Lot-sizing Decisions Research Articles

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.

Multi-outsourcing supply chain coordination under yield and demand uncertainties

As a popular production mode, outsourcing enables the cost saving for original equipment manufacturer (OEM) and the acquisition of advanced technology for contract manufacturer (CM). We consider a multi-outsourcing supply chain within the voluntary compliance regime. The game between one OEM facing uncertain demand and arbitrary CMs owning uncertain yields is modeled to study the interactions regarding lot-sizing decisions. A general method is developed to prove the concavity of profit functions, which is formidable to accomplish based on Hessian Matrix. The optimal ordering and production strategies are subsequently characterized. A revenue sharing with surplus purchase contract with great flexibility in parameter selection is proposed to coordinate the supply chain. We find that there exist threshold outsourcing prices beyond which the CMs are motivated to overproduce, otherwise they tend to produce in consistent with the orders. In a competitive market, price war is not a wise strategy for CM due to profit sacrifice. By contrast, if the selling price charged by the OEM is high, a smart CM can appropriately raise its outsourcing price because the order quantity difference among CMs is relatively small. Although supply stability improvement and outsourcing price reduction are both welcomed by the OEM, the latter is more important for it to consider in allocating orders.

The integrated uncapacitated lot sizing and bin packing problem

In the integrated uncapacitated lot sizing and bin packing problem, we have to couple lot sizing decisions of replenishment from single product suppliers with bin packing decisions in the delivery of client orders. A client order is composed of quantities of each product, and the quantities of such an order must be delivered all together no later than a given period. The quantities of an order must all be packed in the same bin, and may be delivered in advance if it is advantageous in terms of costs. We assume a large enough set of homogeneous bins available at each period. The costs involved are setup and inventory holding costs and the cost to use a bin as well. All costs are variable in the planning horizon, and the objective is to minimize the total cost incurred. We propose mixed integer linear programming formulations and a combinatorial relaxation where it is no longer necessary to keep track of the specific bin where each order is packed. An aggregate delivering capacity is computed instead. We also propose heuristics using different strategies to couple the lot sizing and the bin packing subproblems. Computational experiments on instances with different configurations showed that the proposed methods are efficient ways to obtain small optimality gaps in reduced computational times.

Pricing and lot-sizing decision for fresh goods when demand depends on unit price, displaying stocks and product age under generalized payments

The demand decreases as the price increases. Also, the demand rate for fresh goods depends on its freshness, which can be assessed by its sell-by date. Furthermore, a large pile of consumer goods displayed in a supermarket often provokes consumers to buy more due to its freshness, variety, or visibility. With larger business transactions, the supplier usually requests a good-faith deposit (or an advance payment) to discourage the order cancelation. On the other hand, the retailer likes to hold some portion of the purchase amount until the business transaction is completed and satisfactory (or a credit payment). Consequently, an advance-cash-credit (ACC) payment scheme, which is a combination of advance, cash, and credit payments, becomes commonly used in business transactions. In this study, we explore an inventory system in which: demand curve depends on unit price, displayed volume, and sell-by date; and the supplier and the retailer agree with an ACC payment scheme. The objective for grocery stores is to maximize the total profit by finding its price and order cycle simultaneously. We derive theoretical results, and then propose an algorithm. Finally, we perform sensitivity analyses to gain the following managerial insights: (i) An increase in stock efficiency significantly reduces selling price and cycle time while tremendously raising total profit. (ii) Credit payment generates the lowest selling price but the highest profit among all payment types. (iii) An increase in shelf life through preservation technology investment significantly raises total profit while slightly increasing the unit price and cycle time.

A two-level optimisation-simulation method for production planning and scheduling: the industrial case of a human–robot collaborative assembly line

In this work, a novel optimisation-simulation based on the Recursive Optimisation-Simulation Approach (ROSA) methodology is developed to provide effective decision-support for integrated production planning and scheduling. The proposed iterative approach optimises production plans while satisfying complex scheduling constraints, such as robots' allocation in collaborative tasks. The plans are determined through a two-level MILP model and are iteratively evaluated by a detailed discrete-event simulation model to guarantee capacity-feasible solutions at the scheduling level. Through an industrial case study of a multistage assembly line design collaboratively operated by humans and mobile shared robots, near-optimal solutions comprise lot-sizing decisions, the release schedule of production orders, the allocation of tasks to humans or robots, and the number of robots per period. Moreover, by addressing a set of propositions to assess the methodology, the results highlight the advantages of the hybrid approach to converge into optimised operational decisions and analyse the process dynamics.

Optimal reorder level and lot size decisions for an inventory system with defective items

In this paper an (r,Q) stock policy is used to control an inventory system with deterministic demand and defective items. The fraction of defective items in a batch is a random variable whose variance maybe dependent and independent of the order quantity. Considering the numerous benefits of selling high quality items (noting the vital importance for firms like medical device companies), a full inspection of the order quantity is conducted. Although full inspection reduces returns, ensures high quality products and improves customer satisfaction, it requires a substantial inspection time. Consequently, during this time a non-zero probability of undesirable (unplanned) shortages exists. The dominant suggestion in the literature, is to set the reorder level equal to the total demand during inpection period. This paper suggests an alternative reorder level to prevent shortages during this period and to formulate the backlogging cost in the case that unplanned shortages are permitted. Then, theoretical results ensure the determination of the reorder level and the order quantity that minimize the average total cost of the inventory system. Comparisons, through numerical examples and simulation, demonstrate that the proposed model leads to significant reduction both in cost and probability of shortages (if allowed), in relation to other existing models.

A two-stage sequential approach for scheduling with lot-sizing decisions in the context of plastic injection systems

This study addresses a planning problem for plastics injection systems, where final products are assembled of compatible plastic pieces. Additionally, the production of pieces requires the use of dedicated molds that are installed on compatible injection machines (leading to long setup times). Then, our Manufacturing Planning Problem determines the lot-size of each product in terms of the number of cycles for each piece–mold–machine combination and the scheduling of molds for each injection machine in a single planning period to maximize the total profit. We consider constraints for machines’ available time and avoid mold overlapping on different machines. Moreover, we define a mixed-integer linear program for our problem, and we obtain high-quality solutions for medium-size instances using a commercial solver in minutes. Still, the convergence to optimal solutions for large instances is slower. In response, we propose a two-stage sequential approach that takes advantage of the mathematical formulation to obtain better solutions for large instances in short computational times.

Lot-Sizing Decisions in Manufacturer-Retailer Inventory System under Carbon Emissions Reduction

This paper develops a mathematical model for supply chain inventory system of a manufacturer and a retailer under stochastic demand and carbon reduction policy. The objective of the model is to obtain the decisions regarding inventory management in supply chain system by minimizing the joint total cost. The model contributes to the current stochastic joint lot-sizing literature by allowing the investigation of the impact of carbon cap policy to inventory decisions. The carbon mainly emitted from several activities in the supply chain system, including the production, transportation and storage activities. The regulator, for example the government, uses a carbon cap policy to restrict the amount of emissions generated from the system. As we use strict carbon cap, the carbon emissions generated from the system must not exceed the carbon emission level set by regulator. A list of procedure is proposed to obtain the optimal solutions, which are ordering lot, safety factor, number of shipments and production rate. A numerical instance is provided to show the applicability of the model and to explore the impact of different levels of carbon cap on inventory decisions.

Optimal lot-size decision for deteriorating items with price-sensitive demand, linearly time-dependent holding cost under all-units discount environment

Due to the competitive market dynamics and globalisation, it gradually becomes challenging for the manufacturers or suppliers to entice any potential customer and turn into actual one. In this connection, to attract the retailers’ consideration, manufacturers or suppliers offer an order size dependent discount policy where the per unit purchase cost is measured as a descending step function of the amount of order quantity. A profit maximising inventory model is constructed for a deteriorating product where demand of the product is dependent on selling price, per unit carrying cost as linearly time varying and also proportional to the per unit purchase price under all-units discount environment. After examining the optimality of the decision variables a solution algorithm is suggested to achieve the optimal values of the selling price per unit and replenishment cycle length along with the optimal order size for optimising the net profit. Two computational experiments are executed along with observing the applicability of the proposed algorithm. Finally, a sensitivity analysis is performed for the model by changing the values of all parameters and hence a conclusion is made regarding the current study.

Optimal selling price, replenishment cycle and payment time among advance, cash, and credit payments from the seller's perspective.

Although research on pricing and lot-sizing decisions concerning payment types has been extensive, almost all of it has been done from the buyer’s perspective. In this study, we incorporate the following relevant and essential facts. If a seller allows a buyer to pay on credit, it increases sales volume. But if a seller asks a buyer for an advance payment, it decreases sales volume. Sometimes, a seller offers a buyer a price discount for an advance payment to increase sales and profitability. Asking a buyer for an advance payment bears interest earned and has no default risk. If a seller offers a buyer the opportunity to pay on credit, then a longer credit period may mean a higher the sales volume, but the default risk is higher, too. The seller wants to set an optimal selling price, replenishment time, and payment method simultaneously so that the profit per unit time is maximized. To achieve this, we develop and compare the seller’s profit per unit time under each of three payment methods—advance, cash and credit. Through numerical analyses, the following managerial insights are highlighted: (1) Under certain conditions, a specific payment type obtains the seller’s highest profit among all three payment types. (2) If advance payment is required, then it is critically important for a seller to offer a price discount. (3) An advance payment generates more profit than a credit payment provided that sales volume from a credit payment to an advance payment declines insignificantly and vice versa.

Pricing and lot-sizing decisions for perishable products when demand changes by freshness

<p style='text-indent:20px;'>Perishable products like dairy products, vegetables, fruits, pharmaceuticals, etc. lose their freshness over time and become completely obsolete after a certain period. Customers generally prefer the fresh products over aged ones, leading the perishable products to have a decreasing demand function with respect to their age. We analyze the inventory management and pricing decisions for these products, considering an age-and-price-dependent stochastic demand function. A stochastic dynamic programming model is developed in order to decide when and how much inventory to order and how to price these products considering their freshness over time. We prove the characteristics of the optimal solution of the developed model and extract managerial insights regarding the optimal inventory and pricing strategies. The numerical studies show that dynamic pricing can lead to significant savings over static pricing under certain parameter settings. In addition, longer replenishment cycles are seen under dynamic pricing compared to static pricing, even though similar quantities are ordered in each replenishment.</p>

The multi resource leveling and materials procurement problem: an integrated approach

PurposeThis paper proposes an integrated approach that seeks to jointly optimize project scheduling and material lot sizing decisions for time-constrained project scheduling problems.Design/methodology/approachA mixed integer linear programming model is devised, which utilizes the splitting of noncritical activities as a mean toward leveling the renewable resources. The developed model minimizes renewable resources leveling costs along with consumable resources related costs, and it is solved using IBM ILOG CPLEX optimization package. A hybrid metaheuristic procedure is also proposed to efficiently solve the model for larger projects with complex networks structure.FindingsThe results confirmed the significance of the integrated approach as both the project schedule and the material ordering policy turned out to be different once compared to the sequential approach under same parameter settings. Furthermore, the integrated approach resulted in substantial total costs reduction for low values of the acquiring and releasing costs of the renewable resources. Computational experiments conducted over 240 test instances of various sizes, and complexities illustrate the efficiency of the proposed metaheuristic approach as it yields solutions that are on average 1.14% away from the optimal ones.Practical implicationsThis work highlights the necessity of having project managers address project scheduling and materials lot sizing decisions concurrently, rather than sequentially, to better level resources and minimize materials related costs. Significant cost savings were generated through the developed model despite the use of a small-scale example which illustrates the great potential that the integrated approach has in real life projects. For real life projects with complex network topology, practitioners are advised to make use of the developed metaheuristic procedure due to its superior time efficiency as compared to exact solution methods.Originality/valueThe sequential approach, wherein a project schedule is established first followed by allocating the needed resources, is proven to yield a nonoptimized project schedule and materials ordering policy, leading to an increase in the project's total cost. The integrated approach proposed hereafter optimizes both decisions at once ensuring the timely completion of the project at the least possible cost. The proposed metaheuristic approach provides a viable alternative to exact solution methods especially for larger projects.

Energy Implications of Lot Sizing Decisions in Refrigerated Warehouses

Cold supply chains (CSCs) are responsible for preserving the quality of perishable goods in storage and transport. They consume significant amounts of energy to maintain cooling temperatures constant over time continuously and ubiquitously, which is affected by the surrounding environment and the users’ behavior. The storage filling level is one specific feature of refrigerated warehouses observed in practice: they are more energy efficient when kept full of items, reducing the space that air occupies. Inventory management models that consider energy consumption have received increasing attention recently due to an increase in stakeholders’ awareness of sustainability. Despite this interest, there is no work that jointly considers the effects of the filling level and the temperature inside the warehouse. This study, therefore, integrates those aspects into the economic order quantity model and finds the optimal lot size quantity that minimizes the total system cost, which is the performance measure used herein. It provides numerical results and brings some insights into the behavior of the model proposed.

Solving a flexible job shop lot sizing problem with shared operations using a self-adaptive COA

This paper deals with lot sizing decisions in a flexible job shop manufacturing system considering dependencies between lot sizing decisions. A novel mathematical model is developed to optimise lot sizing and scheduling simultaneously using a product-oriented approach as opposed to the job-oriented approach which has been prevalently considered in the literature. Moreover, the proposed model considers further underlying assumptions such as assembly operations, sequence-dependent setup times, initial inventory and safety stock levels, as well as lots with unequal and variable sizes. The objective is to minimise the total production, setup, and tardiness penalty costs of the system. To solve the formulated problem, a self-adaptive Cuckoo Optimisation Algorithm (COA) embedded with three new immigration mechanisms is developed. Numerical experiments are conducted to demonstrate the validity of the model and investigate the efficiency and effectiveness of the employed optimisation algorithm.

Joint optimization of dynamic pricing and lot-sizing decisions with nonlinear demands: Theoretical and computational analysis

This paper addresses a capacitated lot-sizing problem with pricing decisions. The considered problem consists of planning the production of different products during several time periods with setup costs. Unlike the classical version of the capacitated lot sizing problem, the demand for the products is not fixed but price-sensitive in this problem. The demand function is assumed to be nonlinear. The decisions consist of establishing the best strategy for production and inventory, and the best price policy. We propose an improved mathematical formulation by extending some previous works using new lower and upper bounds to reduce the solution space. We also introduce new heuristic methods to provide near-optimal solutions. These methods are tested on several instances from previous studies. The obtained results illustrate the efficiency of these methods.

Analisis Sistem Persediaan Bahan Baku Air Kelapa pada Industri Nata de Coco (Studi Kasus PT Keong Nusantara Abadi)

Inventory is an important thing for the industrial company, so determination of order quantity must be done optimally and reduce the inventory cost. PT Keong Nusantara Abadi is an industrial company for processing of nata de coco that requires an inventory control system so that the purchase of raw material becomes optimal. Therefore, this research was conducted to analyze inventory control carried out by PT Keong Nusantara Abadi and determine alternative methods of inventory control of raw material for the company. This research use Material Requirement Planning (MRP) system. MRP system has two ways to determine the order quantity called lot sizing decision. Techniques for determining the quantity of order is Economic Order Quantity (EOQ), Lot for Lot (LFL), and Part Periode Balancing (PPB). The technique of determining the order quantity that is best use by PT Keong Nusantara Abadi is EOQ which result in lower inventory cost than inventory control techniques by the company that is saved by 9%, 10%, 7,3% consecutively during 2015, 2016, and 2017.

Optimal Replenishment and Lot Sizing Strategy for Inventory Mechanism with Step-Shaped Demand and Backordering

This paper discusses the inventory mechanism with backordering and with the infinite planning horizon consisting of two stages wherein the demand rate in the first stage is strictly greater than that in the second stage. To minimize the retailer’s inventory cost, we establish a lot sizing decision model. On the basis of the inventory cost analysis, we present a closed-form solution to the model and provide an optimal replenishment and stocking strategy to the retailer. The given numerical experiments show the validity of the model.

Manufacturer's pricing and lot-sizing decisions for perishable goods under various payment terms by a discounted cash flow analysis

In practice, a cash payment is a classical payment term, a credit payment is commonly applied to stimulate sales, and an advance payment is used to avoid order cancellations. A combination of these three payment types is considered an advance-cash-credit (ACC) payment scheme, which is commonly used in business transactions. For instance, a home buyer must pay a homeowner 1% of the listing price as a good-faith deposit (i.e., an advance payment) to start negotiating the price, then pay 10% of the agreed-upon price (i.e., a cash payment) when signing the contract, and then has a delay before the final payment (i.e., a credit payment, a mortgage) is approved. In this paper, we develop an economic production quantity (EPQ) model for perishable goods in a three-echelon supplier-manufacturer-customer chain: (1) demand rate depends on selling price and freshness of a perishable item (i.e., time to sell-by date), and (2) the supplier offers an upstream ACC payment to the manufacturer, while the manufacturer grants a downstream cash-credit payment to customers. Consequently, the manufacturer must determine optimal selling price, production run time, and replenishment cycle time to maximize the present value of total annual profit by using a discounted cash flow (DCF) analysis. The proposed model fits in a general framework that includes numerous previous models as special cases. The numerical results reveal that the present value of total profit is joint concave in both selling price and cycle time. Finally, a sensitivity analysis is performed and managerial insights are also provided.

Common replenishment cycle order policies for multiple products with capacity expansion opportunities and quantity discounts

Many companies face the challenge of maintaining hundreds, if not thousands, of items in inventory, where each product draws from a common resource (e.g., warehouse space). The introduction of this common resource constraint complicates the traditional lot-sizing problem due to the interdependencies of the lot sizes. This challenge intensifies when the firm can adjust the capacity of the common resource and with the presence of quantity discounts, which may encourage the company to increase lot sizes. This paper develops efficient solutions for a constrained, multi-product lot-sizing problem with a common replenishment cycle, all-units quantity discounts, and a flexible common resource capacity. Historically, the common replenishment cycle (similar to the popular periodic review inventory system) excels in convenience but tends to lag behind in cost effectiveness. We extend the traditional common replenishment cycle approach through the introduction of a refinement policy that allows for more flexibility in the lot-sizing decisions. This modification allows companies to leverage quantity discounts for certain items without detrimental increases to the required common resource capacity (by increasing the cycle length). Via numerical examples and sensitivity analysis, we identify situations where the common replenishment cycle approach with refinement policy outperforms the alternative independent cycles approach in both convenience and cost. Good candidate companies include (1) a growing retailer, (2) a factory with limited storage space, (3) a retailer without secondary warehouse space, (4) a company purchasing bulky items, and (5) a firm located in an expensive city with premium storage space costs.

Modeling lot-size with time-dependent demand based on stochastic programming and case study of drug supply in Chile.

The objective of this paper is to propose a lot-sizing methodology for an inventory system that faces time-dependent random demands and that seeks to minimize total cost as a function of order, purchase, holding and shortage costs. A two-stage stochastic programming framework is derived to optimize lot-sizing decisions over a time horizon. To this end, we simulate a demand time-series by using a generalized autoregressive moving average structure. The modeling includes covariates of the demand, which are used as predictors of this. We describe an algorithm that summarizes the methodology and we discuss its computational framework. A case study with unpublished real-world data is presented to illustrate the potential of this methodology. We report that the accuracy of the demand variance estimator improves when a temporal structure is considered, instead of assuming time-independent demand. The methodology is useful in decisions related to inventory logistics management when the demand shows patterns of temporal dependence.