Raf Jans

Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinational optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig-Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples.

Modeling Industrial Lot Sizing Problems: A Review

In this paper we give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems. The focus of this paper is on the modeling of various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production–distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research.

Formulations and Branch-and-Cut Algorithms for Multivehicle Production and Inventory Routing Problems

The inventory routing problem (IRP) and the production routing problem (PRP) are two difficult problems arising in the planning of integrated supply chains. These problems are solved in an attempt to jointly optimize production, inventory, distribution, and routing decisions. Although several studies have proposed exact algorithms to solve the single-vehicle problems, the multivehicle aspect is often neglected because of its complexity. We introduce multivehicle PRP and IRP formulations, with and without a vehicle index, to solve the problems under both the maximum level (ML) and order-up-to level (OU) inventory replenishment policies. The vehicle index formulations are further improved using symmetry breaking constraints; the nonvehicle index formulations are strengthened by several cuts. A heuristic based on an adaptive large neighborhood search technique is also developed to determine initial solutions, and branch-and-cut algorithms are proposed to solve the different formulations. The results show that the vehicle index formulations are superior in finding optimal solutions, whereas the nonvehicle index formulations are generally better at providing good lower bounds on larger instances. IRP and PRP instances with up to 35 customers, three periods, and three vehicles can be solved to optimality within two hours for the ML policy. By using parallel computing, the algorithms could solve the instances for the same policy with up to 45 and 50 customers, three periods, and three vehicles for the IRP and PRP, respectively. For the more difficult IRP (PRP) under the OU policy, the algorithms could handle instances with up to 30 customers, three (six) periods, and three vehicles on a single core machine, and up to 45 (35) customers, three (six) periods, and three vehicles on a multicore machine.

The production routing problem

The production routing problem (PRP) combines the lot-sizing problem and the vehicle routing problem, two classical problems that have been extensively studied for more than half a century. The PRP is solved in an attempt to jointly optimize production, inventory, distribution and routing decisions and is thus a generalization of the inventory routing problem (IRP). Although the PRP has a complicated structure, there has been a growing interest in this problem during the past decade in both academia and industry. This article provides a comprehensive review of various solution techniques that have been proposed to solve the PRP. We attempt to provide an in-depth summary and discussion of different formulation schemes and of algorithmic and computational issues. Finally, we point out interesting research directions for further developments in production routing.

A New Dantzig-Wolfe Reformulation and Branch-and-Price Algorithm for the Capacitated Lot-Sizing Problem with Setup Times

Although the textbook Dantzig-Wolfe decomposition reformulation for the capacitated lot-sizing problem, as already proposed by Manne [Manne, A. S. 1958. Programming of economic lot sizes. Management Sci.4(2) 115--135], provides a strong lower bound, it also has an important structural deficiency. Imposing integrality constraints on the columns in the master program will not necessarily give the optimal integer programming solution. Mannes model contains only production plans that satisfy the Wagner-Whitin property, and it is well known that the optimal solution to a capacitated lot-sizing problem will not necessarily satisfy this property. The first contribution of this paper answers the following question, unsolved for almost 50 years: If Mannes formulation is not equivalent to the original problem, what is then a correct reformulation? We develop an equivalent mixed-integer programming (MIP) formulation to the original problem and show how this results from applying the Dantzig-Wolfe decomposition to the original MIP formulation. The set of extreme points of the lot-size polytope that are needed for this MIP Dantzig-Wolfe reformulation is much larger than the set of dominant plans used by Manne. We further show how the integrality restrictions on the original setup variables translate into integrality restrictions on the new master variables by separating the setup and production decisions. Our new formulation gives the same lower bound as Mannes reformulation. Second, we develop a branch-and-price algorithm for the problem. Computational experiments are presented on data sets available from the literature. Column generation is accelerated by a combination of simplex and subgradient optimization for finding the dual prices. The results show that branch-and-price is computationally tractable and competitive with other state-of-the-art approaches found in the literature.

A review of applications of genetic algorithms in lot sizing

Lot sizing problems are production planning problems with the objective of determining the periods where production should take place and the quantities to be produced in order to satisfy demand while minimizing production, setup and inventory costs. Most lot sizing problems are combinatorial and hard to solve. In recent years, to deal with the complexity and find optimal or near-optimal results in reasonable computational time, a growing number of researchers have employed meta-heuristic approaches to lot sizing problems. One of the most popular meta-heuristics is genetic algorithms which have been applied to different optimization problems with good results. The focus of this paper is on the recent published literature employing genetic algorithms to solve lot sizing problems. The aim of the review is twofold. First it provides an overview of recent advances in the field in order to highlight the many ways GAs can be applied to various lot sizing models. Second, it presents ideas for future research by identifying gaps in the current literature. In reviewing the relevant literature the focus has been on the main features of the lot sizing problems and the specifications of genetic algorithms suggested in solving these problems.

An industrial extension of the discrete lot-sizing and scheduling problem

We propose a model and solution algorithm for an industrial production planning problem at Solideal, an international tire manufacturer. The tires are built in molds and are produced in heaters in large series production runs. Some tires can be cured in two different types of heaters with different efficiencies. Preparing a heater and mold for a specific tire type requires a start up. The resulting lot-sizing problem is an extension of the standard Discrete Lot-Sizing and Scheduling Problem. The specific extensions which complicate the problem are: (i) general start-up times, which can be a fraction of the time bucket; (ii) multiple alternative machines with different efficiencies; (iii) multiple capacitated resources, namely the molds and heaters; and (iv) backlogging. These issues are directly motivated by our real life production planning problem. We propose a column-generation-based algorithm for this problem. The dynamic programming recursion for the subproblem is substantially improved by using valid bounds on the state space and cost function. Further, Lagrange relaxation is used to reduce the degeneracy of the master problem. We test our algorithm on real life data sets with up to 30 products and 30 periods and find good quality solutions and lower bounds within a reasonable computation time. Our best implementation has an overall average gap of 0.25% on these test problems.

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.

Improved lower bounds for the capacitated lot sizing problem with setup times

We present new lower bounds for the capacitated lot sizing problem, applying decomposition to the network reformulation. The demand constraints are the linking constraints and the problem decomposes into subproblems per period containing the capacity and setup constraints. Computational results and a comparison to other lower bounds are presented.

The economic lot-sizing problem with an emission capacity constraint

We consider a generalisation of the lot-sizing problem that includes an emission capacity constraint. Besides the usual financial costs, there are emissions associated with production, keeping inventory and setting up the production process. Because the capacity constraint on the emissions can be seen as a constraint on an alternative objective function, there is also a clear link with bi-objective optimisation. We show that lot-sizing with an emission capacity constraint is NP-hard and propose several solution methods. Our algorithms are not only able to handle a fixed-plus-linear cost structure, but also more general concave cost and emission functions. First, we present a Lagrangian heuristic to provide a feasible solution and lower bound for the problem. For costs and emissions such that the zero inventory property is satisfied, we give a pseudo-polynomial algorithm, which can also be used to identify the complete set of Pareto optimal solutions of the bi-objective lot-sizing problem. Furthermore, we present a fully polynomial time approximation scheme (FPTAS) for such costs and emissions and extend it to deal with general costs and emissions. Special attention is paid to an efficient implementation with an improved rounding technique to reduce the a posteriori gap, and a combination of the FPTASes and a heuristic lower bound. Extensive computational tests show that the Lagrangian heuristic gives solutions that are very close to the optimum. Moreover, the FPTASes have a much better performance in terms of their actual gap than the a priori imposed performance, and, especially if the heuristic’s lower bound is used, they are very fast.