Lot-sizing Decisions Research Articles

MIP models for production lot sizing problems with distribution costs and cargo arrangement

This study presents mixed integer programming (MIP) models for production lot sizing problems with distribution costs using unit load devices such as pallets and containers. Problems that integrate production lot sizing decisions and loading of the products in vehicles (bins) are also modelled, in which constraints such as weight limits, volume restrictions or the value of the cargo loaded in the bins are considered. In general, these problems involve a trade-off between production, inventory and distribution costs. Lot sizing decisions should take into account production capacity and product demand constraints. Distribution decisions are related to the loading and transport of products in unit load devices. The MIP models are solved by the branch-and-cut method of an optimization package and the results show that these approaches have the potential to address different practical situations.

Modeling of product sales promotion and price discounting strategy using fuzzy logic in a retail organization

PurposeThe purpose of this paper is to capture the dynamic variations in sales of a product based upon the dynamic estimation of the time series data and propose a model that imitates the price discounting and promotion strategy for a product category in a retail organization.Design/methodology/approachTime series data relating to sales has been used to model the sales estimates using moving average and proportional and derivative control; thereafter a sales forecast is generated to estimate the sales of a particular product category. This provides valuable inputs for taking lot sizing decisions regarding procurement of the products and selection of suppliers. A hybrid model has been proposed and explained with a hypothetical case, which considerably impacts the sales promotion and intelligent pricing decisions.FindingsA conceptual framework is developed for modeling the dynamic price discounting strategy in retail using fuzzy logic. The model imitates sales promotion and price discounting strategy. This has helped minimize the inventory cost thereby keeping the profitability of the retail organization intact.Research limitations/implicationsThere is no appropriate empirical data to verify the models. In light of the research approach (modeling based upon historical time series data of a particular product category) that was undertaken, there is a possibility that the research results may be valid for the product category that was selected. Therefore, the researchers are advised to test the proposed propositions further for other product categories.Originality/valueThe study provides valuable insight on how to use the real-time sales data for designing a dynamic automated model for product sales promotion and price discounting strategy using fuzzy logic for a retail organization.

Joint optimization of manufacturing/remanufacturing lot sizes under imperfect information on returns quality

The increased quality volatility that characterizes returned used products in reverse supply chains, has led both academic research and industrial practice to support the establishment of, preferably fast and inexpensive, procedures for the quality assessment of returns. To this end, an initial classification of returns is typically performed based on some easily acquirable usage information, which is inherently - yet not always perfectly - related to their quality. Despite the criticality of the returns classification process and its impact on the reverse supply chain operation and profitability, little work has been done on the determination of the optimal threshold values, used to classify returns into different quality categories, taking into account the inevitable quality assessment inaccuracies. Motivated by multiple related reports from industrial applications, in the present paper the problem of optimizing the returns classification process under partial quality information in a hybrid manufacturing/remanufacturing facility is studied. More specifically, analytical models for the joint optimization of the new and returned products lot-sizing decisions are developed, based on usage information of returns which is only indirectly related to their remanufacturability. Our analysis provides useful insights on the value of information regarding the quality of returns under an indicator-based initial classification process. Numerical evidence on the economic superiority of the proposed models under alternative quality patterns of returns is also provided.

Shared resource capacity expansion decisions for multiple products with quantity discounts

When multiple products compete for the same storage space, their optimal individual lot sizes may need to be reduced to accommodate the storage needs of other products. This challenge is exacerbated with the presence of quantity discounts, which tend to entice larger lot sizes. Under such circumstances, firms may wish to consider storage capacity expansion as an option to take full advantage of quantity discounts. This paper aims to simultaneously determine the optimal storage capacity level along with individual lot sizes for multiple products being offered quantity discounts (either all-units discounts, incremental discounts, or a mixture of both). By utilizing Lagrangian techniques along with a piecewise-linear approximation for capacity cost, our algorithms can generate precise solutions regardless of the functional form of capacity cost (i.e., concave or convex). The algorithms can incorporate simultaneous lot-sizing decisions for thousands of products in a reasonable solution time. We utilize numerical examples and sensitivity analysis to understand the key factors that influence the capacity expansion decision and the performance of the algorithms. The primary characteristic that influences the capacity expansion decision is the size of the quantity discount offered, but variability in demand and capacity per unit influence the expansion decision as well. Furthermore, we discover that all-units quantity discounts are more likely to lead to capacity expansion compared to incremental quantity discounts. Our analysis illuminates the potential for significant savings available to companies willing to explore the option of increasing storage capacity to take advantage of quantity discount offerings for their purchased products.

A conceptual model for automation of product dynamic pricing and sales promotion for a retail organization

Purpose– The purpose of this paper is to capture the dynamic variations in sales of a product based upon the dynamic estimation of the time series data and propose a model that imitates the price discounting and promotion strategy for a product category in a retail organization. A modest attempt has been made in the study to capture the relationship between the sales promotion, price discount and the batch procurement strategy of a particular product category to maximize sales volume and profitability.Design/methodology/approach– Time series data relating to sales have been used to model the sales estimates using moving average and proportional and derivative control; thereafter a sales forecast is generated to estimate the sales of a particular product category. This provides valuable inputs for taking lot sizing decisions regarding procurement of the products that considerably impact the sales promotion and intelligent pricing decisions. A conceptual framework is developed for modeling the dynamic price discounting strategy in retail using fuzzy logic.Findings– The model captures the lag effect of sales promotion and price discounting strategy; other strategies have been formulated based upon the sales forecast that was done for taking the lot sizing decisions regarding procurement of products in the selected category. This has helped minimize the inventory cost thereby keeping the profitability of the retail organization intact.Research limitations/implications– There is no appropriate empirical data to verify the models. In light of the research approach (modeling based upon historical time series data of a particular product category) that was undertaken, there is a possibility that the research results may be valid for the product category that was selected. Therefore, the researchers are advised to test the proposed propositions further for other product categories.Originality/value– The study provides valuable insight on how to use the real-time sales data for designing a dynamic automated model for product sales promotion and price discounting strategy using fuzzy logic for a retail organization.

Optimal product pricing and lot sizing decisions for multiple products with nonlinear demands

Most manufacturing firms consider product pricing as a key strategic decision. Production planning and scheduling, on the other hand, are mostly treated as a non-strategic decisions. Most businesses, therefore, follow some sequential decision-making process; where product prices are determined first and then operational plans are made to fulfill the resulting demand at the lowest possible cost. In this paper, we present a coordinated decision model approach for multi-product pricing and lot sizing decisions for a manufacturer who has limited production capacity. Although the presented model is specific to demands that follow constant elasticity of the price, it can easily be extended to other convex demand functions. We show that a coordinated decision-making process where price and production plans are determined simultaneously may lead to substantially higher profits. We propose an efficient solution methodology for finding optimal prices and product quantities and use real-world data to demonstrate the applicability of this research.

A new bi-objective approach for including ergonomic principles into EOQ model

Worldwide, production systems are demanding new procedures and tools towards responsible inventory management, both in theory and in practice. In real industrial environment, manual fmaterial handling activities such as loading/unloading and stocking/picking operations are performed daily when a purchase/production material order is processed into industrial plants, and can be source of work-related musculoskeletal disorders. Only in the last four years, environmental and social aspects of the production systems have been progressively introduced in lot-sizing international theory, mostly in order to address the increasing request of ‘sustainability’ in modern society. Research in the lot-sizing area is fundamentally driven by the first basic model and by a large set of succeeding incremental steps, developed in order to better reflect real industrial problems and constraints. The social impact of economic order quantity policies has not been thoroughly investigated and is often discussed only via a descriptive approach. This work develops a new two-step approach capable of considering the social impact of lot-sizing procedures in terms of ‘ergonomics’ along with traditional annual logistic costs. Such approach divides the lot-sizing decisions in two main types: ‘In-house’, where the most influential decision variable is the size of the packaging units to move within the plant; ‘In-bound’, where the optimal number of ergonomic bins to purchase per order determines the minimum total cost. The outcome is what is defined as the ‘ergonomic lot-size’, which permits companies to reduce the ergonomic risks for their workers, while continuing to be profitable and efficient in assembly line part feeding.

Synchronization, cross-docking, and decoupling in supply chain networks

At any distribution centre (DC), the decision of whether to synchronise inbound and outbound flows for cross-docking, or to decouple these flows by maintaining inventory, has a significant impact on supply chain performance. Key drivers of this decision, in turn, are the sizes of the discrete lots that comprise the flows. Thus, we formulate an original optimisation model that determines order lot-sizing decisions to minimise, for given constant arc flows, the sum of ordering cost and pipeline inventory cost on arcs and buffer inventory at DCs. The model employs an average throughput as a surrogate to estimate buffer inventory at facilities at which synchronisation is not economical and therefore serves to decouple inbound and outbound flows. Perfect lot-for-lot matching of shipments would impose very restrictive constraints on supply chain operations, but equality of average throughput indicates an innovative, relaxed mode of synchronisation. This mode is practicable for cross-docking by means of bulk-breaking or consolidation of shipments. A heuristic approach based on the Lagrangian relaxation and subgradient optimisation is developed for the non-linear mixed-general integer optimisation model, which is illustrated by numerical examples and tested using a benchmark data set.

Multi-item production routing problem with backordering: a MILP approach

The aim of this paper is to present mixed integer linear programming formulations for the production routing problem with backordering (PRP-B) and a new hybrid heuristic to solve the problem. The PRP-B is considered in the context of a supply chain consisting of a production facility with limited production and storage capacities and geographically dispersed points of sale with limited storage capacities. The PRP-B integrates multiple item lot sizing decisions and vehicle routing decisions to the points of sale, where backordering of end customer demands is allowed at a penalty. Two integrated mixed integer programming models are formulated and a solution procedure consisting of a relax-and-fix heuristic combined with a local search algorithm is proposed. The numerical results show that this hybrid heuristic outperforms a state-of-the-art MIP commercial solver, in terms of solution quality and CPU times.

Reverse supply chains: Effects of collection network and returns classification on profitability

Used products collected for value recovery are characterized by higher uncertainty regarding their quality condition compared to raw materials used in forward supply chains. Because of the need for timely information regarding their quality, a common business practice is to establish procedures for the classification of used products (returns), which is not always error-free. The existence of a multitude of sites where used products can be collected, further increases the complexity of reverse supply chain design and management. In this paper we formulate the objective function for a reverse supply chain with multiple collection sites and the possibility of returns sorting, assuming general distributions of demand and returns quality in a single-period context. We derive conditions for the determination of the optimal acquisition and remanufacturing lot-sizing decisions under alternative locations of the unreliable classification/sorting operation. We provide closed-form expressions for the selection of the optimal sorting location in the special case of identical collection sites and guidelines for tackling the decision-making problem in the general case. Furthermore, we examine analytically the effect of the cost and accuracy of the classification procedure on the profitability of the alternative supply chain configurations. Our analysis, which is accompanied by a brief numerical investigation, offers insights regarding the impact of yield variability, number of collection sites, and location and characteristics of the returns classification operation both on the acquisition decisions and on the profitability of the reverse supply chain.

Differential evolution to solve the lot size problem in stochastic supply chain management systems

An advanced resource planning model is presented to support optimal lot size decisions for overall performance improvement of real-life supply chain management systems in terms of either total delivery time or total setup costs. Based on a queueing network, a model is developed for a mix of products, which follow a sequence of operations taking place at multiple interdependent supply chain members. At the same time, various sources of uncertainty, both in demand and process characteristics, are taken into account. In addition, the model includes the impact of parallel servers for multiple resources with period dependent time schedules. The corrupting influence of variabilities from rework and breakdown is also explicitly modeled. This integer non-linear problem is solved by standard differential evolution algorithms. They are able to find each product’s lot size that minimizes its total supply chain lead time. We show that this solution approach outperforms the steepest descent method, an approach commonly used in the search for optimal lot sizes. For problems of realistic size, we propose appropriate control parameters for an efficient differential evolutionary search process. Based on these results, we add a major conclusion on the debate concerning the convexity between lot size and lead time in a complex supply chain environment.

Ergo-Lot-Sizing: Considering Ergonomics in Lot-Sizing Decisions

One aspect that has mostly been overlooked in traditional economic lot-sizing is the implication of decisions on manual tasks and workload. Although lot-sizing decisions can have a significant impact on required manual material handling and on human performance, works that consider ergonomic aspects in lot-sizing are rare. This paper presents a model that integrates ergonomic aspect in terms of human energy expenditure and rest allowance in a traditional lot-sizing model. The behavior of the model is analyzed in a numerical study, showing its ability to reflect manual workload, and that ergonomic aspects should be considered in lot-sizing decisions.

Incorporating coordination costs into purchasing lot size decisions

This paper proposes an addition to the EOQ model in order to take the coordination costs between the supplier and the customer into account. The coordination costs are part of the transaction costs (TC) and vary significantly from one relationship to another. We have incorporated a 'TC factor' into the EOQ formula to take the coordination costs into account when calculating the purchase order size. We have also conducted a pilot study in one industrial company to test the new updated formula with real data. A key issue from a customer point of view is to determine the TC factor uniquely for each supplier.

Modelling and solving of a lot sizing problem for multiple items and multiple periods with shortage

This paper presents a multi–period inventory lot sizing scenario, where there is multiple product and single suppliers. It constructs a goal programming model to determine the order quantities of each product in each period. In this model, three goals are considered: 1) meeting the total available periodic budget, 2) meeting the buyer's maximum acceptable quality, 3) minimise the shortage. Constructing the approach in this way effectively reduces the risk of purchasing. The proposed approach provides flexibility to decision maker in multi–period procurement lot–sizing decisions. In other words, the purchase of any product in any period is determined as by the possibility to get closer to the goals defined in the model. It solves utilising GAMS and meta–heuristic algorithms, and analyses the efficiency of the heuristics. Efficiency of the proposed algorithm by comparing genetic algorithm in large size problems show the proposed algorithm has also better efficiency than genetic algorithm.

Shapley-based side payments and simulated annealing for distributed lot-sizing

The coordination of lot-sizing decisions in an inter-organizational supply chain allows the involved companies to find a global production plan with lower total costs compared to individual and non-coordinated plans. Usually, the involved companies (referred to as agents in the following) are independent, self-interested and have private information. This contribution studies a distributed lot-sizing problem where each item may be produced by at least two agents. A collaborative planning approach based on simulated annealing is used to coordinate the individual plans of the agents. In collaborative planning mechanisms a general design challenge is the acceptance of a new incumbent solution during the negotiation process because it requires the elicitation and aggregation of the agents’ preferences. To deal with this challenge, we integrate the calculation of side payments based on the Shapley value into the negotiation. The computation of the Shapley-based side payments is demanding due to the NP-hardness of the characteristic function, which is represented in terms of the lot-sizing model at hand. Three different ways of computing Shapley-based side payments are presented. Our computational study shows significant differences with respect to the runtime and to a lesser extend with respect to the quality of the found solutions. Most importantly, the approaches using side payments outperform an approach without side payments in 225 out of 272 cases.

A memetic algorithm for a stochastic lot-sizing and sequencing problem

A stochastic multi-product lot-sizing and sequencing problem is considered. Two kinds of uncertainties are integrated into the model: defectives items due to the machines' imperfections and random lead time because of randomly arising breakdowns and uncertain repair time. There are also sequence-dependent set-up times between two items of different types. The optimization problem is to maximize the probability of overall demand satisfying. In the previous work only the lot-sizing part of the problem was considered (a decomposition approach was used). Here we study the entire problem with sequencing and lot-sizing decisions integrated.

Haulage sharing approach to achieve sustainability in material purchasing: New method and numerical applications

Transportation costs are becoming increasingly important in inventory replenishment decisions and, in practice, lot sizing decisions are strongly affected by material handling equipment, transportation flow paths, vehicle capacities and technical constraints. Companies within a global sourcing context daily experience the cost of transportation as playing a major role in total purchasing costs. On the other hand, sustainability issues are always more urgent and environmental impact assessment is becoming a key requirement for materials purchasing and transportation choices. This work is particularly focusing on an interesting new area of research: the haulage-sharing in freight transportation when sustainability considerations are taken into account. The paper develops and discusses a new haulage-sharing lot sizing model in which two partners are cooperating in sharing transportation paths and handling units. A three step methodology based on multi-objective optimization approach is here proposed to permit the complete evaluation of the costs and savings that arise with an horizontal cooperation. The method is described here and applied to two different numerical cases in order to drive useful conclusions and discuss future research steps.

Integrated pricing and lot-sizing decision in a two-echelon supply chain with a finite production rate

In this paper, we consider a supply chain consisting of a supplier and a retailer selling the product manufactured by the supplier in a market, in which the demand for the product is decreasing in the price set by the retailer. Previous research on coordinating the supply chain with price-sensitive demand often assumes that either the supplier purchases the product or the supplier has an infinitely large production rate. We consider explicitly the supplier’s finite production rate in the pricing and lot-sizing decision, which causes the problem much more challenging. We first investigate the optimization problem in the decentralized scenario to find the optimal order quantity and selling price for the retailer as well as the optimal wholesale price and lot size multiplier for the supplier. We then solve the joint optimization problem in the centralized supply chain to find the optimal order quantity, the selling price, and the optimal lot size multiplier. We provide two sequential algorithms to solve the joint optimization problem. Computational experiments show that the algorithms are effective (all 448 tested problems are solved optimally) and efficient (it only takes a few iterations to converge for each tested problem). The supply chain coordination mechanism is then designed through an all-unit quantity discount policy and a franchise fee.

Economic selection of process mean and lot-sizing decisions for multi-newsvendor supply chain

Two related problems are integrated in this paper, the first is the targeting problem and another is production/inventory decisions in a supply chain. The supply chain under consideration consists of a supplier of raw material, a single producer, and multiple newsvendors. The producer can adjust the process mean before starting the production run. Once set to a certain target value, the process mean is not changed until the production lot is completed. At the end of a production run, the producer screens the lot and uses specification limits to evaluate the quality of the item. Nonconforming items are scrapped with no salvage value; however, conforming items are delivered to newsvendors who are subjected to random demand. If demand on a newsvendor in a season is lower than anticipated, surplus items will be returned to the producer at a certain transportation cost. We first develop a mathematical model that maximizes the expected total profit of the supply chain. Then, a table for two special functions is devised to simplify the solution method and is used to find the optimal solution of the proposed model. We also study the significance of this integration by comparing the performance of the proposed model with an independent model where the process mean selection and lot-sizing decisions are found separately.

Lot sizing and lead time decisions in production/inventory systems

Traditionally, lot sizing decisions in inventory management trade-off the cost of placing orders against the cost of holding inventory. However, when these lot sizes are to be produced in a finite capacity production/inventory system, the lot size has an important impact on the lead times, which in turn determine inventory levels (and costs). In this paper we study the lot sizing decision in a production/inventory setting, where lead times are determined by a queueing model that is linked endogenously to the orders placed by the inventory model. Assuming a continuous review (s, S) inventory policy, we develop a procedure to obtain the distribution of lead times and the distribution of inventory levels, when lead times are endogenously determined by the inventory model. This procedure allows to determine the optimal inventory parameters within the class of (s, S) policies that minimize the expected ordering and inventory related costs over time. We numerically show that ignoring the endogeneity of lead times may lead to inappropriate lot sizing decisions and significantly higher costs. This cost discrepancy is very outspoken if the lot size based on the economic order quantity deviates significantly from desirable production lot sizes. In these cases, the endogenous treatment of lead times is of particular importance.