Uncapacitated Lot-sizing Problem Research Articles

Training Demand Prediction Models by Decision Error for Two-Stage Lot-Sizing Problems

Demand prediction to support appropriate production decisions is being actively studied. Many prediction models are designed to minimize the prediction error, which is measured by determining the difference between the predicted and ground-truth demand. However, these models ignore the effect of the prediction error on downstream production decisions. This prompted our study, which focuses on demand prediction models for two-stage uncapacitated lot-sizing problems. In this paper, we present a prediction model that minimizes the decision error, which is measured by the optimization objective of lot-sizing problems. Our model mitigates the impact of prediction errors by leveraging the structure of the lot-sizing problems. To enhance the ability to accommodate imperfect data, such as data based on inaccurate information, we subsequently extend the prediction models to distributionally robust versions. We consider the worst-case formulation in the feature space to enhance the robustness of the model to data imperfection. Numerical experiments demonstrate that the proposed prediction models are significantly superior to the traditional prediction methods when the model being trained is misspecified. In addition, the robust extension enables the models to train well on imperfect datasets while requiring less training data. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This study is motivated by the two-stage lot-sizing problem in manufacturing systems which involves the determination of the time at which (the setup decisions) to produce before the demand is revealed, and the production quantity decisions are adjusted during production processing. Predicting the demand properly while making the set-up decisions helps to decrease the production cost. This paper proposes a novel method that aims to minimize the total production cost to train demand prediction models. Practitioners can deploy the proposed method to train popular prediction models including linear prediction, decision tree, and neural network models. It is shown that the production cost associated with the decisions based on our models is lower than that associated with the traditional prediction models. Furthermore, our method enhances the ability of the model to handle model misspecification and imperfect data. Practitioners can apply our method to overcome the problems caused by imperfect data such as those containing contaminated and inaccurate measurements.

A Mixed-integer Linear Programming Model for Solving Multi-item Uncapacitated Lot-sizing Problems with Storage Capacities Limitations: A Case Study of an Apparel Manufacturing Company

This study aimed to use a mixed-integer linear programming model to solve the multi-item uncapacitated lot size problem with storage capacity limitations for apparel manufacturing. In real-world operations, sometimes the storage capacity of cloth rolls can be insufficient at any given time. This would be because the sum of the remaining cloth rolls from the previous period and the receiving cloth rolls from the existing period would be a vast number. Furthermore, the extra cloth on the rolls would have to be carried and kept in the production line. Consequently, this caused the late issue of the distribution of the material, the loss of the cloth rolls, and the failure to count the cloth rolls. As a result, we conducted a multi-item cloth roll replenishment plan to meet the production line's demand and keep all the inventory and receiving cloth rolls. Using mixed-integer linear programming, the multi-item lot size of each period and the sum of the cloth, ordering, and holding costs was calculated, including the number of occupied cloth racks that did not exceed the number of available cloth racks in the cloth storage. Then, this result was compared with the calculation of a mixed-integer linear programming model based on the procurement policies of a case study company. This study used OpenSolver and GAMS/CPLEX software to calculate the solutions. The results found that in using the procurement policy, we could provide the maximum number of occupied cloth racks, about 368 racks, or 2.78 times the racks on the available shelves of the cloth storage. The result of the proposed model indicated that the number of occupied cloth racks to be installed did not exceed the number of available racks on the shelves. The total cost of the model of the procurement policy was less than the proposed model by about 0.26 percent because ordering occurred only about 25 times.

A lot-sizing problem in deliberated and controlled co-production systems

We consider an uncapacitated lot-sizing problem in co-production systems, in which it is possible to produce multiple items simultaneously in a single production run. Each product has a deterministic demand to be satisfied on time. The decision is to choose which items to co-produce and the amount of production throughout a predetermined planning horizon. We show that the lot-sizing problem with co-production is strongly NP-Hard. Then, we develop various Mixed-Integer Linear Programming (MILP) formulations of the problem and show that LP relaxations of all MILPs are equal. We develop a separation algorithm based on a set of valid inequalities, lower bounds based on a dynamic lot-sizing relaxation of our problem and a constructive heuristic that is used to obtain an initial solution for the solver, which form the basis of our proposed Branch & Cut algorithm for the problem. We test our models and algorithms on different data sets and provide the results.

Combining Polyhedral Approaches and Stochastic Dual Dynamic Integer Programming for Solving the Uncapacitated Lot-Sizing Problem Under Uncertainty

We study the uncapacitated lot-sizing problem with uncertain demand and costs. The problem is modeled as a multistage stochastic mixed-integer linear program in which the evolution of the uncertain parameters is represented by a scenario tree. To solve this problem, we propose a new extension of the stochastic dual dynamic integer programming algorithm (SDDiP). This extension aims at being more computationally efficient in the management of the expected cost-to-go functions involved in the model, in particular by reducing their number and by exploiting the current knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. The algorithm is based on a partial decomposition of the problem into a set of stochastic subproblems, each one involving a subset of nodes forming a subtree of the initial scenario tree. We then introduce a cutting plane–generation procedure that iteratively strengthens the linear relaxation of these subproblems and enables the generation of an additional strengthened Benders’ cut, which improves the convergence of the method. We carry out extensive computational experiments on randomly generated large-size instances. Our numerical results show that the proposed algorithm significantly outperforms the SDDiP algorithm at providing good-quality solutions within the computation time limit. Summary of Contribution: This paper investigates a combinatorial optimization problem called the uncapacitated lot-sizing problem. This problem has been widely studied in the operations research literature as it appears as a core subproblem in many industrial production planning problems. We consider a stochastic extension in which the input parameters are subject to uncertainty and model the resulting stochastic optimization problem as a multistage stochastic integer program. To solve this stochastic problem, we propose a novel extension of the recently published stochastic dual dynamic integer programming (SDDiP) algorithm. The proposed extension relies on two main ideas: the use of a partial decomposition of the scenario tree and the exploitation of existing knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. We provide the results of extensive computational experiments carried out on large-size randomly generated instances. These results show that the proposed extended algorithm significantly outperforms the SDDiP at providing good-quality solutions for the stochastic uncapacitated lot-sizing problem. Although the paper focuses on a basic lot-sizing problem, the proposed algorithmic framework may be useful to solve more complex practical production planning problems.

A Stochastic Dual Dynamic Integer Programming for the Uncapacitated Lot-Sizing Problem with Uncertain Demand and Costs

We study the uncapacitated lot-sizing problem with uncertain demand and costs. We consider a multi-stage decision process and rely on a scenario tree to represent the uncertainty. We propose to solve this stochastic combinatorial optimization problem thanks to a new extension of the stochastic dual dynamic integer programming algorithm. Our results show that this approach can provide good quality solutions in a reasonable time for large-size instances.

Column generation approach for solving uncapacitated dynamic lot-sizing problems with time-varying cost

Column generation approach for solving uncapacitated dynamic lot-sizing problems with time-varying cost

Dantzig-Wolfe decomposition for the facility location and production planning problem

There are several mathematical models proposed for the facility location and production planning problem in the literature. However, some of these models disregard what products each customer has ordered and neglect critical production-related constraints and setup decisions while some others do not well define the connection cost between customers and facilities. In this study, we propose two mathematical models to overcome the disadvantages aforementioned, along with their reformulations by item decomposition to improve lower bounds. We demonstrate that the pricing subproblems of the item decomposition are related to uncapacitated lot-sizing problems with the Wagner-Whitin property. This property is employed to enhance the performance of column generation for the item decomposition. Our computational results show that this item decomposition method can improve lower bounds over other classical lower bounding techniques, such as linear programming relaxation and model reformulation. Additionally, we implement the proposed item decomposition method to other benchmark problems in the literature and observe that our proposed method can improve the benchmark solutions with a statistical significance.

A double auction mechanism for coordinating lot-sizing in supply chains

This paper presents an auction mechanism for the coordination of dynamic, uncapacitated lot-sizing problems in two-party supply chains where parties’ local data are private information and no external mediator is involved. In the first stage of the mechanism, the buyer generates supply proposals which potentially lead to system-wide improvements, and communicates them to the supplier. Then both parties evaluate the cost impact of the different proposals for their local planning domains. In the second stage, buyer and supplier submit sealed bids on each proposal. Then the bids are opened and the proposal with the greatest surplus is implemented. The procedure for proposal generation is efficient and scalable. It identifies the system-wide optimum in special cases, finds close-to-optimal solutions for general settings in computational tests, while the number of needed proposals increases linearly with the number of time periods and items supplied. Further analytical and numerical analyses indicate that the auction in the second stage is highly efficient.

Two-level uncapacitated lot-sizing problem considering the financing cost of working capital requirement

During financial crisis, companies constantly need free cash flows to efficiently react to any uncertainty, thus ensuring solvency. Working capital requirement (WCR) has been recognized as a key factor for releasing tied up cash in companies. However, in literatures related to lot-sizing problem, WCR has only been studied in the single-level supply chain context. In this paper, we initially adopt WCR model for a multi-level case. A two-level (supplier-customer) model is established on the basis of the classic multi-level lot-sizing model integrated with WCR financing cost. To tackle this problem, we propose sequential and centralized approaches to solve the two-level case with a serial chain structure. The ZIO (Zero Inventory Ordering) property is further confirmed valid in both cases. This property allows us to establish a dynamic programming-based algorithm, which solves the problem in O(T4). Finally, numerical tests show differences in optimal plans obtained by both approaches and the influence of varying delays in payment on the WCR of both actors.

Competitive uncapacitated lot-sizing game

We study the strategical behaviour of firms facing a lot-sizing problem with Cournot competition. Each player is a firm with her own production facility, modeled as an uncapacitated lot-sizing problem (i.e., production incurs set-up and variable costs and inventories are allowed). A Cournot competition is played in each time period (market) with each player deciding the quantity of product to place on it. The market price of that product in each time period depends on the total quantity placed in the market.We show that this is a potential game with possibly multiple pure Nash equilibria. We then investigate the plausibility of these equilibria to predict the game outcome by evaluating the difficulty of computing them. If the game has a single period, we prove that an equilibrium can be found in polynomial time, but it is weakly NP-hard to find an optimal pure Nash equilibrium (with respect to a given equilibrium refinement). If the game has no variable production and inventory costs, we prove that a pure Nash equilibrium can be computed in polynomial time.

Two-level lot-sizing with inventory bounds

We study a two-level uncapacitated lot-sizing problem with inventory bounds that occurs in a supply chain composed of a supplier and a retailer. The first level with the demands is the retailer level and the second one is the supplier level. The aim is to minimize the cost of the supply chain so as to satisfy the demands when the quantity of item that can be held in inventory at each period is limited. The inventory bounds can be imposed at the retailer level, at the supplier level or at both levels. We propose a polynomial dynamic programming algorithm to solve this problem when the inventory bounds are set on the retailer level. When the inventory bounds are set on the supplier level, we show that the problem is NP-hard. We give a pseudo-polynomial algorithm which solves this problem when there are inventory bounds on both levels. In the case where demand lot-splitting is not allowed, i.e. each demand has to be satisfied by a single order, we prove that the uncapacitated lot-sizing problem with inventory bounds is strongly NP-hard. This implies that the two-level lot-sizing problems with inventory bounds are also strongly NP-hard when demand lot-splitting is considered.

The impact of service level constraints in deterministic lot sizing with backlogging

This paper studies the impact of service level constraints in the context of the capacitated and uncapacitated lot sizing problems with deterministic demand and backlogging over a discrete and finite time horizon. Because the standard formulation of these problems cannot distinguish between a backlog and a backorder, we use a reformulation to explicitly make this distinction. We then formulate several service level constraints adapted from the stochastic inventory control literature and introduce new ones. We analyze the impact of these constraints on the structure of the optimal solutions in terms of the number of backorders and total backlog, and on the performance of these solutions in terms of various service level indicators (proportion of periods with no backlog, proportion of backorders over total demand, maximum delay, average unit waiting time for an item). We also consider the introduction of various types of backordering costs (both fixed and duration dependent) and study their impact. We find that they tend to stabilize the structure of solutions, regardless of the service level constraints imposed. Finally, we analyze the impact of a first-in first-out order management policy and show that it helps improving most solutions.

Coordination of dynamic lot-sizing in supply chains

This paper presents a new scheme for the coordination of dynamic, uncapacitated lot-sizing problems in two-party supply chains where parties’ local data are private information and no external or central entity is involved. This coordination scheme includes the following actions: At first, the buyer generates a series of different supply proposals using an extension of her local lot-sizing problem. Then the supplier calculates his cost changes that would result from the implementation of the buyer’s proposals. Based on these information, parties can identify the best proposal generated. The scheme identifies the system-wide optimum in different settings—for instance in a two-stage supply chain where the supplier’s costs for holding a period’s demand in inventory exceed his setup costs.

A mixed integer linear programming model for the multi-item uncapacitated lot-sizing problem: a case study in the trailer manufacturing industry

A mixed integer linear programming model for the multi-item uncapacitated lot-sizing problem: a case study in the trailer manufacturing industry

A mixed integer linear programming model for the multi-item uncapacitated lot-sizing problem: a case study in the trailer manufacturing industry

In this paper, a mixed integer linear programming model for the multi-item uncapacitated lot-sizing problem is presented. The considered factors for formulating the proposed model are the monthly demand of the selected product from the case study company, type of parts used in the product and their consumption coefficients based on the bill of materials, lead time to receive parts from outsourcing suppliers, the costs of ordering, purchasing, and holding, and the amount of safety stock for each part. Accordingly, several forecasting techniques are tried to determine future demands. The prices of the selected parts are estimated using linear regression method. The optimal safety stock for each part is calculated based on variance in demand, lead time and the target service level. Material requirements planning are also performed to obtain the economic purchasing schedule of parts. LINGO and GAMS software are used to solve and validate the suggested model respectively. The results show that the proposed model can find the optimum order quantity for each part per period, which minimises the total cost.

Formulations and heuristics for the multi-item uncapacitated lot-sizing problem with inventory bounds

We consider the multi-item uncapacitated lot-sizing problem with inventory bounds, in which a production plan for multiple items has to be determined considering that they share a storage capacity. We present (a) a shortest path formulation and (b) a formulation based on the a priori addition of valid inequalities, which are compared with a facility location formulation available in the literature. Two easy-to-implement mixed integer programming heuristic frameworks are also presented, (a) a rounding scheme and (b) a relax-and-fix approach performed in a time partitioning fashion. Computational experiments are performed to evaluate the different approaches. The numerical results show that the proposed relax-and-fix heuristic outperforms all other approaches. Its solutions are within 4.0% of optimality in less than 10 minutes of running time for all tested instances, with mean gaps in the order of 2.1 and 1.8% for instances with more relaxed and tighter capacities, respectively. The obtained solutions were always better than those obtained by a commercial MIP solver running for one hour using any of the available formulations.

Erratum: A tight formulation for uncapacitated lot-sizing with stock upper bounds

For an n-period uncapacitated lot-sizing problem with stock upper bounds, stock fixed costs, stock overload and backlogging, we present a tight extended shortest path formulation of the convex hull of solutions with O$$(n^2)$$(n2) variables and constraints, also giving an O$$(n^2)$$(n2) algorithm for the problem. This corrects and extends a formulation in Section 4.4 of our article "Lot-sizing with production and delivery time windows", Mathematical Programming A, 107:471---489, 2006, for the problem with just stock upper bounds.

Extended formulations for stochastic lot-sizing problems

In this paper, extended formulations for stochastic uncapacitated lot-sizing problems with and without backlogging are developed in higher dimensional spaces that provide integral solutions. Moreover, physical meanings of the decision variables in the extended formulations are explored and special cases with more efficient formulations are studied.

Minimum concave cost flow over a grid network

The minimum concave cost network flow problem (MCCNFP) is NP-hard, but efficient polynomial-time algorithms exist for some special cases such as the uncapacitated lot-sizing problem and many of its variants. We study the MCCNFP over a grid network with a general nonnegative separable concave cost function. We show that this problem is polynomially solvable when all sources are in the first echelon and all sinks are in two echelons, and when there is a single source but many sinks in multiple echelons. The polynomiality argument relies on a combination of a particular dynamic programming formulation and an investigation of the extreme points of the underlying flow polyhedron. We derive an analytical formula for the inflow into any node in an extreme point solution, which generalizes a result of Zangwill (Manag Sci 14(7):429---450, 1968) for the multi-echelon lot-sizing problem.

An inverse economic lot-sizing approach to eliciting supplier cost parameters

Recent literature on supply chain coordination offers a wide range of game theoretic and optimization approaches that ensure efficient planning in the supply chain, but assume that the involved parties have complete information about each other. However, in reality, complete information is rarely available, and those models alone do not present any incentive for the parties to reveal their private information, e.g., the cost parameters that they use when solving their planning problems.This paper proposes an inverse lot-sizing model for eliciting the cost parameters of a supplier from historic demand vs. optimal delivery lot-size pairs, gathered during repeated earlier encounters. It is assumed that the supplier solves a single-item, multi-period, uncapacitated lot-sizing problem with backlogs to optimality to calculate its lot-sizes, and the buyer is aware of this fact. The inverse lot-sizing problem is reformulated to an inverse shortest path problem, which is, in turn, solved as a linear program. This model is used to compute the ratios of the supplier's cost parameters, i.e., the setup, the holding, and the backlog cost parameters consistent with all the historic samples.The elicited cost parameters can be used as input for various game theoretic or bilevel optimization models for supply chain coordination. Computational experiments on randomly generated problem instances indicate that the approach is very efficient in predicting future supplier actions from the historic records.