A network flow model for the capacitated lot-sizing problem

A heuristic lot sizing algorithm for a GT cell

Nonconvex network flow models are used in a wide variety of problem domains involving discounting or economies of scale. In this paper, we present two “enhancements” to the traditional branch-and-bound procedure for solving minimum cost network flow problems with concave arc cost functions. The first enhancement, “conditional penalties”, is analogous to the “up and down” penalties used in integer programming. We show how conditional penalties can be applied to problems with arbitrary concave objective functions. The second enhancement, “capacity improvement”, systematically reduces the feasible region. Previous capacity improvement techniques were based on a linear relaxation of the problem. In this paper, we base capacity improvement on a non-convex relaxation that is tighter than the linear relaxation. For a series of network flow test problems with concave quadratic objective functions, these enhanced methods found the global optimal solutions from two to sixty times faster than the traditional approach.

A new dual simplex type algorithm for the Minimum Cost Network Flow Problem (MCNFP) is presented. The proposed algorithm belongs to a special 'exterior- point simplex type' category. Similarly to the classical network dual simplex algorithm (NDSA), this algorithm starts with a dual feasible tree-solution and reduces the primal infeasibility, iteration by iteration. However, contrary to the NDSA, the new algorithm does not always maintain a dual feasible solution. Instead, the new algorithm might reach a basic point (tree-solution) outside the dual feasible area (exterior point - dual infeasible tree).

The multicommodity flow problem is a generalization of the minimum cost network flow problem. In this problem, several commodities governed by their own network flow constraints share the same underlying network. Given the supplies and demands of the different commodities, the shared capacity of each arc in the network, and cost of flow of each commodity on each arc, the objective is to determine the flow that minimizes the total cost. The multicommodity flow problem has been extensively applied to solve problems in areas such as transportation, telecommunications, and scheduling. In this article, we introduce the multicommodity flow problem, outline some of its applications, and describe efficient methods to solve the problem.

The lot‐sizing problem in capacitated multi‐stage systems with a serial product structure is addressed. This is a complex optimization problem that is part of the decision set in material requirements planning (MRP) systems. The mathematical model that describes the problem uses the concept of echelon stock and includes lead times. Setup times are taken into account, which implies that the problem of finding a feasible solution is NP‐Complete. This paper proposes a heuristic method that provides a production plan in order to minimize inventory, production, and setup costs. The heuristic starts from a solution for the uncapacitated problem, which is given by the sequential application of the Wagner‐Whitin algorithm. Feasibility is then attempted by shifting production amounts between periods. Computational tests conducted in 1,800 instances with up to 40 components and 18 periods have shown that feasible solutions were obtained in 83.7% of the instances. For the infeasible instances, on average, the heuristic is able to find solutions with very low capacity excess. The solutions' quality is evaluated through a lower bound provided by Lagrangean relaxation and on average the gap is less than 10%.

Two strongly polynomial cut cancelling algorithms for minimum cost network flow

In this paper we examine an integer programming formulation of the resource constrained shortest path problem. This is the problem of a traveller with a budget of various resources who has to reach a given destination as quickly as possible within the resource constraints imposed by his budget. A lagrangean relaxation of the integer programming formulation of the problem into a minimum cost network flow problem (which in certain circumstances reduces to an unconstrained shortest path problem) is developed which provides a lower bound for use in a tree search procedure. Problem reduction tests based on both the original problem and this lagrangean relaxation are given. Computational results are presented for the solution of problems involving up to 500 vertices, 5000 arcs, and 10 resources.

Finding the largest suborder of fixed width in a partially ordered set is an interesting combinatorial problem with applications in combinatorial optimization and scheduling. We present a polynomial time solution for this problem by transforming it into a minimum cost network flow problem in an appropriate auxiliary network.

High-level synthesis tools automatically produce RTL designs from algorithmic specifications. These designs, however, are not necessarily easy to test. In this paper we present TBINET, an algorithm for module and register binding, which generates RTL designs having low testability overheads. It obtains a heuristic solution to the binding problem by mapping it onto a sequence of minimum cost network flow problems which can be solved very quickly. A cost function that considers the testability of the design is defined in the paper. The results of experiments on various benchmarks show that the designs produced by our binding algorithm are indeed easier to test as compared to circuits designed without testability considerations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The allocation of tools to machines determines potential part routes in flexible manufacturing systems. Given production requirements and a minimum feasible set of tools, the decision of how to fill vacant slots in tool magazines to maximize routing flexibility is shown to be a minimum cost network flow problem for the cases when routing flexibility is a function of the average workload per tool aggregated over tool types, or of the number of possible routes through the system. A linear programming model is then used to plan a set of routes for each part type so as to minimize either the material handling requirement or the maximum workload on any machine. The impact of these tool addition strategies on the material handling and workload equalization is investigated and computational results presented. The advantage of the overall approach is computational simplicity at each step and the ability to react to dynamic changes.

In DiffServ networks QoS-based routing for only one QoS measure (e.g. constrained bandwidth) is known as the restricted shortest path (RSP) problem. It can be treated as the minimum cost network flow problem in the multicommodity single source multiple destination network. For constrained-based path selection in QoS routing with N quality of service levels in function to serve predicted traffic demands, an efficient heuristic algorithm is being developed. The algorithm approach is of BFH (backward-forward heuristic) type. The optimal flow theory enables separation of these extreme flows which can be a part of an optimal expansion solution from those which cannot be. Algorithm is compared with algorithm based on exact approach. In all numerical test-examples the best possible result is achieved, but with significant reduction of complexity and computation savings.

A multi-unit tender award process: The case of Transantiago

Applying steepest-edge techniques to a network primal-dual algorithm

A memetic algorithm for a multistage capacitated lot-sizing problem

Pricing, relaxing and fixing under lot sizing and scheduling