Multi-item Lot-sizing Problem Research Articles

DESMILS: a decision support approach for multi-item lot sizing using interactive multiobjective optimization

We propose a decision support approach, called DESMILS, to solve multi-item lot sizing problems with a large number of items by using single-item multiobjective lot sizing models. This approach for making lot sizing decisions considers multiple conflicting objective functions and incorporates a decision maker’s preferences to find the most preferred Pareto optimal solutions. DESMILS applies clustering, and items in one cluster are treated utilizing preferences that the decision maker has provided for a representative item of the cluster. Thus, the decision maker provides preferences to solve the single-item lot sizing problem for few items only and not for every item. The lot sizes are obtained by solving a multiobjective optimization problem with an interactive method, which iteratively incorporates preference information and supports the decision maker in learning about the trade-offs involved. As a proof of concept to demonstrate the behavior of DESMILS, we solve a multi-item lot sizing problem of a manufacturing company utilizing their real data. We describe how the supply chain manager as the decision maker found Pareto optimal lot sizes for 94 items by solving the single-item multiobjective lot sizing problem for only ten representative items. He found the solutions acceptable and the solution process convenient saving a significant amount of his time.

Optimization Multi-Item Lot Sizing Model involve Transportation and Capacity Constraint under Stochastic Demand using Aquila Optimizer

Inventory issues are often a major concern as they significantly impact operating costs. Lot sizing is one of the key decisions in managing inventory. However, in a real context, the demand for each item is often uncertain. It is supplied from the same supplier, requiring product orders to be placed in the same period. In addition, limited vehicle capacity and transportation costs are important factors to consider in making multi-item lot sizing decisions. The purpose of this study is to propose a new inventory model multi-item lot sizing model involving transportation cost and capacity constraints under stochastic demand. The decision variables involved in the model are each item's ordering cycle and safety factor with the objective function of minimizing the total inventory cost. To optimize the inventory model, this study also offers the advanced procedure of the Aquila Algorithm. This study also presents sensitivity analysis to the appropriate policy for optimizing the multi-item lot-sizing inventory problem involving transportation cost and capacity constraint under stochastic demand.

Motivations and analysis of the capacitated lot-sizing problem with setup times and minimum and maximum ending inventories

This paper first analyzes the negative impact of the end-of-horizon effect when solving the capacitated multi-item lot-sizing problem with setup costs and times on a rolling horizon. Maximum ending inventories for items and a global minimum ending inventory are considered to define a new optimization problem whose optimal solutions are much less impacted by the end-of-horizon effect. Then, a generation scheme is proposed to create new instances with initial inventories and ending inventories. This scheme relies on the analysis of the cyclical production planning problem to derive relevant parameters. Computational experiments are carried out to compare the solutions obtained for original instances of the literature and for the new instances, and to analyze the relevance of the new instances on a rolling horizon.

Predictive Search for Capacitated Multi-Item Lot Sizing Problems

For capacitated multi-item lot sizing problems, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework. Advanced analytics can predict the probability that an event will happen and has been applied to pressing industry issues, such as credit scoring, risk management, and default management. Although little research has applied such technique for lot sizing problems, we observe that advanced analytics can uncover optimal patterns of setup variables given properties associated with the problems, such as problem attributes, and solution values yielded by linear programming relaxation, column generation, and Lagrangian relaxation. We, therefore, build advanced analytics models that yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to partition the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. The discussion is followed by computational tests, where comparisons with other methods indicate that our approach can obtain better results for the benchmark problems than other state-of-the-art methods. Summary of Contribution: In this study, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework for capacitated multi-item lot sizing problems. The advanced analytics models are used to yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to divide the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. Through computational tests based on benchmark problems, we observe that the proposed approach can obtain better results than other state-of-the-art methods.

Multi-Item Production Lot Sizing with Postponement, External Source for Common Parts, and Adjustable Rate for End Products

This study considers a multi-item production lot-size problem incorporating postponement, an external source for common parts, and an adjustable-rate for end products. Dealing with product variety, timely requirements, and limited in-house capacity has led production managers to seek manufacturing schemes and utilization-reduction strategies that can help them meet customer needs, smoothen fabrication schedules, and lower overall manufacturing expenses. We propose a two-stage manufacturing scheme. The first stage produces common parts for multiproduct incorporating a partial supply from an outside contractor to reduce utilization/uptime. Stage two fabricates all end products using an adjustable-rate to reduce the uptime further. We build a model to characterize the problem’s features and use optimization methods to derive the optimal rotation cycle time in order to help managers make cost-effective lot-size decisions and allow manufacturers to gain competitive advantages. A numerical illustration validates the model’s capability and applicability. This study makes two important contributions: (1) It offers a decision-support model for studying such a particular batch-size problem and deciding the optimal rotation cycle time, and (2) it identifies the individual/collective influence of dual uptime-reduction strategies on the operating policy and various performance indexes to help facilitate managerial decision-making.

A fix-and-optimize heuristic for the capacitated multi-item stochastic lot-sizing problem

This study addresses the stochastic multi-item capacitated lot-sizing problem. Here, it is assumed that all items are produced on a single production resource and unmet demands are backlogged. The literature shows that the deterministic version of this problem is NP-Hard. We consider the case where period demands are time-varying random variables. The objective is to determine the minimum expected cost production plan so as to meet stochastic period demands over the planning horizon. We extend the mixed integer programming formulation introduced in the literature to capture the problem under consideration. Further, we propose a fix-and-optimize heuristic building on an item-period oriented decomposition scheme. We then conduct a numerical study to evaluate the performance of the proposed heuristic as compared to the heuristic introduced by Tempelmeier and Hilger [16]. The results clearly show that the proposed fix-and-optimize heuristic arises as both cost-efficient and time-efficient solution approach as compared to the benchmark heuristic.

The value of aggregate service levels in stochastic lot sizing problems

Dealing with demand uncertainty in multi-item lot sizing problems poses huge challenges due to the inherent complexity. The resulting stochastic formulations typically determine production plans which minimize the expected total operating cost while ensuring that a predefined service level constraint for each product is satisfied. We extend these stochastic formulations to a more general setting where, in addition to the individual service level constraints, an aggregate service level constraint is also imposed. Such a situation is relevant in practical applications where the service level aggregated from a variety of products must be collectively satisfied. These extended formulations allow the decision maker to flexibly assign different individual service levels to different products while ensuring that the overall aggregate service level is satisfied and these aggregate service level measures can be used in conjunction with the commonly adopted individual service levels. Different mathematical formulations are proposed for this problem with different types of service levels. These formulations are a piece-wise linear approximation for the β, γ, and δ service levels and a quantile-based formulation for the αc service level. We also present a receding horizon implementation of the proposed formulations which can be effectively used in a dynamic environment. Computational experiments are conducted to analyze the impact of aggregate service levels and demonstrate the value of the proposed formulations as opposed to standard service levels imposed on individual items.

Constant Approximation Algorithm for Nonuniform Capacitated Multi-Item Lot Sizing via Strong Covering Inequalities

We study the nonuniform capacitated multi-item lot-sizing problem. In this problem, there is a set of demands over a planning horizon of T discrete time periods, and all demands must be satisfied on time. We can place an order at the beginning of each period s, incurring an ordering cost Ks. In this order, we can order up to Cs units of products. On the other hand, carrying inventory from time to time incurs an inventory holding cost. The goal of the problem is to find a feasible solution that minimizes the sum of ordering and holding costs. Levi et al. [Levi R, Lodi A, Sviridenko M (2008) Approximation algorithms for the capacitated multi-item lot-sizing problem via flow-cover inequalities. Math. Oper. Res. 33(2):461–474.] gave a two-approximation for the problem when the capacities Cs are the same. Extending the result to the case of nonuniform capacities requires new techniques as pointed out in the discussion section of their paper. In this paper, we solve the problem by giving a 10-approximation algorithm for the capacitated multi-item lot-sizing problem with general capacities. The constant approximation is achieved by adding an exponential number of new covering inequalities to the natural facility location–type linear programming (LP) relaxation for the problem. Along the way of our algorithm, we reduce the lot-sizing problem to two generalizations of the classic knapsack-covering problem. We give LP-based constant approximation algorithms for both generalizations via the iterative rounding technique.

Эвристика снижения чередующихся окрестностей для задачи определения величины разнородной партии в случае повторного производства

Эвристика снижения чередующихся окрестностей для задачи определения величины разнородной партии в случае повторного производства

Application of a genetic algorithm for multi-item inventory lot-sizing with supplier selection under quantity discount and lead time

This study presents an application of genetic algorithm (GA) for solving the multi-item inventory lot-sizing problem with supplier selection under discounts and lead time constraints. A mixed-integer linear programming (MILP) model is developed for proposed problem. To solve the problem, a genetic algorithm (GA) with two additional operations is proposed for handling the effect of the problem size. An adaptor for adjusting a chromosome data before the evaluation process and a penalty step for deterring an infeasible solution are developed. Finally, numerical examples are generated to evaluate the performance of the proposed GA, and the comparison with MILP approach about the solution quality and time is presented.

Optimizing make-to-stock policies through a robust lot-sizing model

In this paper we consider a practical lot-sizing problem faced by an industrial company. The company plans the production for a set of products following a Make-To-Order policy. When the productive capacity is not fully used, the remaining capacity is devoted to the production of those products whose orders are typically quite below the established minimum production level. For these products the company follows a Make-To-Stock (MTS) policy since part of the production is to fulfill future estimated orders. This yields a particular lot-sizing problem aiming to decide which products should be produced and the corresponding batch sizes. These lot-sizing problems typically face uncertain demands, which we address here through the lens of robust optimization. First we provide a mixed integer formulation assuming the future demands are deterministic and we tighten the model with valid inequalities. Then, in order to account for uncertainty of the demands, we propose a robust approach where demands are assumed to belong to given intervals and the number of deviations to the nominal estimated value is limited. As the number of products can be large and some instances may not be solved to optimality, we propose two heuristics. Computational tests are conducted on a set of instances generated from real data provided by our industrial partner. The heuristics proposed are fast and provide good quality solutions for the tested instances. Moreover, since they are based on the mathematical model and use simple strategies to reduce the instances size, these heuristics could be extended to solve other multi-item lot-sizing problems where demands are uncertain.

Lot sizing and supplier selection with multiple items, multiple periods, quantity discounts, and backordering

Abstract A general model is presented for a realistic multi-item lot-sizing problem with multiple suppliers, multiple time periods, quantity discounts, and backordering of shortages. Mixed integer programming (MIP) is used to formulate the problem and obtain the optimum solution for smaller problems. Due to the large number of variables and constraints, the model is too hard to solve optimally for practical problems. In order to tackle larger problem sizes, two heuristic solution methods are proposed. The first method is developed by modifying the Silver-Meal heuristic, and the second one by developing a problem-specific Genetic Algorithm (GA). Both heuristic methods are shown to be effective in solving the lot-sizing problem, but the GA is generally superior.

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.

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

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.

A biased random key genetic algorithm approach for inventory-based multi-item lot-sizing problem

In this article, we have explored multi-item capacitated lot-sizing problem by addressing the backlogging and associated high penalty costs incurred. At the same time, penalty cost for exceeding the resource capacity has also been taken into account. Penalty cost related to both backlogging and overutilizing capacity has been included in main objective function. The main objective is to achieve such a solution that minimizes the total cost. The ingredients of total cost are the setup cost, production cost, inventory holding cost and aforementioned both the penalty costs. To solve this computationally complex problem, a less explored algorithm “biased random key genetic algorithm” has been applied. To the best of our knowledge, this research presents the first application of biased random key genetic algorithm to a lot-sizing problem. To test the effectiveness of proposed algorithm, extensive computational tests are conducted. The encouraging results show that the proposed algorithm is an efficient tool to tackle such complex problems. A comparative study with other existing heuristics shows the supremacy of proposed algorithm in terms of quality of the solution, number of generation and computational time.

Relaxations for two-level multi-item lot-sizing problems

We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.

Decomposing a Multi-Plant, Multi-Item Lot-Sizing Problem with Transfer Costs/Capacities into Facility Location and Flow Sub-Problems

This paper describes the way a multi-item, multi-plant Lot-Sizing problem with transfer costs/capacities may be expressed according to the multi-commodity flow formalism, and decomposed into a master Facility Location problem, and a slave minimal cost flow problem. It describes the way this decomposition framework gives rise in a natural way to the design of a heuristic relax/project algorithm, and it ends with numerical experiments.

A bilinear programming model and a modified branch-and-bound algorithm for production planning in steel rolling mills with substitutable demand

In this paper, we address an instance of the dynamic capacitated multi-item lot-sizing problem (CMILSP) typically encountered in steel rolling mills. Production planning is carried out at the master production schedule level, where the various end items lot sizes are determined such that the total cost is minimised. Through incorporating the various technological constraints associated with the manufacturing process, the integrated production–inventory problem is formulated as a mixed integer bilinear program (MIBLP). Typically, such class of mathematical models is solved via linearisation techniques which transform the model to an equivalent MILP (mixed integer linear program) at the expense of increased model dimensionality. This paper presents an alternative branch-and-bound based algorithm that exploits the special structure of the mathematical model to minimise the number of branches and obtain the bound at each node. The performance of our algorithm is benchmarked against that of a classical linearisation technique for several problem instances and the obtained results are reported.

Polyhedral and Lagrangian approaches for lot sizing with production time windows and setup times

In this paper, we solve the capacitated multi item lot-sizing problem with non-customer specific production time windows and setup times using two approaches: (i) using a Lagrangian relaxation-based heuristic and (ii) using reformulations and a commercial software. The results of the two approaches are analyzed and compared based on randomly generated data sets. The results show that the first approach finds feasible solution more rapidly but a steady state is reached very quickly. On the other hand the second approach quickly finds good lower bounds and finds good feasible solutions if more CPU time is allowed. It turns out that, for a wide variety of instances varying in size and other parameters, we can obtain feasible solutions within 1–5% of optimal within 10 s and also obtain solutions that are guaranteed within 1–2% of optimal within 60–120 s.