Economic Lot Size Research Articles

Development of a Joint Economic Lot Size Model for Vendor-Buyer with Normal Demand and Empirical Delivery Time with Service level and Incremental Discount

This study addresses the single-product joint vendor-buyer lot-sizing problem with normal and empirical distributions for demand and delivery time respectively. A service level constraint is assumed for the buyer in which unsatisfied demand might be lost or backordered. The vendor produces products at a certain rate and as a fraction of the received order and, subsequently, ships the produced product to the buyer. The shipments are assumed to be equal in size. Also, to encourage the buyer to place further orders, the vendor adopts the incremental discount policy. The purpose is to determine the number of shipments delivered by the vendor to the buyer, as well as buyer’s reorder point, buyer’s order quantity, and buyer’s safety stock, so as to minimize the average costs incurred by both the buyer and the vendor. Once the problem has been formulated, the particle swarm optimization (PSO) algorithm was implemented to solve it.

Joint economic lot size (JELS) model for single-vendor single-buyer with variable production rate and partial backorder

This paper presents joint economic lot size (JELS) models under stochastic demand for single-vendor single-buyer system by synchronising ordering, production and raw material procurement cycles. We consider variable production rate and partial backorder in the proposed model. An iterative procedure is employed to determine simultaneously safety factor, delivery lot size, delivery frequency, production batch, raw material lot size and production rate for minimising total cost. We investigate the behaviour of the model with the use of numerical analysis. Our study indicates that considering variable production rate in the model results in lower total cost compared to the case where the production rate is assumed to be fixed. Our study also shows that the increase in demand uncertainty leads to higher total costs, but the buyer absorbed most of the increase. In addition, we also show that the minimum total cost is obtained when the shortages are fully, rather than partially, backordered.

A dynamic method for the production lot sizing with machine failures

This paper presents a dynamic method to deal with two economic lot sizing models with exponential machine failures. Its main difference from the static method is that the machine failure is separately treated from the perspective of failure realisation. The optimal lot size is derived through the equation between the incurred average cost and the marginal cost, which has to be performed for each lot due to the random failures. In the first no-resumption model, the simulation shows that two methods have the similar performance as the lot number increases. For the resumable model, we propose a zero-inventory resumption policy that always resumes the production after each machine failure but delays the resumption until the on-hand inventory is depleted. The simulation result indicates that the new policy by the dynamic method outperforms the initial abort/resume policy, and it also shows the convergence as the production continues.

Ford Whitman Harris's economical lot size model

Here we celebrate the centennial of Ford Whitman Harris's model for determining economical lot sizes, which was published in the A.W. Shaw Company's magazine Factory, The Magazine of Management in February 1913. The square-root formula derived by Harris has become one of the most cited and applied results in production and operations management. We examine the circumstances under which this result was derived, and explore the probable causes for the later obscurity of Harris's paper, which was forgotten for many years.

Optimal policy for an inventory system with power demand, backlogged shortages and production rate proportional to demand rate

This article analyzes an inventory system for items with time-varying demand. The way by which demand occurs during the inventory cycle follows a power demand pattern. The production of items allows to add stock in the inventory during a replenishment period. The production rate is proportional to demand rate. Shortages are allowed and completely backlogged. Holding cost, shortage cost and order cost are the costs considered in the inventory model. The aim consists of the minimization of the total cost per inventory cycle. An efficient approach is developed to obtain the optimal scheduling period and the optimal reorder point. In addition, the economic lot size and the minimum cost of the inventory management is determined. Some numerical examples are presented for illustrating the proposed inventory model.

A single-vendor single-buyer joint economic lot size model subject to budget constraints

In current competitive business environment, firms have realized that integrated scheduling and managing their supply chain through its different levels is substantial. On the other hand, the supply chain parties act under some restrictions, and ignoring these limitations results in infeasible solutions in many real-world case studies. The purpose of this paper is to present a joint economic lot size (JELS) model, which considers the buyer's and the vendor's budget limitations to achieve a more reliable model solution. The idea is to present an extension of the model originally proposed by Ben-Daya et al. (Eur J Oper Res 185:726–742, 14). In addition, a solution algorithm is presented to make the solution procedure of proposed constrained model less complex. Numerical analysis is conducted to show the effect of applying the budget constraints.

Economic lot sizing: The capacity reservation model

Capacity reservation contracts allow a consumer to purchase up to a certain capacity at a unit price lower than that of the spot market, while the consumer’s excess orders are realized at the spot price. In this paper, we consider a lot sizing problem where the consumer places orders following a capacity reservation contract. In particular, we study the general problem and the polynomial time solvable special cases of the problem and propose corresponding algorithms for them.

A Vendor-Buyers Integrated Inventory Model Involving Quality Improvement Investment in a Supply Chain

Today's supply chain environment requires a new spirit of cooperation between the buyer and the vendor. So the characteristics of current manufacturing systems are consistent high quality, small lot sizes, frequent delivery, and close supplier ties. Just-in-time (JIT) production is an integrated set of activities to achieve those objects by adopting continuous quality improvement and developing a long-term partnership. This paper presents the optimum joint economic lot size in a case where multiple buyers are demanding one type of item from a single vendor by summing the ordering/setup cost, holding cost and quality improvement investment. This model is useful particularly for integrated inventory systems where the vendor and the buyer form a strategic alliance for profit sharing.

Vendor-managed inventory with consignment stock agreement for single vendor–single buyer under the emission-trading scheme

This paper presents a joint economic lot size (JELS) model for coordinated inventory replenishment decisions under the vendor-managed inventory (VMI) with consignment stock (CS) agreement and an emission-trading scheme. The paper assumes a single product that flows along a two-level supply chain system, with a single vendor and a single buyer. The total cost of the system is the performance measure, which is the sum of the vendor’s and the buyer’s total costs. The total cost includes the set-up and order costs, inventory holding costs, greenhouse gases (GHG) emissions tax and penalty costs. A mathematical model is proposed to determine: (1) the vendor’s production lot size quantity; (2) the number of shipments sent by the vendor to the buyer in a cycle; and (3) the production rate that minimises the total cost of the supply chain. Some numerical examples are carried out, as well as comparisons with the traditional JELS model for a classic two-level supply chain. Results show that the performance of the system is better when it is operated under a VMI with CS agreement, which is capable of reducing the traditional inventory holding costs and, for some values of given parameters, the GHG emissions tax and penalty costs.

PENENTUAN UKURAN LOT GABUNGAN DENGAN BARGAINING GAME DAN CONSIGNMENT UNTUK PEMANUFAKTUR DAN PEMBELI TUNGGAL

This paper considers the integrated inventory model, where the ordering lot is no longerdetermined by the buyer, but it is determined by both parties in order to minimize the totalinventory cost. There are three approaches in determining the joint lot size: Joint Economic LotSize (JELS), Joint Economic Lot Size with Bargaining Game, and Joint Economic Lot Size withConsignment. On JELS with Bargaining Games, the manufacturer will offer rebates to thebuyer, since buyer will be at the disadvantage side if the manufacturer only offers JELS.Furthermore, on model JELS with Consignment, the manufacturer will own the product until itis used by the buyer. Using JELS, manufacturer makes a savings of 8.73% while the buyersuffered a loss of 3.37%. The result of using the JELS with Bargaining Game will give themanufacturer a savings of 6.63%, while the inventory cost for the buyer is the same with theinitial condition (individually). This paper concludes that using the JELS with consignment willprovide savings to the manufacturer as well as for the whole system.

Coordinating supplier-retailer using multiple common replenishment epochs with retailers’ choices

Purpose: Provide a coordination strategy using multiple common replenishment epochs (MCRE) for a single-supplier multi-retailer supply chain. Design/methodology/approach: The demand of a product occurs only with a group of heterogeneous and independent retailers with constant rates, whereas all their order requests are fulfilled by the supplier. The supplier decides a set of MCREs with general price and extra bonus to entice the retailers to join any one of the MCREs, or to let them remain with their original order time epochs. A retailer is willing to participate in a CRE as long as the retailer’s cost increase is within its tolerance. This paper provide a mixed integer programming to determine the MCRE strategies in order to minimize the total costs of the supplier. Findings: The results illustrate that MCRE model provided in the paper can generate a better replenishment coordination scheme than single CRE models. Practical implications: Replenishment coordination is one of the most important mechanisms to improve the efficiency in supply chains, e.g., chain convenience stores in the modern retail industry. Originality/value: This is a follow-up research on Joint Economic Lot Size (JELS) models with a focus on multiple retailers with their replenishment coordination.

A multi-stage joint economic lot size model with lead time penalty costs

This paper studies the case of a multi-stage supply chain where equal- and unequal-sized batches are transported between the stages and where penalty costs are imposed for long lead times. Unlike earlier works, this paper does not restrict the number of stages to a given value, but rather analyses a general case of the model. First, an analytical model of the supply chain is developed and the properties of the model are discussed. Secondly, a simulation study is conducted to analyse the behaviour of the model. The results of our analysis indicate that the performance of upstream stages of the supply chain is more critical to the performance of the supply chain than the performance of downstream stages. Furthermore, our findings show that a supply chain stage may improve its position significantly by coordinating its behaviour with its direct neighbours in the supply chain.

Reexamination of “Combining an alternative multi-delivery policy into economic production lot size problem with partial rework” Using an Alternative Approach

A conventional approach for solving an economic production quantity (EPQ) model with multi-delivery policy andquality assurance is to use mathematical modeling for deriving the system cost function and to employ differentialcalculus for proving convexity of the cost function before determining the optimal operating policy. This paperpresents a simplified solution procedure to reexamine a specific EPQ model [1] and demonstrates that both, theoptimal manufacturing lot size and the long-run cost function can be derived without derivatives. The proposedapproach may enable practitioners—with little knowledge of calculus—to understand such a real life vendor-buyerintegrated system with ease.

Design of contract parameters in a closed-loop supply chain

We study an extended joint economic lot size problem which incorporates the return flow of remanufacturable used products. The supply chain under consideration consists of a single supplier and a single buyer. The buyer orders a single product from the supplier, uses it for her own needs, and collects the remanufacturable items after use. The ordered items are shipped from the supplier to the buyer in the lot-for-lot fashion by a vehicle which also returns the collected used items from the buyer to the supplier for remanufacturing and subsequent service of the buyer’s demand in the next order cycle. For satisfying the total demand, the supplier manufactures new items or remanufactures used ones received from the buyer. For given demand, productivity, collection rate, disposal cost, setup cost, order cost, holding cost for serviceable and nonserviceable products at the supplier as well as the buyer the lot size (order size) for the supplier (buyer) has to be found which minimizes the total cost. Furthermore, we address a decentralised decision making of the parties under a two-part tariff and determine their equilibrium strategies within the Nash framework.

An economic lot-size model with non-linear holding cost hinging on time and quantity

This paper develops an economic lot size inventory model where the demand rate depends on the stock level and the cumulative holding cost is non-linear on both the quantity and the time they are stored. More concretely, it is supposed that the demand rate is a concave potential function of the inventory level and the holding cost is potential on both time and quantity. Moreover, shortages are not allowed. A general procedure to determine the optimal lot size and the maximum inventory profit is developed. Also, some results about the profitability of the inventory system are presented. This work extends several inventory models previously considered in the literature. Finally, numerical examples, which help us to understand the theoretical results, are also given.

Optimal economic order quantity for buyer–distributor–vendor supply chain with backlogging derived without derivatives

In this article, we first complement an inappropriate mathematical error on the total cost in the previously published paper by Chung and Wee [2007, ‘Optimal the Economic Lot Size of a Three-stage Supply Chain With Backlogging Derived Without Derivatives’, European Journal of Operational Research, 183, 933–943] related to buyer–distributor–vendor three-stage supply chain with backlogging derived without derivatives. Then, an arithmetic–geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we provide a closed-form solution to integral number of deliveries for the distributor and the vendor without using complex derivatives. Furthermore, our method can solve many cases in which their method cannot, because they did not consider that a squared root of a negative number does not exist. Finally, we use some numerical examples to show that our proposed optimal solution is cheaper to operate than theirs.

A joint economic lot-sizing model for the hospital’s supplier with capacitated warehouse constraint

In a vertically integrated supply chain of healthcare groups such as Chang Gung memorial hospitals in Taiwan, the vendor of special medical items and the hospitals are owned by the same business conglomerate; besides, warehouse capacity is one of the scarcest resources that greatly affect the inventory control policies. Taking the vendor’s limited warehouse capacity into account, this article first proposes a generic joint economic lot sizing (JELS) model in determining the optimal production and shipment policy for a two-layer supply chain. The vendor manufactures the requested products in lots and delivers them in batches to the buyer with partial shipment policy. A threshold value which defines those existing JELS problems as a special case of our proposed model, while the manufacturer’s warehouse capacity exceeds it, is attained. Finally, a numerical example is used to characterize the optimal production and replenishment lot-size decisions for the supply chain.

An economic production lot size model for defective items with stochastic demand, backlogging and rework

An economic production lot size model for defective items with stochastic demand, backlogging and rework

A note on an economic lot size model for price-dependent demand under quantity and freight discounts

Abstract In this study, we revisit research contributed by Burwell et al. (1997. International Journal of Production Economics 48, 141–155), where an economic lot size model for price-dependent demand under quantity and freight discounts was proposed. Specifically, for the cases of mixed discounts in which the quantity discounts are either of the incremental or all-units variety and the freight discounts are of the opposite type, we first provide counterexamples to show that adopting the existing algorithm to determine overall optimal lot size and selling price may not achieve the goal of maximizing profit. We then propose an easy to understand computational algorithm to obtain an exact solution.

Optimal production control policy in unreliable batch processing manufacturing systems with transportation delay

This paper considers the problem of production planning of unreliable batch processing manufacturing systems. The finished goods are produced in lots, and are then transported to a storage area in order to continuously meet a constant demand rate. The main objective of this work is to jointly determine the optimal lot sizing and optimal production control policy that minimise the total expected cost of inventory/backlog and transportation, over an infinite time horizon. The decision variables are the lot sizing and the production rate. The problem is formulated with a stochastic dynamic programming model and the impulse control theory is applied to establish the Hamilton–Jacobi–Bellman (HJB) equations. Based on a numerical resolution of the HJB equations, it is shown that the optimal control policy is governed by a base stock policy for production rate control and economic lot size for batch processing. A thorough analysis and practical issues are addressed with a simulation-based approach. Thus, a combined discrete–continuous simulation model is developed to determine the optimal parameters of the proposed policy when the failure and repair times follow general distributions. The results are illustrated with numerical examples and confirmed through sensitivity analysis.