The reorder point formula in the Economic Order Quantity (EOQ) model traditionally assumes that demand during the lead-time follows a normal distribution. However, this assumption presents several challenges. First, alternative probability distributions may better capture demand patterns for specific products and markets. Second, historical data may not always be available to predict these distributions; in such cases, fuzzy set theory can be used to estimate demand based on expert opinions and judgments. Third, the conventional reorder point formula overlooks important factors, such as unit wholesale price and unit shortage costs. For instance, when unit shortage or goodwill costs are high, increasing the reorder point can help minimize the risk of stockouts. To address these issues, we reformulate the inventory problem during the lead-time as a newsvendor problem and derive closed-form solutions for the optimal reorder point. In this model, demand during the lead-time is represented using both fuzzy numbers (to capture possibility) and probability distributions, allowing us to incorporate factors like unit wholesale price and shortage or goodwill costs. Additionally, we provide managerial insights through numerical analysis, helping to guide decisions on reorder point adjustments. • Develop a closed-form solution for reorder point optimization under uncertainty. • Integrate probabilistic and fuzzy methods for demand estimation. • Enhance decision-making by incorporating cost factors in inventory models. • Provide practical insights through numerical analysis and case studies. • Offer a flexible approach that is adaptable to various industries and markets.

The fractional calculus significantly contributes to accounting and inventory control for the memory effect. This study introduces an inventory model with a price-influenced demand pattern and a complete backlog shortage, including the memory effect under a fuzzy environment. The proposed lot-sizing model uses the fractional-order derivative to include the memory effect. The primary objectives of this paper are summarized as follows: (i) the impact of the fractional rate of change in inventory level on the fractional inventory system is established, and (ii) price-dependent demand is considered as a triangular fuzzy number, and the total average cost is calculated with the help of two different defuzzification methods, such as signed distance (SD) method as well as the graded mean integration (GMI) method. The impact of the fractional-order rate of change in the proposed inventory system is verified through numerical examples and graphical displays. The unit shortage cost is sensitive to the strong and poor memory effects of SD and GMI methods. The results indicate that, in the SD method and the memory-less case, businesses run a long time to obtain the minimum total average cost, whereas, in the GMI method and the memory-less case, businesses run comparatively significantly little time to obtain the minimum total cost. Finally, some suggestions are given for making another type of fractional model.

The paper focuses on developing an inventory model for deteriorated item when demand is time- and inventorydependent. Deteriorated items can usually be found in items such as vegetables, fruit, milk, chemical product, pharmaceutical and it needs special attention in managing this kind of inventory. We model the inventory control for these items by a mathematical model involving time- and inventory-dependent demand and considering the backorder policy in handling shortages. The developed model aims to find the optimal time between replenishment and when inventory drops to zero, which minimizes the total inventory cost. The total inventory cost consists of the purchase cost, the order cost, the holding cost, and the shortage cost. Sensitivity analysis is performed to analyse the effect of changing the parameters’ values to the time between replenishment, when inventory drops to zero, the order quantity, and the total inventory cost. The finding shows that changing the parameters’ values of deterioration rate, demand, unit holding cost, unit holding cost and unit shortage cost will have an impact on the time between replenishment, time when inventory drops to zero, order quantity, and inventory cost.

Efficient inventory management, including optimal safety-stock levels, is crucial for operational continuity and cost-effectiveness in various industries. This study seeks the optimal inventory management strategy to minimize costs and determine ideal safety-stock levels. It compares five approaches: the company’s (STAR) current “number of days” method, two alternative models from the literature (the theory of constraints (TOC) replenishment model and the service-level approach), and two newly developed hybrid methodologies (the TOC replenishment model with ABC–XYZ classification and the service-level approach with ABC–XYZ classification). The analysis focused on financial performance, considering inventory holding and shortage costs. Monthly production plans were established and fixed as constant based on predetermined optimum month-end inventory levels derived from each method. Through simulation, actual month-end inventory levels were assessed, comparing total inventory costs (TICs). While unit holding costs (UHCs) were documented in financial records in the company, unit shortage costs (USCs) were not; thus, USCs were examined in three scenarios. The results show that the second proposed hybrid model consistently outperformed the other four methods, including the company’s current approach, significantly reducing TIC. The analysis emphasizes the importance of demand variation in setting safety stocks and demonstrates the second hybrid methodology’s effectiveness in optimizing safety-stock strategies and improving overall inventory management efficiency.

A location-production-routing problem for distributed manufacturing platforms: A neural genetic algorithm solution methodology

This paper derives the optimal solution for a distributionally robust multi-product newsvendor problem, in which different products are produced under a capacity constraint, while only the mean and variance of the product demand are known. The problem aims to find a capacity allocation scheme to minimize the system cost of the worst-case among all possible demand distributions. When the total capacity is among certain ranges, the optimal solution has a closed-form. For other ranges, the optimal solution can be derived by solving one equation. The solution shows that there exist a number of threshold values for the capacity, below which some products are neglected (allocated with zero capacity). Besides, in the optimal solution, which products are prioritized for production is determined by the order of an index measured by the unit production cost, unit shortage cost, demand mean, and demand variance. For a special case where the cost structures for different products are identical, a closed-form solution is derived for any total capacity value. For this special case, the index order is simply determined by the coefficient of variation of each product demand. Sensitivity analysis shows that when the capacity is abundant, a larger demand variability of one product may cause a higher production quantity for this product; while if the capacity is tight, a larger demand variability cause a lower production quantity. For a set of test problems, the performance of the robust optimization solution is quite close to that of the stochastic optimization solution.

Emergency decision-making model of suppliers with updating information in cases of sudden accidents

The inventory control of turnover parts is the key and difficult point for aircraft spare parts management, which is directly related to the benefit level of airlines in aviation industry.On the basis of considering the depreciation cost, repair cost, inventory cost and shortage cost, this paper analyzes the annual replacement times and turnover cycle, and puts forward the optimization model of Economic Order Quantity (EOQ) of aircraft turnover parts. The feasibility of the method is verified by an example. Through sensitivity analysis, it is concluded that the strong sensitive factors affecting EOQ are annual replacement times and unit shortage cost, while the purchase price has little influence on EOQ.

Water retailer managed inventory is a classical and inevitable inventory management mode in present economic society. Stochastic models can more clearly explain demand uncertainty and are closely related to water supply chains. Risk preferences are widely valued in behavioral operation management. Related to the risk preferences in inventory management, the research on risk aversion is dominant, while risk-seeking is insufficient. Based on the model assumptions, the risk-seeking retailer’s optimal decision-making inventory model with stochastic demand in a water supply chain is studied. The risk-seeking retailer’s optimal inventory quantity, optimal inventory cost, supplier profit, retailer profit, and the profit of the entire water supply chain are derived. The validity of the equations is proved. The sensitivity analysis of the risk-seeking retailer’s optimal inventory decision-making is carried out. The risk level effects on the five dimensions, the retail price, wholesale price, unit shortage cost, unit inventory cost, and unit residual value, are displayed through numerical simulation. The optimal inventory quantity and optimal inventory cost of the risk-seeking retailer are obtained.

This article proposes a new approach to sales forecasting for new products (stock-keeping units [SKUs]) with long lead time but short product life cycle. These SKUs are usually sold for one season only, without any replenishments. An exponential factorization machine (EFM) sales forecast model is developed to solve this problem which not only takes into account SKU attributes, but also pairwise interactions. The EFM model is significantly different from the original Factorization Machines (FM) from two fold: (1) the attribute-level formulation for explanatory/input variables; and (2) exponential formulation for the positive response/output/target variable. The attribute-level formation excludes infeasible intra-attribute interactions and results in more efficient feature engineering comparing with the conventional one-hot encoding, while the exponential formulation is demonstrated more effective than the log-transformation for the positive but not skewed distributed responses. In order to estimate the parameters, percentage error squares (PES) and error squares (ES) are minimized by a proposed adaptive batch gradient descent method over the training set. To overcome the over-fitting problem, a greedy forward stepwise feature selection method is proposed to select the most useful attributes and interactions. Real-world data provided by a footwear retailer in Singapore are used for testing the proposed approach. The forecasting performance in terms of both mean absolute percentage error (MAPE) and mean absolute error (MAE) compares favorably with not only off-the-shelf models but also results reported by extant sales and demand forecasting studies. The effectiveness of the proposed approach is also demonstrated by two external public datasets. Moreover, we prove the theoretical relationships between PES and ES minimization, and present an important property of the PES minimization for regression models; that it trains models to underestimate data. This property fits the situation of sales forecasting where unit-holding cost is much greater than the unit-shortage cost (e.g., perishable products).

PurposeThe purpose of this paper is to develop a decision support system to consider geographic information, logistics information and greenhouse gas (GHG) emission information to solve the proposed green inventory routing problem (GIRP) for a specific Taiwan publishing logistics firm.Design/methodology/approachA GIRP mathematical model is first constructed to help this specific publishing logistics firm to approximate to the optimal distribution system design. Next, two modified Heuristic-Tabu combination methods that combine savings approach, 2-opt and 1-1 λ-interchange heuristic approach with two modified Tabu search methods are developed to determine the optimum solution.FindingsSeveral examples are given to illustrate the optimum total inventory routing cost, the optimum delivery routes, the economic order quantities, the optimum service levels, the reorder points, the optimum common review interval and the optimum maximum inventory levels of all convenience stores in these designed routes. Sensitivity analyses are conducted based on the parameters including truck loading capacity, inventory carrying cost percentages, unit shortage costs, unit ordering costs and unit transport costs to support optimal distribution system design regarding the total inventory routing cost and GHG emission level.Originality/valueThe most important finding is that GIRP model with reordering point inventory control policy should be applied for the first replenishment and delivery run and GIRP model with periodic review inventory control policy should be conducted for the remaining replenishment and delivery runs based on overall simulation results. The other very important finding concerning the global warming issue can help decision makers of GIRP distribution system to select the appropriate type of truck to deliver products to all retail stores located in the planned optimal delivery routes depending on GHG emission consumptions.

ABSTRACTStochastic inventory system simulation is often the tool of choice by industry practitioners who struggle with the evaluation of the quality of proposed inventory targets using service levels. However, driving simulations with unknown input distribution parameters has its own challenges. In this paper, we focus on the newsvendor problem and quantify the amount of demand parameter uncertainty – the uncertainty around the unknown demand distribution parameters which are estimated from the limited historical demand data – in the confidence interval of the mean service level. We use this quantification to understand how the variance of the mean service level, due to the amount of the demand parameter uncertainty in the simulation output process, changes with the choice of Type-1 and Type-2 service-level criteria, the historical data length, the ratio of the unit shortage cost to the unit holding cost, and the distributional shape of the demand’s density function.

This paper attempts to create a supply chain inventory model for perishable goods that simultaneously considers delayed payment and shortage. For this purpose, a two-layer supply chain system was described, and a perishable goods inventory model was created for two types of grace periods. Then, the optimal distribution cycle and quantity were computed in the mode of single ordering and multiple distributions. Finally, the author investigated the effect of unit shortage cost, deterioration rate and grace period on the total cost of the supply chain. The model was validated through a case study. The research findings provide valuable references for inventory management of supply chain enterprises.

Petrol is a kind of strategic natural resources. Provide legitimate transportation plans for the petrol secondary distribution are the key links to guarantee the petrol provision. If the total supply is insufficient, some petrol stations can’t avoid shortage because their demands could not be met. So the shortage cost will appear. This paper studies the problem of how to arrange the transportation plan in order to minimize the total cost when the total volume of supply is insufficient. Given the storage volume, the sales rate and the unit shortage cost of every petrol station, considering the full loading constraints of the compartment vehicle, a mixed integer programming model for minimizing the total cost of petrol secondary distribution is established. A Lingo program is compiled for solving the model. Finally, simulation on an example has been done and a reasonable transportation plan is obtained. The model and algorithm in this paper can provide a theoretical basis for dispatching department to make transportation plan.

Decision-makings in safety investment: An opportunity cost perspective

Estimating the optimal order quantity and the maximum expected profit for single-period inventory decisions

We deal with the problem of a profit-maximizing vendor selling a perishable product. At the beginning of a planning cycle, the vendor determines a minimum committed order per period. During the cycle, he may also place a supplemental order in each period based on the observed demand signal in that period. Moreover, the vendor is committed to a specific service target evaluated over the planning cycle. This is a complex problem, and we, as an approximation, offer a single-period, two-stage modeling approach. Under this approach, the vendor determines a first-stage order as the minimum committed order with the possibility of supplementing it based on a demand signal observed at the second stage. The problem is to maximize his expected profit subject to a constraint on his overall service performance across all possible values of the demand signal. We characterize the optimal policy for in-stock rate and fill-rate targets, and make comparisons. Whereas in the classical newsvendor model a service target can be replaced by a single unit shortage cost, it is not so in our model. Instead, a set of unit shortage costs are imputed—one for each demand signal. The imputed shortage costs reflect trade-offs among the profits under different demand signals in meeting the service targets. We also show that under a given ordering policy, the in-stock rate is lower (higher) than the fill rate when the demand has an increasing (decreasing) hazard rate. This result suggests that the vendor can infer a fill-rate measure from the corresponding in-stock rate without the difficult task of tracking lost sales. Furthermore, we analyze how the order quantity varies according to the observed signal, which allows us to formalize the notion of a valuable demand signal.

This study addresses the production planning problem for perishable products, in which the cost and shortage of products are minimised subject to a set of constraints such as warehouse space, labour working time and machine time. Using the concept of postponement, the production process for perishable products is differentiated into two phases to better utilise the resources. A two-stage stochastic programming with recourse model is developed to determine the production loading plan with uncertain demand and parameters. A set of data from a toy company shows the benefits of the postponement strategy: these include lower total cost and higher utilisation of resources. The impact of unit shortage cost under different probability distribution of economic scenarios on the total cost is analyzed. Comparative analysis of solutions with and without postponement strategies is also performed.

Production planning problems play a vital role in the supply chain management area, by which decision makers can determine the production loading plan—consisting of the quantity of production and the workforce level at each production plant—to fulfil market demand. This paper addresses the production planning problem with additional constraints, such as production plant preference selection. To deal with the uncertain demand data, a stochastic programming approach is proposed to determine optimal medium-term production loading plans under an uncertain environment. A set of data from a multinational lingerie company in Hong Kong is used to demonstrate the robustness and effectiveness of the proposed model. An analysis of the probability distribution of economic demand assumptions is performed. The impact of unit shortage costs on the total cost is also analysed.

The problem of inventory control of spare parts in a hierarchical two-level system in uncertain environment is considered. The hierarchical two-level system consists of one central depot and N branch warehouses. A demand of each kind of treated spare parts is uncertain and modeled by fuzzy numbers. A new procedure of classification according to the fuzzy ABC method is developed. The control models are determined for spare parts of great importance. A new fuzzy mathematical model for determining the optimum order quantities of the central depot and branch warehouses is presented. The values of unit procurement price, unit holding costs and unit shortage costs are deterministic. The developed procedure for the classification and the algorithm for determining the optimum order quantities is illustrated by an example.