Economic Lot Size Research Articles

Replenishment policy in a logistics system with environmental considerations

One of the most applicable models in inventory and production planning is the economic lot sizing model, which recently extended by environmental considerations and the sustainability concept. The theoretical sustainable lot sizing model is only a basic model that ignores many real-world conditions in the modern cloud manufacturing environments, for example, the possibility of shortage and stockout in the inventory system. In this work, we have presented new sustainable lot-sizing models in which different shortage conditions are considered. We developed a basic model for the time that shortage in not allowed, and full lost sale, full backordering and partial backordering models when shortages are allowed. We proposed an integrated procedure for determining the optimal values of the variables of the proposed models. The models and proposed procedure application are illustrated with numerical examples based on an Iranian petrochemical company's data. Finally, applying sensitivity analysis, the model's results have been analysed and discussed. These results show that the partial backordering case is a general and more realistic model which can be used in many real cases.

Replenishment policy in a logistics system with environmental considerations

One of the most applicable models in inventory and production planning is the economic lot sizing model, which recently extended by environmental considerations and the sustainability concept. The theoretical sustainable lot sizing model is only a basic model that ignores many real-world conditions in the modern cloud manufacturing environments, for example, the possibility of shortage and stockout in the inventory system. In this work, we have presented new sustainable lot-sizing models in which different shortage conditions are considered. We developed a basic model for the time that shortage in not allowed, and full lost sale, full backordering and partial backordering models when shortages are allowed. We proposed an integrated procedure for determining the optimal values of the variables of the proposed models. The models and proposed procedure application are illustrated with numerical examples based on an Iranian petrochemical company's data. Finally, applying sensitivity analysis, the model's results have been analysed and discussed. These results show that the partial backordering case is a general and more realistic model which can be used in many real cases.

Development of a Joint Economic Lot Size Model with Stochastic Demand Within non-equal shipments

The majority of the researches in the integrated vendor-buyer inventory problem assume that the shipments are equal. In this paper, shipments are considered to be non-equal. Both demand and delivery times are also assumed to be stochastic. Moreover, unsatisfied demand can be backordered and lost, as well asconsidering a service level constraint. The objective is to minimize both buyer and vendor costs at the same time. The problem is solved by an exact heuristic algorithm. To validate the algorithm’s performance, the results are compared with that of LINGO solver. Finally, a set of numerical problems is applied to compare the result in the integrated and independent forms.

Incorporating logistic costs into a single vendor–buyer JELS model

Inventory replenishment decisions in supply chains usually benefit from economies of scale to minimize logistics costs; however, this approach may create a conflict between truckload utilization and economic ordering decisions. This paper proposes two Joint Economic Lot Size models that incorporate logistic costs (a function of shipping weight and distance) with two transportation modes: truckload (TL) and less-than-truckload (LTL) shipments. Additionally, a heuristic model is proposed to provide the solution when the mode of transportation is limited to TL and LTL shipments. The procedures for solving each model are developed based on two challenges in transportation: incapacitated shipments and capacitated shipments. Moreover, numerical examples are given to illustrate the proposed models based on four aspects: total costs, order quantity, the number of deliveries, and production batch quantity. The three proposed models are subsequently compared to reveal results for managers to consider for improving the performance of their supply chain.

Forward and backward stocking policies for a two-level supply chain with consignment stock agreement and stock-dependent demand

We consider a vendor-managed inventory with consignment stock agreement applied to the integrated vendor–buyer system, in which the vendor manufactures a single product in batches and delivers it in equal-sized transfer lots to the buyer. Some of the delivered items are presented to the end customers in the buyer's display area, while the rest of the items are kept in the buyer's backroom warehouse. Demand is assumed to be positively dependent on the amount of stock displayed. We propose a new joint economic lot sizing (JELS) model, taking into account the vendor's stocking policy, to maximize the total profit for the coordinated system. This paper first proves that, for any stock-dependent demand, a minimum restocking level at the buyer's sales floor is a more profitable strategy than the traditional run-out replenishment policy. It then shows that when the unit inventory holding cost decreases as stock moves downstream the supply chain, the vendor ought to adopt the forward stocking policy, in which product is pushed forward to the buyer's warehouse as soon as possible. Finally, it derives the analytical formulations for the maximum inventory levels at different stocking points, while the vendor adopts either forward or backward stocking policy. Numerical examples are also provided for illustration.

Duality Approaches to Economic Lot‐Sizing Games

Sharing common production, resources, and services to reduce cost are important for not for profit operations due to limited and mission‐oriented budget and effective cost allocation mechanisms are essential for encouraging effective collaborations. In this study, we illustrate how rigorous methodologies can be developed to derive effective cost allocations to facilitate sustainable collaborations in not for profit operations by modeling the cost allocation problem arising from an economic lot‐sizing (ELS) setting as a cooperative game. Specifically, we consider the economic lot‐sizing (ELS) game with general concave ordering cost. In this cooperative game, multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost. When both the inventory holding cost and backlogging cost are linear functions, it can be shown that the core of this game is non‐empty. The main contribution of this study is to show that a core allocation can be computed in polynomial time under the assumption that all retailers have the same cost parameters. Our approach is based on linear programming (LP) duality. More specifically, we study an integer programming formulation for the ELS problem and show that its LP relaxation admits zero integrality gap, which makes it possible to analyze the ELS game by using LP duality. We show that there exists an optimal dual solution that defines an allocation in the core. An interesting feature of our approach is that it is not necessarily true that every optimal dual solution defines a core allocation. This is in contrast to the duality approach for other known cooperative games in the literature.

The two-level economic lot sizing problem with perishable items

We present an economic lot sizing model of a supply chain for the procurement and distribution of a perishable item. We assume that the consumers always buy the item that lasts longer. We show that determining optimal procurement and transfer plan is NP-hard, and present polynomial time solvable special cases.

Supply Chain Management with Online Customer Selection

We consider new online variants of supply chain management models, where in addition to production decisions, one also has to actively decide on which customers to serve. Specifically, customers arrive sequentially during a selection phase, and one has to decide whether to accept or reject each customer upon arrival. If a customer is rejected, then a lost-sales cost is incurred. Once the selection decisions are all made, one has to satisfy all the accepted customers with minimum possible production cost. The goal is to minimize the total cost of lost sales and production. A key feature of the model is that customers arrive in an online manner, and the decision maker does not require any information about future arrivals. We provide two novel algorithms for online customer selection problems, which are based on repeatedly solving offline subproblems that ignore previously made decisions. For many important settings, our algorithms achieve small constant competitive ratio guarantees. That is, for any sequence of arriving customers, the cost incurred by the online algorithm is within a fixed constant factor of the cost incurred by the respective optimal solution that has full knowledge upfront on the sequence of arriving customers. Finally, we provide a computational study on the performance of these algorithms when applied to the economic lot sizing and joint replenishment problems with online customer selection.

A joint economic lot size model with financial collaboration and uncertain investment opportunity

Establishing long-term relationships among the members of a supply chain has become necessary to enhance the supply chain's competitiveness in a globalized environment. Besides coordinating operational decisions, such as how much and when to produce or to order, the members of a supply chain may also share financial resources or act jointly on the capital market. This is important especially when companies have unequal access to capital, for example because they are located in countries with different economic conditions and banking policies and/or they have different credit ratings. The joint financing of investments across the supply chain may thus ensure the stability of production and of the flow of products to the customers. In addition, it strengthens the established relationships among the supply chain members. The paper at hand takes up these issues and presents a joint economic lot size model that allows investments financed cooperatively by the members of the supply chain. In particular, it considers a two-stage single-vendor single-buyer supply chain and assumes that the vendor has the option to invest in increasing its production rate. The outcome of these attempts to improve production capabilities, however, is uncertain and subject to an investment success probability. Due to different access to capital, the vendor and the buyer may also share the investment and the outcome uncertainty, which can be beneficial to both parties.

Optimal contract design in the joint economic lot size problem with multi-dimensional asymmetric information

Previous work has studied the classical joint economic lot size model as an adverse selection problem with asymmetric cost information. Solving this problem is challenging due to the presence of countervailing incentives and two-dimensional information asymmetry, under which the classical single-crossing condition does not need to hold. In the present work we advance the existing knowledge about the problem on hand by conducting its optimality analysis, which leads to a better informed and an easier problem solution: First, we refine the existing closed-form solution, which simplifies problem solving and its analysis. Second, we prove that Karush–Kuhn–Tucker conditions are necessary for optimality, and demonstrate that the problem may, in general, possess non-optimal stationary points due to non-convexity. Third, we prove that certain types of stationary points are always dominated, which eases the analytical solution of the problem. Fourth, we derive a simple optimality condition stating that a weak Pareto efficiency of the buyer’s possible cost structures implies optimality of any stationary point. It simplifies the analytical solution approach and ensures a successful solution of the problem by means of conventional numerical techniques, e.g. with a general-purpose solver. We further establish properties of optimal solutions and indicate how these are related with the classical results on adverse selection.

Model Persediaan Pemasok-Pembeli Dengan Produk Cacat Dan Kecepatan Produksi Terkontrol.

In this paper, we consider a joint economic lot size (JELS) model consisting of single vendor single buyerwith single product. We intend to study the impact of defective items and controllable production rate onthe model. The demand in buyer side is assumed to be stochastic. The delivery of lot from vendor to buyer is conducted under equal size shipment and the lead time is assumed to be constant. The defective items founded by the inspector in buyer side are carried in buyer’s storage until the next shipment and will be returned to the vendor. The goal of the proposed model is to determine optimal delivery frequency, review period and production rate by minimizing the joint total cost. A sensitivity analysis is performed to show the impact of the changes of the decision variables on model’s behavior. The result from the sensitivity analysis shows that the joint total cost is sensitive to the changes of defect rate, demand uncertainty and demand rate.

Efficient approximation schemes for Economic Lot-Sizing in continuous time

We consider a continuous-time variant of the classical Economic Lot-Sizing (ELS) problem. In this variant, the setup cost is a continuous function with lower bound Kmin>0, the demand and holding costs are integrable functions of time and arbitrary replenishment policies are allowed. Starting from the assumption that certain operations involving the setup and holding cost functions can be carried out efficiently, we show that this variant admits a simple approximation scheme based on dynamic programming: if the optimal cost of an instance is OPT, we can find a solution with cost at most (1+ϵ)OPT using no more than O(1ϵ2OPTKminlogOPTKmin) of these operations. We argue, however, that this algorithm could be improved on instances where the setup costs are generally “very large” compared with Kmin. This leads us to introduce a notion of input-size parameter σ that is significantly smaller than OPT/Kmin on instances of this type, and then to define an approximation scheme that executes O(1ϵ2σ2log2(OPTKmin)) operations. Besides dynamic programming, this second approximation scheme builds on a novel algorithmic approach for Economic Lot Sizing problems.

A surcharge pricing scheme for supply chain coordination under JIT environment

Just-In-Time (JIT) system involves frequent shipments of smaller batch sizes from the supplier to the buyer. For the buyer, it results in the reduction of the inventory holding cost. However, it is often accompanied by an increase in the set up cost for the supplier. Thus, the supplier may be reluctant to switch to the JIT mode unless he is assured of some form of compensation. In this paper, we introduce a pricing scheme where the buyer offers the supplier an increase in the wholesale price, to encourage the supplier to switch to the JIT mode. Such a pricing scheme may be termed as a surcharge. We develop the economics of surcharge pricing as a supply chain coordinating mechanism under JIT environment. We also establish the equivalence of surcharge pricing with other common coordination mechanisms like Quantity Discount (QD) and Joint Economic Lot Sizing (JELS) model.

An EPQ (economic production lot size) model for product life cycle (maturity stage) of deteriorating items with increasing demand rate and shortages

A product life cycle is the life span of a product which the period begins with initial product specification and ends with the withdrawal from the market of both the product and its support. A product life cycle can be divided into several stages characterised by the revenue generated by the product. The purpose of this paper is to investigate the optimal production lot size and total cost during the product life cycle which consists of introduction, growth, maturity and decline stages. The defective rate is considered as a variable of known proportions. The relevant model is built and it is also solved. Necessary and sufficient conditions for a unique and optimal solution are derived. A numerical example is provided using LINGO software.

An integrated inventory model for single-vendor single-buyer system with freight rate discount and stochastic demand

This paper deals with joint economic lot sizing model (JELS) in the context of two stage supply chain comprising of single-vendor and single-buyer. The shipment from vendor to buyer is conducted in equally sized and the demand in buyer side is assumed to be normally distributed. In this paper, freight cost is explicitly incorporated in the model and formulated by all-weight freight discount model and incremental freight discount model. We provide two procedures to find the optimal shipment size, safety factor, production batch and discount level on both models. Numerical examples are presented to illustrate the benefit of the model. The result shows that incorporating freight discount into inventory model may result in significant saving on total cost. The increase in the number of discount levels will lead to the decrease in buyer cost and total cost. Moreover, it also shows that the changes in ordering cost, purchase cost, setup cost and production rate in incremental freight discount model give significant impacts on shipment size and safety factor.

A note on the joint economic lot size model with stochastic demand and variable lead time

This paper considers the problem of choosing optimal production policy in an integrated production-inventory system comprising a single manufacturer and a single buyer. Two production scenarios are compared, namely continuous and lot-for-lot production policies, using a mathematical modelling approach and the expected total joint cost is considered as the performance measure. In addition, the lost opportunity cost due to production downtime in the lot-for-lot policy, including the cost of unemployment of human resources and stoppage of the production line, is incorporated into the model. Numerical experiment and sensitivity analysis in terms of setup and lost opportunity costs are provided to illustrate the theory and the percentage of difference in the expected total cost of the two policies is analysed. The presented approach helps managers to make the right decisions on the optimal production policy under different conditions, especially where expert and technical human resources and/or expensive production equipments are utilised.

A review of mathematical inventory models for reverse logistics and the future of its modeling: An environmental perspective

The responsible management of product return flows in production and inventory environments is a rapidly increasing requirement for companies. This can be attributed to economic, environmental and/or regulatory motivations. Mathematical modeling of such systems has assisted decision-making processes and provided a better understanding of the behavior of such production and inventory environments. Therefore, this paper reviews the literature on the modeling of reverse logistics inventory systems that are based on the economic order/production quantity (EOQ/EPQ) and the joint economic lot size (JELS) settings, so as to systematically analyze the mathematics involved in capturing the main characteristics of related processes. The literature is surveyed and classified according to the specific issues faced and modeling assumptions. Special attention is given to environmental issues. In addition, there are indications of the need for the mathematics of reverse logistics models to follow current trends in ‘greening’ inventory and supply-chain models. Finally, the modeling of waste disposal, greenhouse-gas emissions and energy consumption during production is considered as the most pressing priority for the future of reverse logistics models. An illustrative example for modeling reverse logistics inventory models with environmental implications is presented.

Comments on ‘vendor-managed inventory with consignment stock agreement for single vendor–single buyer under the emission-trading scheme’

Zanoni, Mazzoldi, and Jaber [Zanoni, S., L. Mazzoldi, and M. Y., Jaber. 2014. Vendor-managed inventory with consignment stock agreement for single vendor–single buyer under the emission-trading scheme. International Journal of Production Research 52 (1): 20–31] consider a joint economic lot size problem under the vendor-managed inventory with consignment stock agreement and an emission-trading scheme. They show that the total cost of the system is a jointly convex function by simply showing that every element of the Hessian is positive. Noticing this mistake, we analyse the same problem in this technical note. We first provide a closed-form solution when the production rate is given. In order to avoid a complete search over all possible production rates, we then develop an efficient continuous approximation algorithm. Computational experiment shows that the approximation algorithm is effective and efficient.

A new hybrid algorithm of scatter search and Nelder–Mead algorithms to optimize joint economic lot sizing problem

This paper introduces a hybrid algorithm of Nelder–Mead and scatter search algorithms called SSNM. Numerical results for some known test problems with varying dimensions from 2 to 100 variables prove its great performance for a bounded or constrained problem. This paper also has a considerable contribution to the joint economic lot sizing (JELS) problem literature. This problem is considered for a two-stage supply chain with price-sensitive demand for geometric, geometric-then-equal size and optimal shipment policies. Solution procedures are also developed which use the SSNM algorithm in their steps. These models and solution procedures are novel in the JELS literature.

The algorithm for the optimal cycle time and pricing decisions for an integrated inventory system with order-size dependent trade credit in supply chain management

A given inventory problem consists of two parts: (1) the modeling part and (2) the solution procedure part. The modeling part can provide insight into the solution of the inventory problem and the solution procedure part involves the implementation of the inventory model. Both the modeling part and the solution procedure part of the inventory problem are equally important. Recently, Ouyang et al. [17] developed an integrated inventory model with a price-sensitive demand rate and determined both the economic lot size of the buyer’s ordering and the supplier’s production batch in order to maximize the total profit per unit time. Basically, their modeling is correct and interesting. They developed an algorithm based upon the first-order condition and the second-order condition to locate the optimal solution. However, the fundamentals of mathematics and the numerical examples which are considered in this paper illustrate that their algorithm based upon the first-order condition and the second-order condition to locate the optimal solution has several shortcomings. These shortcomings are shown here to influence the accuracy of the implementation of the inventory model. Since there exist reasons and motivations to present the correct solution procedures to the targeted readers, the main purpose of this paper is to adopt the rigorous methods of mathematical analysis in order to develop the complete solution procedures to locate the optimal solution for removing shortcomings in the earlier investigation by Ouyang et al. [17].