates or priorities) affects the quality of final schedules and the global productive efficiency. This feature is particularly important when a feasible solution does not exist; in this case, from a managerial viewpoint, it is required to the system to support the user in building a feasible solution. From a mathematical perspective we considered scheduling as a Constraint Satisfaction Problem (CSP) [6], concerned with the assignment of values to variables subject to a set of constraints. The solution of the CSP (a feasible set of values assumed by the variables) depends on the constraints defined on the problem: by changing the constraints, the solution will change, too. In solving a CSP, two decisions have to be made at each cycle, i.e. which variable to instantiate next and which value to assign to that variable. This formalization highlights the importance associated to constraints analysis: • they allow the quick computation of a feasible solution, without exploring the entire problem domain. • Through their analysis we can obtain information about how they can be modified opportunistically during scheduling. Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 Artificial Intelligence in Engineering 629 We distinguished between two classes of constraints [3], considering the possibility for production managers to change their values in a very short time horizon: Restrictions: they must be always respected and cannot be easily and quickly changed; they can be furtherly divided into causal constraints (they represent conditions to be satisfied before starting an activity), physical constraints (each equipment has specific capabilities that restrict the type and the speed of operations it can accomplish), and resources unavailability. Preferences: restrictions leave scheduling problem underconstrained. Preferences represent the set of managerial constraints and the definition of their values is subject to a decision process. Some examples are: defining orders due date and priority, meeting due-dates, minimizing WIP, maximizing resource utilization, the definition of operational preferences (an order may follow alternative production paths in shop floor), etc. The role of constraints can be underlined considering a second classification, which concerns the links among activities. In a manufacturing environment orders (made of many activities) are processed contemporaneously; moreover many of them may require the same productive resources. These connections cause two kinds of constraints [9]: • intra-order: they represent explicit links existing among activities belonging to an order (for example temporal relations [1]. These constraints are explicitly defined in the problem. • Inter-order: they represent implicit links existing among activities belonging to different orders. For example activities belonging to different orders may require contemporaneously the same productive resource. These constraints remain implicit after the problem definition. THE SOS SCHEDULING PROCESS The goal of our system is to find solutions that satisfies a given set of constraints as quick as possible. If a solution does not exist, the system relaxes only preferences, minimizing the overall effects on final schedule. The scheduling policies followed in SOS can be summarized as follows: 1. Scheduling is made allocating one activities at a time, consider at each step the effects of intraand inter-order constraints; Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 630 Artificial Intelligence in Engineering 2. Most constrained and least-impact policies are implemented at each step. First the most-constrained policy selects dynamically on which agent must be focused scheduling attention. Then, the least impact policy chooses for that agent a value whose impact on the rest of the non-scheduled agents is as small as possible. The strategic goal is the identification of critical activities that heavily rely on the possession of highly contended temporal intervals or resources because of intra-order and inter-order interactions (look-ahead strategy). These two policies are based on numeric indexes which account for the particular structure of a problem. They give a measure of the interaction among activities and resources in terms of variable looseness (it is a measure of constraint degree) and value goodness (it is a measure of the impact of a scheduling decision on non-scheduled activities) [9]. Finally -we decided to avoid the utilization of backtracking in SOS systems because of the particular field of application that we identified. In real problems, we have no a priori guarantees concerning the existence of a solution: in this case backtracking is completely useless. Moreover, in order to meet our basic requirements, we prefer to look for a satisfying solution in a short time, instead than to wait for the best solution (which may not exist). V)PI ( Predictive pre-analysis 1 constraints modification / jtraints modification W Result and jus sy ' xi' :•:':':•:•::•:: •;::-:•: j @ Sch, Pr X^x:: :.:••::.::.-::: s display jtification stem

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.