Lot Size Model Research Articles

Forecast horizon for dynamic lot sizing model under product substitution and perishable inventories

This study investigates the optimal forecast horizon for a two-product dynamic lot sizing problem under the following conditions: (1) stock deterioration rates and holding costs in every period depending on the age of the stock and (2) possibility of one-way product substitution (i.e., one product may be used to satisfy the demand of the other product but not vice versa). We use two structural properties of the optimal solution as bases to develop a forward dynamic programming algorithm that solves the problem. The complexity of the algorithm is O(T5), where T is the length of the planning horizon. We provide sufficient conditions to obtain the forecast horizon results by establishing the monotonicity of the order point. On an extensive test bed, we demonstrate the influence of a joint setup and substitution costs on a fixed-length rolling horizon that will enable the operations manager to dynamically update the duration of its horizon.

A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities

In the real-world production process, the firms need to determine the optimal production planning under minimum production quantity constraint in order to achieve economies of scale. However, the inventory cost will hugely increase when there is a very large amount of production in a period and also a large amount of total demands for the next few periods. This paper considers a single-item dynamic lot sizing problem with production-or-outsourcing decisions. In each period, the production level cannot be lower than a given quantity in order to make full use of resources, but the outsourcing is unrestricted. The demands in a period can be backlogged. The production and outsourcing costs are fixed-plus-linear, and the inventory and backlogging costs are linear. We establish a mathematical programming model according to the real problem in the firm. We explore some structural properties of the optimal solution and use them to develop a dynamic programming algorithm to solve the proposed problem. We further present a special case with stationary production and outsourcing costs which can be solved with reduced computational complexities. In the end, we use three numerical instances to show how to obtain the optimal solutions by using the dynamic programming algorithm. Furthermore, we show that the policy of backlogging or outsourcing can reduce the total cost.

An integrated lot-sizing model with supplier and carrier selection and quantity discounts considering multiple products

Because of growing competition in the global markets and the vital role of suppliers in business success, the subject of supplier selection has attracted many researchers and practitioners during the recent years. In addition to the supplier selection, the order quantity discount provided by the suppliers, is considered through a new mixed-integer linear programming (MILP) model involving a manufacturer with multiple products and multiple purchasing items over multiple periods. According to the proposed model, the manufacturer purchases different amount of raw materials from selected suppliers in order to produce different products. Customers’ demands are fulfilled by minimizing the total purchase, inventory, production, and transportation costs over a multi-period planning horizon. Since the problem is NP-hard, an efficient genetic algorithm (GA) is used to solve the large-scale real-world instances. The results are compared with results from the exact approach wherever possible in order to investigate the efficiency of the algorithm.

EPL models with fuzzy imperfect production system including carbon emission: a fuzzy differential equation approach

The paper outlines the production policies for maximum profit of a firm producing imperfect economic lot size with time-dependent fuzzy defective rate under the respective country’s carbon emission rules. Generally in economic production lot-size models, defective production starts after the passage of some time from production commencement. So the starting time of producing defective units is normally uncertain and imprecise. Thus, produced defective units are fuzzy, partially reworked instantly and sold as fresh units. As a result, the inventory level at any time becomes fuzzy and the relation between the production, demand and inventory level becomes a fuzzy differential equation (FDE). Nowadays, different governments have made environmental regulations following the United Nations Framework Convention on Climate Change to reduce carbon emission. Some governments use cape and trade policy on emission. Due to this, firms are in fix how to optimize the production. If the firms produce more, the profit increases along with more emission and corresponding tax. Here, models are formulated as profit maximization problems using FDE, and the corresponding inventory and environmental costs are calculated using fuzzy Riemann integration. An $$\alpha $$-cut of average profits is obtained and the reduced multi-objective crisp problems are solved using intuitionistic fuzzy optimization technique. The models are illustrated numerically and results are presented graphically. Considering different carbon regulations, an algorithm for a firm management is presented to achieve the maximum profit. Real-life production problems for the firms in Annex I and developing countries are solved.

Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities

We consider the dynamic lot size problem for perishable inventory under minimum order quantities. The stock deterioration rates and inventory costs depend on both the age of the stocks and their periods of order. Based on two structural properties of the optimal solution, we develop a dynamic programming algorithm to solve the problem without backlogging. We also extend the model by considering backlogging. By establishing the regeneration set, we give a sufficient condition for obtaining forecast horizon under without and with backlogging. Finally, based on a detailed test bed of instance, we obtain useful managerial insights on the impact of minimum order quantities and perishability of product and the costs on the length of forecast horizon.

Economic order quantity model for growing items with incremental quantity discounts

Certain inventory items are living organisms, for example livestock, and are therefore capable of growing during the replenishment cycle. These items often serve as various saleable food items downstream in supply chains. The purpose of this paper is to develop a lot sizing model for growing items if the supplier of the items offers incremental quantity discounts. A mathematical model is derived to determine the optimal inventory policy which minimises the total inventory cost in both the owned and rented facilities. A solution procedure for solving the model is developed and illustrated through a numerical example. Sensitivity analysis is performed to demonstrate the response of the order quantity and total costs to some key input parameters. Incremental quantity discounts result in reduced purchasing costs; however, ordering very large quantities has downsides as well. The biggest downsides include the increased holding costs, the risks of running out of storage capacity and item deterioration since the cycle time increases if larger quantities are purchased. Owing to the importance of growing items in the food supply chains, the model presented in this article can be used by procurement and inventory mangers when making purchasing decisions.

Economic production quantity model with learning in production, quality, reliability and energy efficiency

Learning has become essential for many companies whose manufacturing environment, where tasks have increased in complexity, requires a workforce of specific skills. Training of workers and learning-by-doing reduce the unit cost as the production rate speeds and improve product and process quality. Better quality means less rework and machine downtime. All these aspects indirectly influence energy efficiency, which has become a concern for manufacturing companies due to the stringent targets imposed by legislators, the significance of energy costs, and the increasing customers’ awareness about the environment. A company, however, could invest in some alternatives to improve its energy efficiency. This paper, therefore, proposes a lot sizing model for a manufacturing company that includes the relevant learning outcomes, directly and indirectly, affecting its energy efficiency. The main result of the study is to show how learning in production and energy efficiency affects each other and the optimal lot size quantity.

Ergonomics as a Social Component of Sustainable Lot-sizing: A Review

Lot-sizing has an increased attention in recent years. In the area of production planning and control, this trend has given rise to the development of lot-sizing models that considers sustainability issues besides the optimization of total operational cost. The study is based on tertiary study that is ensured to analyze the total work have been published. The research was conducted by the definition of appropriate keywords for understanding sustainability issues and ergonomics as a social component in lot-sizing. The paper at hand attempts to understand the development of sustainability issues in lot-sizing and ergonomics as a social component in lot-sizing. We observe that studies focusing on all three dimensions of sustainability are comparatively scarce. However, only a few of the studies have been covered the social sustainability aspect. It is observed that studies addressing ergonomics issues are scarce, and more focus is required on the social sustainability impacts along the supply chain and lot-sizing. Most of ergonomic assessment covered relaxation allowance and energy expenditure rate, OWAS, NIOSH and another consideration about ergonomic lot-sizing is the motion types investigated by authors which were picking, storing as a lifting and carrying motions and did not covered pushing, pulling, bending and other hand motions which have positive relationship with work related musculoskeletal disorders. Finally, we propose future directions to extend research on the ergonomics in lot-sizing.

The single-manufacturer single-retailer integrated production-delivery lot sizing model with production capacity under stochastic lead time demand

This paper develops an integrated production-delivery lot sizing model in which a manufacturer supplies a product a retailer. The manufacturer produces the product at a finite rate and the production rate is assumed to be smaller than market demand rate. Meanwhile, we assume that the lead time demand is stochastic and that shortages are permitted. The lead time and the reorder point are in this model as decision variables. The objective is to determine the optimal ordering quantity, reorder point, lead time and the delivery number during each production cycle which minimize the expected total cost per unit time of the manufacturer and the retailer by giving an effective iterative algorithm. A numerical example is given as an illustration of this algorithm for the integrated system.

Achieving cost efficiency through increased inventory leanness: Evidences from oriented strand board (OSB) industry

Impact of inventory leanness on financial performance has been well-documented by scientific community for various industries. However, its application in value-added wood products industry is relatively scarce when compared to other industry segments. The objective of the study was to investigate impact of increased inventory leanness on financial performance of a value-added wood products manufacturer. Outcomes of the study was also expected to contribute to the intersection zone of three research streams, namely Lean Inventory Management, Procurement Lot-Sizing and Demand Forecasting. To achieve stated objectives, research question and hypotheses were constructed around an un-capacitated single item raw material lot-sizing problem in the context of an SME. A six-step systematic optimization approach (SOA) was developed and followed to improve cost efficiency throughout the study. Time series and Regression based demand forecasting coupled with Dynamic and Statistical lot-sizing models to derive the optimal solution to the problem. At the end of exhaustive simulations, an optimized procurement strategy that could achieve 32.87%–60.90% raw material inventory management cost savings depending on desired buffer level was proposed. Consequently, increased inventory leanness led to less costly procurement activities within target supply chain, which indicated enhanced overall financial performance of the firm through proper risk management and mitigation.

Sustainability in Material Purchasing: A Multi-Objective Economic Order Quantity Model under Carbon Trading

Sustainability in material purchasing is a growing area of research. Goods purchasing decisions strongly affect transportation path flows, vehicle consolidation, inventory levels, and related obsolescence costs. These choices have an economic impact on the supply chain, in terms of different logistic costs, and an environmental impact, in terms of the carbon emissions produced during goods transportation, storage and final recovery. In this paper, we initially analyze and compare the environmental economic policies established by the International Governments in relation to the carbon trading systems adopted. Then, we overcome a traditional single objective formulation, by developing a bi-objective lot-sizing model in which costs and emissions are kept separated and analyzed by using a Pareto frontier subject to a Cap and Trade mitigation policy. The model is useful in practice to support managers in understanding the Pareto frontier shape linked to a specific purchasing problem, defining the cost-optimal and emission-optimal solutions and identifying a sustainable quantity to purchase when a Cap and Trade mitigation policy is present. We further analyze the model behavior according to variation in market carbon price and we finally analytically demonstrate that today carbon prices are still far too low to motivate managers towards sustainable purchasing choices: there is still a gap of about 79%.

Optimization of a Joint Economic Lot Size Model for a First-tier Supplier with Sequential Processes Using a Genetic Algorithm

As corporate competition intensifies in the 21st century, optimal in the integrated supply chain is more important than optimization of individual company. Our research examines a series of integrated supply chain systems comprising a single raw material supplier, a first-tier supplier with multiple processes, and a single original equipment manufacturer in the Korean automotive industry. Unlike other papers, we have studied the situation in which the first-tier supplier has an assembly process. We have also analyzed the situation in which the demand for semi-finished products occurs in the first-tier supplier process or in which semi-finished products are purchased from subcontractors and put into production. The objective function is to minimize the sum of production costs, inventory holding costs, ordering costs, and setup costs in the integrated supply chain. To solve this problem, we formulated nonlinear programming, and developed a genetic algorithm. The results showed that using a dynamic lot size is cheaper than using a fixed lot size. In addition, the lower the setup cost, the smaller the lot size, and when a certain level is reached, the lot size will be the same even if the cost changes.

The Uncapacitatied Dynamic Single-Level Lot-Sizing Problem under a Time-Varying Environment and an Exact Solution Approach

The dynamic lot-sizing problem under a time-varying environment considers new features of the production system where factors such as production setup cost, unit inventory-holding cost, and unit price of manufacturing resources may vary in different periods over the whole planning horizon. Traditional lot-sizing theorems and algorithms are no longer fit for these situations as they had assumed constant environments. In our study, we investigated the dynamic lot-sizing problem with deteriorating production setup cost, a typical time-varying environment where the production setup is assumed to consume more preparing time and manufacturing resources as the production interval lasts longer. We proposed new lot-sizing models based on the traditional lot-sizing model considering the changing setup cost as a new constraint, called uncapacitatied dynamic single-level lot-sizing under a time-varying environment (UDSLLS-TVE for short). The UDSLLS-TVE problem has a more realistic significance and higher research value as it is closer to reality and has higher computational complexity as well. We proposed two mathematical programming models to describe UDSLLS_TVE with or without nonlinear components, respectively. Properties of the UDSLLS-TVE models were extensively analyzed and an exact algorithm based on forward dynamic programming (FDP) was proposed to solve this problem with a complexity of O (n2). Comparative experiments with the commercial MIP solver CPLEX on synthesized problem instances showed that the FDP algorithm is a global optimization algorithm and has a high computational efficiency.

Production lot-sizing with dynamic capacity adjustment

In this paper we study a single-item lot-sizing model in which production capacity can be adjusted from time to time. There are a number of different production capacity levels available to be acquired in each period, where each capacity level is assumed to be a multiple of a base capacity unit. To reduce the waste of excess of capacity but guarantee meeting the demand, it is important to decide which level of capacity should be acquired and how many units of the item should be produced for every period in the planning horizon. Capacity adjustment cost incurs when capacity acquired in the current period differs from the one acquired in the previous period. Capacity acquisition costs, capacity adjustment costs, and production costs in each period are all time-varying and depend on the capacity level acquired in that period. Backlogging is allowed. Both production costs and inventory costs are assumed to be general concave. We provide optimal properties and develop an efficient exact algorithm for the general model. For the special cases with zero capacity adjustment costs or fixed-plus-linear production costs, we present a faster exact algorithm. Computational experiments show that our algorithm is able to solve medium-size instances for the general model in a few seconds, and that cost can be reduced significantly through flexible capacity adjustment.

Tactical production planning for physical and financial flows of supply chain in a multi-site context

In financial crisis, companies always need free cash flow to efficiently react to any uncertainties to ensure solvency. Thus, this thesis serves as an interface between operations and finance to develop tactical production planning models for joint management of physical and financial flows in the supply chain. In these models, the financing cost of operation-based working capital requirement (WCR) is integrated as a new financial aspect never before considered in the lot-sizing literature. We first focus on extending the classic EOQ model by considering the financing cost of WCR with a profit maximization objective. The optimal analytic production quantity formula is derived as well as sensitivity analysis of this model. Moreover, a comparison with the EOQ model and with the formula which considers the cost of capital are discussed. Secondly, a dynamic lot-sizing-based, discounted cash flow model is established based on Uncapacitated lot-sizing model. The zero-inventory ordering property is proven valid for this case and a polynomial-time algorithm can thus be established. Thirdly, multi-level and infinite capacity scenario is investigated with both sequential and centralized approaches. The ZIO property is demonstrated valid in both cases. Dynamic-programming based algorithms are constructed in order to obtain an optimal solution. This thesis should be considered as a first, but significant setup of combining production planning and working capital management. It is shown the significant financial consequences of lot-sizing decision on production planning. The cases investigated in this thesis may be tackled as subproblems in the study of more realistic scenarios.

Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks

Price is an important factor affecting demand rate based on economic and marketing theory. In addition, today's health-conscious consumers prefer a product further from its expiration date because it is fresher and can be stored longer. Furthermore, a large display of goods in a supermarket induces customers to buy more due to its visibility, variety, and seasonality. As a result, we develop a joint pricing and lot-sizing model for retailers selling perishable products in which the demand depends on not only the selling price and reference price but also the product freshness linked to expiration date and displayed stock level. We then formulate the problem as a deterministic model in which the retailer determines the optimal selling prices and ending inventory levels to maximize the discounted total profit. Furthermore, we discuss the retailer's optimal decisions under two scenarios of demand behavior: loss neutrality and loss aversion. Our theoretical analysis shows that the optimal pricing and ending inventory level strategies are both monotonic in time, and the equilibrium of the system is unique for these two scenarios of demand behavior. Finally, a set of structural properties and sensitivity analysis is also developed to characterize the impacts of model parameters on the optimal decisions.

Testing the robustness of deterministic models of optimal dynamic pricing and lot-sizing for deteriorating items under stochastic conditions

Many models within the field of optimal dynamic pricing and lot-sizing models for deteriorating items assume everything is deterministic and develop a differential equation as the core of analysis. Two prominent examples are the papers by Rajan et al. (Manag Sci 38:240–262, 1992) and Abad (Manag Sci 42:1093–1104, 1996). To our knowledge, nobody has ever tested whether the optimal solutions obtained in those papers are valid if the real system is exposed to randomness: with regard to demand process as well as with regard to the deterioration process. The motivation is that although the real world is indeed stochastic, it is often more convenient to work with a deterministic decision model providing a nice closed form solution. The crucial thing is of course whether the results derived in the deterministic setting are robust when tested in a stochastic environment. Therefore, in this paper, we will try to expose the model by Abad (1996) and Rajan et al. (1992) to stochastic inputs; however, designing these stochastic inputs such that they as closely as possible are aligned with the assumptions of those papers. We do our investigation through a numerical test where we test the robustness of the numerical results reported in Rajan et al. (1992) and Abad (1996) in a simulation model. Our numerical results seem to confirm that the results stated in these papers are indeed robust when being imposed to stochastic inputs.

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.

Green logistics under imperfect production system: A Rough age based Multi-Objective Genetic Algorithm approach

Abstract Imperfect economic production lot size (EPL) models are considered with time dependent defective rate. Here, defective production starts after the passage of some time from production commencement. Produced defective units are partially reworked and sold as fresh units. Under the environmental regulation, a cost (carbon tax) is charged by the government to mitigate global warming by reducing carbon emission (CE). Management also uses carbon trading when upper limit of carbon emission is given by the government. These costs bring a contradiction to a production manager. For more profit, if more production is decided, then CE and tax due to that are more. The models are formulated as single- and multi-objective profit maximization problems and solved using Rough age based single- and multi-objective Genetic Algorithms (RMOGAs). Numerical experiments are performed and graphical presentation of the results are depicted to illustrate the models. An algorithm with example for a firm management to achieve the maximum profit is also presented.

Determination of Economic Lot Size between Suppliers and Manufacturers for Imperfect Production System with Probabilistic Demand

This study aims to develop a combined economical lot size model between supplier and manufacturer for imperfect production processes with probabilistic demand patterns and constant lead times. The supplier side produces the product within a certain time interval then sent to the manufacturer with a certain amount of lot size. Imperfect supplier production systems are characterized by the probability of defective product (γ). The model decision variables are the lot size of the manufacturer's ordering, supplier lot size, and the reorder point of the manufacturer. The optimal decision variables are obtained by minimizing the total expected cost of the combined costs between the suppliers and the manufacturers borne by both parties. The model is built compared to the transactional partnership model, in which the supplier does not participate in the efficiency of its inventory system. A numerical example is given as an illustration of the JELS model and the transactional partnership model. Sensitivity analysis of the model is done by changing the parameters aimed at analyzing the behavior of the developed model.