Single-item Capacitated Lot-sizing Problem Research Articles

Learning Optimal Solutions via an LSTM-Optimization Framework

In this study, we present a deep learning-optimization framework to tackle dynamic mixed-integer programs. Specifically, we develop a bidirectional Long Short Term Memory (LSTM) framework that can process information forward and backward in time to learn optimal solutions to sequential decision-making problems. We demonstrate our approach in predicting the optimal decisions for the single-item capacitated lot-sizing problem (CLSP), where a binary variable denotes whether to produce in a period or not. Due to the dynamic nature of the production and inventory levels that must meet the periodic demand, the CLSP can be treated as a sequence labeling task where a recurrent neural network can capture the problem’s temporal dynamics. Computational results show that our LSTM-Optimization (LSTM-Opt) framework significantly reduces the solution time of benchmark CLSP problems without much loss in feasibility and optimality. For example, the predictions at the 85% level reduce the CPLEX solution time by a factor of 9 on average for over 240000 test instances with an optimality gap of less than 0.05% and 0.4% infeasibility in the test set. Also, models trained using shorter planning horizons can successfully predict the optimal solution of the instances with longer planning horizons. For the hardest data set, the LSTM predictions at the 25% level reduce the solution time of 70 CPU hours to less than 2 CPU minutes with an optimality gap of 0.8% and without any infeasibility. The LSTM-Opt framework outperforms classical ML algorithms, such as the logistic regression and random forest, in terms of the solution quality, and exact approaches, such as the ( $$\ell$$ , S) and dynamic programming-based inequalities, with respect to the solution time improvement. Our machine learning approach could be beneficial in tackling sequential decision-making problems similar to CLSP, which need to be solved repetitively, frequently, and in a fast manner.

Compiling capacitated single-item lot-sizing problem in a CostMDD

This paper deals with capacitated single-item lot sizing problem (CLSP) in an interactive support system context. The interaction is done thanks to queries to the system: The decision-maker makes partial choices and asks about the consequences in terms of costs but also about possible inventory levels. Hence, three fundamental queries are investigated: Find production plans with a cost less than K; Find possible plans with conditioning on a pair of variables (production and inventory); Find the production plans that simultaneously satisfy the previous queries. A knowledge compilation approach is used and composed of two phases: offline and online. Offline, a top-down algorithm computes cost multivalued decision diagrams. While online, we show how cost multivalued decision diagrams can improve the reactivity of answers to queries.The proposed approach is compared to classical constraint programming with CP Optimizer. Tests indicate that our algorithm is more efficient than CP re-solving in terms of computation time.

Fuzzy Mathematical Model For A Lot-Sizing Problem In Closed-Loop Supply Chain

The aim of lot sizing problems is to determine the periods where production takes place and the quantities to be produced in order to satisfy the customer demand while minimizing the total cost. Due to its importance on the efficiency of the production and inventory systems, Lot sizing problems are one of the most challenging production planning problems and have been studied for many years with different modeling features. In this paper, we propose a fuzzy mathematical model for the single-item capacitated lot-sizing problem in closed-loop supply chain. The possibility approach is chosen to convert the fuzzy mathematical model to crisp mathematical model. The obtained crisp model is in the form of mixed integer linear programming (MILP), which can be solved by existing solver in crisp environment to find optimal solution. Due to the complexity of the problems harmony search (HS) algorithm and genetic algorithm (GA) have been used to solve the model for fifteen problem. To verify the performance of the algorithm, we computationally compared the results obtained by the algorithms with the results of the branch-and-bound method. Additionally, Taguchi method was used to calibrate the parameters of the meta-heuristic algorithms. The computational results show that, the objective values obtained by HS are better from GA results for large dimensions test problems, also CPU time obtained by HS are better than GA for Large dimensions.

A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand

We study the single-item single-resource capacitated lot-sizing problem with stochastic demand. We propose to formulate this stochastic optimization problem as a joint chance-constrained program in which the probability that an inventory shortage occurs during the planning horizon is limited to a maximum acceptable risk level. We investigate the development of a new approximate solution method which can be seen as an extension of the previously published sample approximation approach. The proposed method relies on a Monte Carlo sampling of the random variables representing the demand in all planning periods except the first one. Provided there is no dependence between the demand in the first period and the demand in the later periods, this partial sampling results in the formulation of a chance-constrained program featuring a series of joint chance constraints. Each of these constraints involves a single random variable and defines a feasible set for which a conservative convex approximation can be quite easily built. Contrary to the sample approximation approach, the partial sample approximation leads to the formulation of a deterministic mixed-integer linear problem having the same number of binary variables as the original stochastic problem. Our computational results show that the proposed method is more efficient at finding feasible solutions of the original stochastic problem than the sample approximation method and that these solutions are less costly than the ones provided by the Bonferroni conservative approximation. Moreover, the computation time is significantly shorter than the one needed for the sample approximation method.

Solving a capacitated flow-shop problem with minimizing total energy costs

In this paper, a single-item capacitated lot-sizing problem in a flow-shop system with energy consideration is addressed. The planning horizon is split into T periods where each one is characterized by a duration, an electricity cost, a maximum peak power and a demand. This problem is NP-hard, since its simple version is known to be NP-hard. Therefore, to deal with the complexity and to find good quality solutions in a reasonable time, a fix-and-relax heuristic and a genetic algorithm are developed. Computational experiments are performed on different instances to show the efficiency of these proposed heuristics. To evaluate their performances, problems of different scales have been studied and analyzed.

Lot-sizing in a multi-stage flow line production system with energy consideration

In this paper, a single-item capacitated lot-sizing problem in a flow-shop system with energy consideration is studied. The planning horizon is defined by a set of periods where each one is characterised by a length, an allowed maximal power, an electricity price, a power price and a demand. The objective is to determine the quantities to be produced by each machine at each period while minimising the production cost in terms of electrical, inventory, set-up and power required costs. For medium- and large-scale problems, lot-sizing problems are hard to solve. Therefore, in this study, two heuristics are developed to solve this problem in a reasonable time. To evaluate the performances of these heuristics, computational experiments are presented and numerical results are discussed and analysed.

A Two-Phase Iterative Heuristic Approach for the Production Routing Problem

This paper investigates the integrated optimization of production, distribution, and inventory decisions related to supplying multiple retailers from a central production facility. A single-item capacitated lot-sizing problem is defined for optimizing production decisions and inventory management. The optimization of daily distribution is modeled as a traveling salesman problem or a vehicle routing problem depending on the number of vehicles. A two-phase iterative method, from which several heuristics are derived, is proposed that iteratively focuses on lot-sizing and distribution decisions. Computational results show that our best heuristic outperforms existing methods.

A Lagrangian heuristic for capacitated single item lot sizing problems

This paper presents a new Lagrangian heuristic to solve the general capacitated single item lot sizing problem (CSILSP) where the total costs of production, setup, and inventory are to be minimized. We introduce a pre-smoothing procedure to transform the problem into a CSILSP with non-customer specific time windows and relax constraints that are specific to the CSILSP. The resulting uncapacitated single item problems with non-customer specific production time windows can be solved in polynomial time. We also analyze the performance of the Lagrangian heuristic for solving the CSILSP with non-customer specific time windows. Finally, the heuristic is adapted to the customer specific case. The introduction of pre-smoothing and the relaxation of CSILSP-specific constraints help to decrease the gap between lower bounds and upper bounds from 26.22 to 1.39 %, on average. The heuristic can be used to solve aggregate production planning problems, or can be integrated into a general procedure to solve more complex lot sizing problems.

An O(n log n) algorithm for a single-item capacitated lot-sizing problem with linear costs and no backlogging

For the single-item capacitated lot-sizing problem with linear costs and no backlogging, we suggest a straightforward time algorithm, which does not need a polymatroid or min-cost flow problem reformulation.

Single item lot sizing problems

A state-of-the-art of a particular planning problem, the Single Item Lot Sizing Problem (SILSP), is given for its uncapacitated and capacitated versions. First classes of lot sizing problems are briefly surveyed. Various solution methods for the Uncapacitated Single Item Lot Sizing Problem (USILSP) are reviewed. Four different mathematical programming formulations of the classical problem are presented. Different extensions for real-world applications of this problem are discussed. Complexity results of the Capacitated Single Item Lot Sizing Problem (CSILSP) are given together with its different formulations and solution techniques.

A Greedy Algorithm for Capacitated Lot-Sizing Problems

We show that the convex hull of the set of feasible solutions of single-item capacitated lot-sizing problem (CLSP) is a base polyhedron of a polymatroid. We present a greedy algorithm to solve CLSP with linear objective function. The proposed algorithm is an effective implementation of the classical Edmonds' algorithm for maximizing linear function over a polymatroid. We consider some special cases of CLSP with nonlinear objective function that can be solved by the proposed greedy algorithm in O ( n ) time.

Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models.

An Algorithm for the Single-item Capacitated Lot-sizing Problem with Concave Production and Holding Costs

An Algorithm for the Single-item Capacitated Lot-sizing Problem with Concave Production and Holding Costs

An Algorithm for the Single-item Capacitated Lot-Sizing Problem with Concave Production and Holding Costs

We consider a multi-period single-item production scheduling problem with a deterministic, time-varying demand pattern and concave cost functions. Optimal production lot sizes, so as to minimize the total costs of production, set-up, and inventory, are determined subject to dynamic production capacity and no backlogs. The proposed algorithm was tested extensively by solving several randomly generated problems with varying degrees of complexity. The proposed algorithm appears to perform quite reasonably for practical applications.