The capacitated lot sizing problem with setup carry-over

Although there is a significant amount of literature on the capacitated lot sizing problem, there has been insufficient consideration of planning problems in which it is possible for a lot size, or production run, to continue over consecutive time periods without incurring multiple setups. While there are papers that consider this feature, they typically restrict production to at most one product in each period. We present a set of mixed integer linear programs for the capacitated lot sizing problem that incorporate setup carry-over without restricting the number of products produced in each time period. Efficient reformulations are developed for finding optimal solutions, and a Lagrangian decomposition heuristic is provided that quickly generates near-optimal solutions. The computational results demonstrate that incorporating setup carry-over has a significant effect on both cost and lot sizes.     

A framework for modelling setup carryover in the capacitated lot sizing problem

This paper addresses the single-level capacitated lot sizing problem (CLSP) with setup carryover. Specifically, we consider a class of production planning problems in which multiple products can be produced within a time period and significant setup times are incurred when changing from one product to another. Hence, there might be instances where developing a feasible schedule becomes possible only if setups are carried over from one period to another. We develop a modelling framework to formulate the CLSP with setup times and setup carryovers. We then extend the modelling framework to include multiple machines and tool requirements planning. The need for such a model that integrates both planning and lot sizing decisions is motivated by the existence of a similar problem in a paper mill. We apply the modelling framework to solve optimally, an instance of the paper mill's problem.     

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.     

A Branch and Bound Algorithm for a Single-Item Multi-Source Dynamic Lot Sizing Problem with Capacity Constraints

In this paper we present a branch and bound algorithm for solving a single-item, multi-source dynamic lot sizing problem with time-varying capacity constraints. We consider a single-item production (and/or procurement) environment where there are multiple sources of supply with different cost structures. Computational results are presented.     

Capacitated lot sizing with parallel machines,sequence-dependent setups,and a common setup operator

In this paper, we consider a special variant of a capacitated dynamic lot sizing problem which, despite its practical relevance, has not been treated sufficiently in the literature. In addition to the significant complexity introduced by the sequence dependency of setup times, the problem is further complicated by the fact that there is only one single setup operator which is responsible for all setups on all machines. Hence, the multi-machine problem cannot be decomposed into multiple single machine problems, as the setup operations must be coordinated among the different machines. In addition, lots are produced in batches whereby the processing time of a lot is a step-wise function of the number of batches per lot. Due to perishability and quarantine issues, time windows for the production are given. We formulate a big-bucket lot sizing model and apply MIP-based heuristics to two industrial data sets.     

Efficient heuristics for dynamic lot sizing

Dynamic lot sizing heuristics commonly implemented in standard MRP software perform well under conditions of systematic demand only. A survey and numerical test of methods proposed in the literature shows that techniques are available which cope efficiently with both systematic and erratic demand. The paper discusses theoretical and practical ramifications of the dynamic lot sizing problem, surveys heuristics published in the U.S. and West German literature, reports summarily on the findings of extensive simulation experiments involving these heuristics, and, as a conclusion from analytical and numerical results, proposes an algorithm which enhances performance under conditions of erratic demand.     

A hybrid approach for the capacitated lot sizing problem with setup carryover

The capacitated lot sizing problem with setup carryover deals with the issue of planning multiple products on a single machine. A setup can be carried over from one period to the next by incorporating the partial sequencing of the first and last product. This study proposes a novel hybrid approach by combining Genetic Algorithms (GAs) and a Fix-and-Optimise heuristic to solve the capacitated lot sizing problem with setup carryover. Besides this, a new initialisation scheme is suggested to reduce the solution space and to ensure a feasible solution. A comparative experimental study is carried out using some benchmark problem instances. The results indicate that the performance of the pure GAs improves when hybridised with the Fix-and-Optimise heuristic. Moreover, in terms of solution quality, promising results are obtained when compared with the recent results in the literature.     

A dynamic lot sizing problem with multiple customers: customer-specific shipping and backlogging costs

This paper considers a dynamic lot sizing problem faced by a producer who supplies a single product to multiple customers. Characterized by their backorder costs as well as shipping costs, a customer with a high backorder cost has a greater need for the product than a customer with a low backorder cost. We show that the general problem with time-varying customer-dependent backlogging and shipping costs is NP-hard in the strong sense. We then develop an efficient dynamic programming algorithm for an important instance of the problem when there is no speculative motive for backlogging. We also establish forecast horizon results for the case of stationary production and shipping costs, which help the decision maker determine a proper forecast horizon in a rolling-horizon planning process.     

Rapid risk assessment using probability of fracture nomographs

Traditional risk-based design process involves designing the structure based on risk estimates obtained during several iterations of an optimization routine. This approach is computationally expensive for large-scale aircraft structural systems. Therefore, this paper introduces the concept of risk-based design plots that can be used for both structural sizing and risk assessment for fracture strength when maximum allowable crack length is available. In situations when crack length is defined as a probability distribution the presented approach can only be applied for various percentiles of crack lengths. These plots are obtained using normalized probability density models of load and material properties and are applicable for any arbitrary load and strength values. Risk-based design plots serve as a tool for failure probability assessment given geometry and applied load or they can determine geometric constraints to be used in sizing given allowable failure probability. This approach would transform a reliability-based optimization problem into a deterministic optimization problem with geometric constraints that implicitly incorporate risk into the design. In this paper, cracked flat plate and stiffened plate are used to demonstrate the methodology and its applicability.     

Joint Lot Sizing and Inspection Policy in an EOQ Model with Random Yield

We study a joint lot sizing and inspection policy under an EOQ model where a random proportion of units are defective. Those unitscan be discovered only through costly inspections. The problem is thus bivariate: both lot size and fraction to inspect are to be chosen. We first analyze a model in which the only penalty for uninspected defectives is financial, and then consider a model where defective units cannot be used and thus must be replaced by non-defective ones.     

Lot streaming and scheduling multiple products in two-machine no-wait flowshops

We consider the problem of minimizing makespan in two-machine no-wait flowshops with multiple products requiring lot streaming. A “product” (or lot) consists of many identical items. Lot streaming (lot sizing) is the process of creating sublots (or transfer batches to move the completed portion of a production )sublot to downstream machines so that operations can be overlapped. The number of sublots for each product is fixed. When the flowshop produces only a single product, we obtain optimal continuous-sized sublots. It is shown that these sublot sizes are also optimal for the problem of simultaneous lot streaming and scheduling of multiple products. The optimal scheduling of products can be accomplished by application of the algorithm due to Gilmore and Gomory [1]. Then, we devise an efficient heuristic for the problem of simultaneous lot streaming (finding optimal integer-sized sublots) and scheduling of multiple products. Computational results indicate that this heuristic can consistently deliver close-to-optimal solutions for the problem. A comparison of this heuristic is also made with a heuristic that first divides items belonging to each product into nearly equal-sized sublots and then constructs a schedule for such sublots. Finally, we extend our solution procedures to a traditional and more general lot streaming model, where the number of sublots for each product is a decision variable.     

A note on capacitated lot sizing with setup carry over

The “capacitated lot sizing problem with setup carry over” is based on the well known “capacitated lot sizing problem” and incorporates the possibility of preserving a setup state between successive periods. The approach at hand is to decompose the problem using Lagrangean relaxation. Subproblems are to be solved optimally employing dynamic programming techniques. Subgradient optimization guides the approach to heuristic solutions of the original problem. The present paper shows that this algorithm does not necessarily provide the optimal solution to the subproblems. The algorithm's flaw is corrected such that it allows to solve the subproblems optimally.     

A simple heuristic for perishable item inventory control under non-stationary stochastic demand

In this paper, we study the single-item single-stocking location non-stationary stochastic lot sizing problem for a perishable product. We consider fixed and proportional ordering cost, holding cost and penalty cost. The item features a limited shelf life, therefore we also take into account a variable cost of disposal. We derive exact analytical expressions to determine the expected value of the inventory of different ages. We also discuss a good approximation for the case in which the shelf-life is limited. To tackle this problem, we introduce two new heuristics that extend Silver’s heuristic and compare them to an optimal Stochastic Dynamic Programming policy in the context of a numerical study. Our results demonstrate the effectiveness of our approach.     

A Dual-Cost Heuristic For The Capacitated Lot Sizing Problem

This paper examines a mathematical programming method of accounting for capacity costs for the deterministic, multi-item, single operation lot sizing problem. With the capacity constraints of CLSP removed with Lagrangian relaxation, the problem decomposes into a set of uncapacitated single product lot sizing problems which are solved with dynamic programming. The Lagrangian dual costs are updated by subgradient optimization. Feasible solutions (production plans within the capacity limitations) are constructed with a heuristic smoothing procedure. The dual-cost heuristic gave solutions which were better on average than the other algorithms tested (and was faster than some comparable algorithms).     

Capacitated production and subcontracting in a serial supply chain

This paper extends a series of recent results regarding the polynomial solvability of single and two-echelon deterministic lot sizing models. In particular, we consider a two-echelon supply chain in which production, inventory, transportation, backlogging, and subcontracting decisions are integrated. We allow for arbitrary concave cost functions and for stationary production and subcontracting capacities. We develop dynamic programming algorithms for various problems in this class that run in polynomial time in the planning horizon of the problem.     

基于eM-Plant对注塑企业的生产线建模仿真

分析了注塑企业生产流程,并建立其生产工序的数学模型和仿真程序。根据企业的订单数量、车间生产线的规模等因素,通过对生产过程中批量的控制,研究批量和在制品库存水平的关系,提出了通过控制批量实现控制在制品的数学关系。     

PERFORMANCE OF A MYOPIC LOT SIZE POLICY WITH LEARNING IN SETUPS

We consider the problem of lot sizing when learning results in decreasing setup costs. Finding optimal lot sizes requires information about future setup costs and also the horizon length, which can be difficult to forecast. We analyze an intuitively appealing and well known myopic policy (Part Period Balancing). This policy sets the current lot size such that the current setup cost equals the holding cost for the current lot. It is easy to implement and does not require information on future setup costs. It is shown that the number of setups in the myopic policy is at most one greater than the optimal number of setups. Using this bound, we show that the myopic policy costs no more than 6/(3 + min(l, 1.5R)) times the optimal cost, where R is the ratio of the minimum setup cost to the initial setup cost. Computational experiments show that its average performance is good even for horizons as short as eight times the initial reorder interval. Further, our study shows that the average performance improves with R.     

一种多品种多加工中心FMS中的批调度问题研究

针对一种比较典型的多品种,多加工中心柔性加工企业（ＦＭＳ）的生产调度问题建立了动态模型,并给出了一种多级递阶算法,利用这种算法,可以在较少的计算量基础上获得近优解。     

The G-group heuristic to solve the multi-product,sequencing, lot sizing and scheduling problem in flow shops

This paper presents a new heuristic to solve the problem of making sequencing, lot sizing and scheduling decisions for a number of products manufactured in a flow shop environment, so as to minimize the sum of setup and inventory holding costs while a given demand is fulfilled without backlogging. The proposed solution method first determines sequencing decisions then lot sizing and scheduling decisions are simultaneously determined. This heuristic, called the G-group method, divides the set of products into G groups and requires that products belonging to the same group have the same cycle time. Also, the cycle time of each group is restricted to be an integer multiple of a basic period. For each basic period of the global cycle, the products to be produced during this period and the production sequence to be used are chosen. Then, a non-linear program is solved to determine lot sizes and to construct a feasible production schedule. To evaluate its performance, the G-group method was compared to four other methods. Numerical results show that the proposed heuristic outperforms all these methods.     

Optimal inspection sublots for deteriorating batch production

This paper analyzes a problem of optimal inspection intervals for a failure-prone batch production facility. The model makes a seemingly minor modification to similar models that have appeared recently, but which results in a significant change in the form of the optimal policy. Optimal solutions are developed for both continuous and integer valued sublot sizes. The paper also examines the effects of inspection sublots in an economic lot sizing context and shows that a simple, exact modification of the Economic Production Quantity formula is often possible in the continuous relaxation of the problem.