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.     

Minimizing the number of workers in a paced mixed-model assembly line

We study a problem of minimizing the maximum number of identical workers over all cycles of a paced assembly line comprised of m stations and executing n parts of k types. There are lower and upper bounds on the workforce requirements and the cycle time constraints. We show that this problem is equivalent to the same problem without the cycle time constraints and with fixed workforce requirements. We prove that the problem is NP-hard in the strong sense if $m=4$ and the workforce requirements are station independent, and present an Integer Linear Programming model, an enumeration algorithm and a dynamic programming algorithm. Polynomial in k and polynomial in n algorithms for special cases with two part types or two stations are also given. Relations to the Bottleneck Traveling Salesman Problem and its generalizations are discussed.     

Experiments in mixed-integer linear programming using pseudo-costs

This paper discusses heuristic branch and bound methods for solving mixed integer linear programming problems. The research presented on here is the follow on to that recorded in [3].After a resumé of the concept of pseudo-costs and estimations, new heuristic rules for generating a tree which make use of pseudo-costs and estimations are presented. Experiments have shown that models having a low percentage of integer variables behave in a radically different way from models with a high percentage of integer variables. The new heuristic rules seem to apply generally to the first type of model.Later, other heuristic rules are presented that are used with models having a high percentage of integer variables and with models having a special structure (models including special ordered sets.)The rules introduced here have been implemented in the IBM Mathematical Programming System Extended/370. They are used to solve large mixed integer linear programming models.Numerical results that permit comparisons to be made among the different rules are provided and discussed.     

Optimization of air vehicles operations using mixed-integer linear programming

A scenario where multiple air vehicles are required to prosecute geographically dispersed targets is considered. Furthermore, multiple tasks are to be successively performed on each target, that is, the targets must be classified, attacked, and verified as destroyed. The optimal, for example, minimum time, performance of these tasks requires cooperation among the vehicles such that critical timing constraints are satisfied, that is, a target must be classified before it can be attacked, and an air vehicle is sent to a target area to verify its destruction only after the target has been attacked. In this paper, the optimal task assignment/scheduling problem is posed as a mixed-integer linear program (MILP). The solution of the MILP assigns all tasks to the vehicles and performs the scheduling in an optimal manner, including staged departure times. Coupled tasks involving timing and task order constraints are automatically addressed. When the air vehicles have sufficient endurance, the existence of a solution is guaranteed.     

Controlling mixed-model assembly lines in real-time by using distributed systems

The production of large quantities is frequently handled by the use of assembly lines. These systems yield significant reductions of variable costs for the production of homogenous products. But due to increasing competition and differentiated demands, today it is no longer sufficient to offer only standardized products. To combine the wishes and requirements of the customers with the advantages of an efficient flow line production, different product variants are produced simultaneously on the same mixed-model assembly line. Therefore, the task of controlling such production processes is becoming more complex since an efficient production execution has to be realized in spite of the occurring oscillating work content of the different variants. Additionally, unforeseen disturbances in the production process can compromise its planned execution. To deal with these problems, this paper proposes a new approach for an adaptive real-time control of assembly lines that extends the pure sequencing problem by the integration of specific line-balancing aspects and the mapping of consequences of possible disturbance scenarios. Such a scenario could be, for instance, the loss of a worker, a material bottleneck or a machine breakdown. To guarantee an efficient continuation of the production process, the controlling instrument reacts instantly to a disturbance by adapting the current plan in a very short time, consisting of only a few cycle times, to reduce the additional costs caused by the disturbances.     

Bicriteria sequencing methods for the mixed-model assembly line in just-in-time production systems

A mixed-model manufacturing facility operating in a pull production environment can be controlled by setting a production schedule only for the last process in the facility which is usually an assembly line of mixed-model type. In the mixed-model sequencing problems, two major goals are considered: (1) smoothing the workload on each workstation on the assembly line, and (2) keeping a constant rate of usage of all parts used on the assembly line. In this study, first, some well-known solution approaches with goal 2 are analyzed through minimizing the sum-of-deviations of actual production from the desired amount. The approaches that are found to be performing better than the others are extended for the bicriteria problem considering goals 1 and 2, simultaneously. It is also shown that the bicriteria problem with the sum-of-deviations type objective function can also be formulated as an assignment problem, and the optimal solution to the small-sized problems can thus be obtained by solving the assignment problem. Finally, the conditions when it is important to take the workload-smoothing goal into consideration are analyzed.     

Experiments in mixed-integer linear programming

This paper presents a branch and bound method for solving mixed integer linear programming problems. After briefly discussing the bases of the method, new concepts called pseudo-costs and estimations are introduced. Then, the heuristic rules for generating the tree, which are the main features of the method, are presented. Numerous parameters allow the user for adjusting the search strategy to a given problem.This method has been implemented in the IBM Extended Mathematical Programming System in order to solve large mixed integer L. P. problems. Numerical results making comparisons between different choices of rules are provided and discussed.This paper was presented at the 7th Mathematical Programming Symposium The Hague, The Netherlands.     

Local cuts for mixed-integer programming

A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane, if one exists, and via substitution this gives a cutting plane in the original space. We extend this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes. Some of these possibilities are explored computationally, both in floating-point arithmetic and in rational arithmetic.     

Lifting for conic mixed-integer programming

Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.     

Lattice-free sets,multi-branch split disjunctions,and mixed-integer programming

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing $n$ -dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with $n$ integer variables is a $t$ -branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value $t$ , for which all facets of polyhedral mixed-integer sets with $n$ integer variables can be generated as $t$ -branch split cuts, grows exponentially with $n$ . In particular, when $n=3$ , we observe that not all facet-defining inequalities are 6-branch split cuts.     

Fenchel decomposition for stochastic mixed-integer programming

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature involving the special structure of knapsack problems with nonnegative left-hand side coefficients. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general. Since Fenchel cuts can be computationally expensive in general and are best suited for problems with special structure, both algorithms exploit the special structure of the test instances by reducing the size of the cut generation problems based on the number of nonzero components in the non-integer solution that needs to be cut off.     

Feasibility in reverse convex mixed-integer programming

In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility (CF), that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility.     

Computing the Pareto frontier of a bi-objective bi-level linear problem using a multiobjective mixed-integer programming algorithm

In this article, we study the bi-level linear programming problem with multiple objective functions on the upper level (with particular focus on the bi-objective case) and a single objective function on the lower level. We have restricted our attention to this type of problem because the consideration of several objectives at the lower level raises additional issues for the bi-level decision process resulting from the difficulty of anticipating a decision from the lower level decision maker. We examine some properties of the problem and propose a methodological approach based on the reformulation of the problem as a multiobjective mixed 0–1 linear programming problem. The basic idea consists in applying a reference point algorithm that has been originally developed as an interactive procedure for multiobjective mixed-integer programming. This approach further enables characterization of the whole Pareto frontier in the bi-objective case. Two illustrative numerical examples are included to show the viability of the proposed methodology.     

Computational study of a family of mixed-integer quadratic programming problems

We present computational experience with a branch-and-cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest real-life problems in a few minutes of run-time.     

Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.     

Testing cut generators for mixed-integer linear programming

In this paper, a methodology for testing the accuracy and strength of cut generators for mixed-integer linear programming is presented. The procedure amounts to random diving towards a feasible solution, recording several kinds of failures. This allows for a ranking of the accuracy of the generators. Then, for generators deemed to have similar accuracy, statistical tests are performed to compare their relative strength. An application on eight Gomory cut generators and six Reduce-and-Split cut generators is given. The problem of constructing benchmark instances for which feasible solutions can be obtained is also addressed. Supported by ONR grant N00014-09-1-0033.     

Sequencing mixed-model assembly lines: Survey,classification and model critique

Manufacturers in a wide range of industries nowadays face the challenge of providing a rich product variety at a very low cost. This typically requires the implementation of cost efficient, flexible production systems. Often, so called mixed-model assembly lines are employed, where setup operations are reduced to such an extent that various models of a common base product can be manufactured in intermixed sequences. However, the observed diversity of mixed-model lines makes a thorough sequence planning essential for exploiting the benefits of assembly line production. This paper reviews and discusses the three major planning approaches presented in the literature, mixed-model sequencing, car sequencing and level scheduling, and provides a hierarchical classification scheme to systematically record the academic efforts in each field and to deduce future research issues.     

Scheduling just-in-time part supply for mixed-model assembly lines

With increasing cost competition and product variety, providing an efficient just-in-time (JIT) supply has become one of the greatest challenges in the use of mixed-model assembly line production systems. In the present paper, therefore, we propose a new approach for scheduling JIT part supply from a central storage center. Usually, materials are stored in boxes that are allotted to the consumptive stations of the line by a forklift. For such a real-world problem, a new model, a complexity proof as well as different exact and heuristic solution procedures are provided. Furthermore, a direct comparison with a simple two-bin kanban system is provided. Such a system is currently applied in the real-world industrial process that motivates our research. It becomes obvious that this policy is considerably outperformed according to the resulting inventory- and α-service levels. Moreover, at the interface between logistics and assembly operations, strategic management implications are obtained. Specifically, based on the new approach, it is the first time a statistical analysis is being made as to whether widespread Level Scheduling policies, which are well-known from the Toyota Production System, indeed facilitate material supply. Note that in the literature it is frequently claimed that this causality exists.     

Safe bounds in linear and mixed-integer linear programming

Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size. Mathematics Subject Classification (2000):primary 90C11, secondary 65G20     

On approximation algorithms for concave mixed-integer quadratic programming

Concave mixed-integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an \(\epsilon \)-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in \(1/\epsilon \), provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless \(\mathcal {P}=\mathcal {NP}\).