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In this paper, a sustainable production inventory model for two decision variables, optimal pricing and production lot size with advertisement and price-dependent demand, is developed. This study develops sustainable production strategy for defective items produced that (1) A small proportion of manufactured goods have defectives; however, quality may be improved by investing in quality improvement cost, (2) Due to manufacturing processes and the stock of products, carbon emissions may be reduced by carbon reduction cost, (3) One of the best methods to increase the popularity of a product throughout all client categories, by using advertisement and price dependent demand. (4) The important role of the manufacturing concern is to fix the price of the product, it may be fixed by using the Price break-even point. (5) In a competitive market, the best price and production lot size are chosen using Bertrand's competitive strategy. The objective of this work to find optimal pricing and lot size for achieving maximum profit. The price-break-even point is determined, the law of demand is verified, and the maximized profit is identified A Mathematical model is suggested to assess the optimal values for two decision variables and the analyzation of sensitivity analysis is performed using a numerical experiment. [ABSTRACT FROM AUTHOR]

In today's competitive market landscape, ensuring product quality is paramount for businesses to maintain customer satisfaction and loyalty. Defective quality items not only lead to financial losses but also tarnish brand reputation. Traditional methods of quality control often fall short in detecting and addressing defects efficiently. This study proposes a novel approach utilizing Artificial Intelligence (AI) for market analysis of defective quality items. The study focuses on leveraging AI techniques, such as machine learning algorithms and computer vision, to analyze patterns and identify defects in products. By harnessing AI, businesses can streamline the quality control process, detect defects with greater accuracy, and ultimately reduce the incidence of defective items reaching consumers. Furthermore, the study explores the implications of defective quality items on the market, including customer perception, brand image, and financial impact. Through comprehensive market analysis, businesses can gain insights into the root causes of defects, identify areas for improvement in the production process, and make data-driven decisions to enhance overall product quality. The proposed AI-driven market analysis framework offers a proactive approach to quality control, enabling businesses to preemptively address defects before they escalate into larger issues. By integrating AI technologies into their quality control strategies, businesses can not only minimize financial losses associated with defective items but also enhance customer satisfaction and loyalty, thus gaining a competitive edge in the market. [ABSTRACT FROM AUTHOR]

This research proposes an innovative modification to the Economic Production Quantity (EPQ) model that integrates Weibull-distributed gradual deterioration, holding costs, variable demands, and effective strategies for managing shortages and backlogs. The extended EPQ model is designed to optimize production quantity and total variable cost for items with slow degradation, particularly perishable goods. The model is based on modeling inventory with three different demand patterns and utilizing analytical methods for solving ordinary differential equations by incorporating Maclurin series approximation techniques. The proposed model addresses deterministic time-varying demand and defective items during the production processes. The research contributes valuable insights into inventory management and production planning, it has significant implications for decision-making in specifically perishable goods production industries. Two examples are given to validate the model's applicability, finally sensitivity analysis and the scope of possible future research are suggested.

Smart production plays a significant role to maintain good business terms among supply chain players in different situations. Adjustment in production uptime is possible because of the smart production system. The management may need to reduce production uptime to deliver products ontime. But, a decrement in production uptime reduces the projected production quantity. Then, the management uses a limited investment for pursuing possible alternatives to maintain production schedules and the quality of products. This present study develops a mathematical model for a smart production system with partial outsourcing and reworking. The market demand for the product is price dependent. The study aims to maximize the total profit of the production system. Even in a smart production system, defective production rate may be less but unavoidable. Those defective products are repairable. The model is solved by classical optimization. Results show that the application of a variable production rate of the smart production for variable market demand has a higher profit than a traditional production (52.65%) and constant demand (12.45%).

Success in the industrial sector is compromised by diverse conditions such as imperfect product production, manufacturing line interruptions, and unscheduled maintenance. The precise use of common practices in production environments is an available solution to eliminate some of these issues. Applying a warm-up period in a manufacturing process is adequate and cost-effective for almost all companies. It improves the equipment's productivity and helps the manufacturing line generate fewer defective products. Even though several inventory management studies have included a warm-up phase in their models, its use in economic production quantity (EPQ) models remains largely unexplored. Adding a warm-up phase to the production cycle minimizes maintenance expenses and defective products and increases the machine's performance. In this study, the dependency between the machine downtime and the warm-up length is examined for the first time. The warm-up time depends on the machine's off-state period: if the machine has a longer operation timeout, then a longer warm-up period is needed. The model includes a function to model the warm-up time relative to the machine downtime and two types of defective products: scrapping and reworking items. The study is concluded with some numerical examples, a sensitivity analysis, and some management insights related to the EPQ. [ABSTRACT FROM AUTHOR]

The intention of the article is to present an imperfect production process inventory system for the decaying items, where the demand rate is depending on advertising cost and price. In this research, an industrial manager produces products in a determined (m) imperfect replenishment cycles and also in each replenishment cycle, the manufacturer manufactures items in three imperfect production cycles and one rework setup, (M/P3/R1) inventory system. Rework is one of the important issues in reverse logistics and green supply chain, since it reduces waste, environmental harms and also the overall inventory cost significantly. Here shortages are considered under complete backlogging strategy. The major objective is to establish the optimal advertising cost, optimal cycle time, optimal lot size quantity and optimum selling price with the intend of minimizing the total inventory cost. A numerical example and a sensitivity analysis are recognized to authenticate the theoretical results, and managerial insights for business managers are provided. [ABSTRACT FROM AUTHOR]

In this paper, the seller offers a credit period to his buyer for more sales and the buyer accepts the seller's policy to gain more profit, and it is assumed that the seller has defective and non-defective items. When the seller provides lots for sale to his buyer then, the buyer separates the whole lots with the help of inspection process into defective and perfect quality items. Further, in this scenario, the percentage of defective items present in the lot follows the S-shape learning curve and it is also considered that the demand rate is imprecise in nature. Here, the demand rate assumes a triangular fuzzy number due to the imprecise nature and it is the model assumption. Based on this assumption, we developed an inventory model with the effect of learning and trade credit strategy under a fuzzy environment for the buyer. The buyer's total profit has been optimized concerning the order quantity in the fuzzy environment where order quantity has been assumed as a decision variable. The results of this model were verified with the help of numerical examples and sensitivity analysis. We compared the buyer's total profit in a crisp and fuzzy environment and the buyer gained more profit in a fuzzy environment compared to the crisp environment. Moreover, we compared the results with and without the effect of learning and trade credit on the buyer's ordering policy and obtained a positive effect on the ordering policy in the numerical section. We determined positive results from the sensitivity analysis, which proved that the trade credit policy will be beneficial for both partners of the supply chain. [ABSTRACT FROM AUTHOR]

Inventory management seeks to improve manufacturing by contracting inventory costs in a similar fashion to raise efficiency and profit. One approach is to develop inventory management models according to actual production systems. Furthermore, governmental policies in many countries impose many regulations on firms to fulfill the growing demand for a reduction in carbon emissions. Warm-up is a familiar concept in industrial applications. It allows the manufacturing system to work at a higher level of productivity and efficiency, as well as decreasing the number of defective items and maintenance costs. Along with fewer poor-quality items, the system has less waste as scrap items entering the environment and also requires less energy and workload to focus on reworking. The economic production quantity (EPQ) problems with a warm-up as an input parameter have been studied in a few works recently. This paper proposes a production-inventory model which considers the warm-up period as a decision variable and investigates its impact on the total cost. Furthermore, the defective rate is a decreasing linear function related to the warm-up period's length. The production-inventory model takes into account the carbon emission tax policy. The main aim of this research is to jointly optimize both the length of the warm-up period and the production cycle in order to minimize the total cost of the production-inventory system and, therefore, reduce emitted carbon emissions. The comparison of tax prices and the effect of the proper warm-up period on the amount of carbon emissions are discussed in the sensitivity analysis. [ABSTRACT FROM AUTHOR]

Global warming and climate change become a sensitive issue worldwide, and many countries try to control the CO2 emissions by investigating in many projects. This study proposes a sustainable inventory model to reduce carbon emission. It is generally assumed that demand is increasing at starting of an inventory level and decreasing at ending for a particular newly launched product. In this situation, demand is usually represented by Normal distribution or imprecise fuzzy demand. However, in reality, those assumptions do not always hold. In most of the inventory models, it is seen that demand is a linearly increasing function of time before starting deterioration; then, the demand decreases inversely proportional to the deterioration rate after starting the deterioration. To describe the possible situations more clearly, we introduce here a new type of demand, called as composite demand. Most of the products have a fixed life span for maintaining the quality in original condition which is known as non-instantaneous deterioration. Therefore, we design an inventory model for non-instantaneous deteriorating items with composite demand function. The deterioration rate is assumed to follow an exponential distribution; the holding cost is considered as time dependent. The model is formulated based on retailer's perspective and, thereafter, the total cost for the retailer is calculated. Numerical examples are provided for better understanding of this model. The results show that the total profit for non-instantaneous deteriorating items is higher than for instantaneously deteriorating items. A sensitivity analysis is conducted based on the important parameters. Finally, the paper ends with conclusions and an outlook to future research. [ABSTRACT FROM AUTHOR]

Economic production quantity (EPQ) models are traditionally used in operations management. Despite the large number of papers that describe the models, the classic EPQ model does not consider either imperfect quality batches or shortages. However, some industries may be able to sell imperfect items for a lower price, reducing the total production cost. This paper proposes an EPQ model with partial backordering and discount for imperfect quality batches and an algorithm that returns optimal values for the problem. From a numerical example, it is possible to analyse how the changes in the variables affect each part of the total cost function, which provides a useful tool for strategic decision-making. We conclude that it is better to sell imperfect items as soon as possible because the savings in holding costs results in a total cost reduction. It is more profitable for the producer to have planned shortages considering that some costumers are willing to wait. Furthermore, the reduction of the goodwill cost does not necessarily reduce the total cost. [ABSTRACT FROM AUTHOR]

Taleizadeh, Noori-daryan, and Tavakkoli-Moghaddam consider pricing and ordering decisions in a three-level supply chain with defective items and inspection processes in their paper published in 2015. They examine three scenarios for defective items at the manufacturer: disposal, rework and selling at a lower price. Unfortunately, this interesting and innovative paper suffers from some shortcomings that are addressed in our note. [ABSTRACT FROM AUTHOR]

This paper integrates maintenance and product quality for a manufacturing system with three failure modes. The system is monitored and can be alerted by a prediction signal before any failure mode. Due to the high setup cost of maintenance activities, their clustering can be beneficial. To reduce the frequency of failures and their impacts on the quality of output products, a clustering model with perfect and imperfect maintenance actions is applied. The objective is to investigate the impacts of clustering maintenance actions on non conforming production costs and to compare cost saving in perfect and imperfect maintenances. Our analysis and results yield valuable insights into when and how to cluster maintenance operations so that the cost for maintenance and non conforming units be minimized. Clustering perfect maintenance can be more optimal than the imperfect one for several reasons, e.g., because perfect maintenance actions restore the system completely to the as-good-as new state and the time between two maintenance is larger, and thus the frequency of maintenance decreases. [ABSTRACT FROM AUTHOR]

This paper develops a three-layer supply chain for defective and non-defective types of produced items by supplier and manufacturer. The condition set by the chain for the suppliers and manufacturers is that after completion of screening, defective items are sold to the supplier to produce other items using the mentioned items. In the subsequent stage, the retailer accepts the non-defective items produced and screened by the manufacturer. Hence, although the retailers receive items of perfect quality for selling to the customers, they consider the effect of the deterioration of items. This model also considers the impact of several business strategies such as optimal order size of raw materials, production rate, unit production cost, idle time costs of the supplier, manufacturers in a collaborative marketing system, etc. to determine the optimum average profit of the integrated model. This study discusses the selling price of the retailer, demand rate of the customer, purchase cost of supplier, and holding cost, which can be a significant breakthrough in expanding the profit of the business in real terms. Numerical examples and sensitivity analysis are presented to illustrate the phenomenon of theoretical study and demonstrate the managerial implication of the model. [ABSTRACT FROM AUTHOR]

This paper develops an integrated fuzzy two-layer supply chain for manufacturers and retailers with defective and non-defective types of products. The manufacturer produces up to a specific time, including both faulty and non-defective items, and post-screening only the non-defective item sends to the retailer. The strategy of the retailer is to do the screening of items received from the manufacturer. Subsequently, the perfect quality items use to fulfill the demand of the customer, and the defective items are reworked. The retailer considers that customer demand is time and reliability dependent. The proposed supply chain considers probabilistic deterioration for the manufacturer and retailers. The optimum solution of the given model is evaluated for both the cases of crisp and fuzzy environments on the strategies such as production rate, unit production cost, idle time, screening, rework, etc. Managerial insights and the effect of parameters on the optimal inventory under fuzziness are presented. [ABSTRACT FROM AUTHOR]

In this article, the economic production and inventory model in a three-layer supply chain including one distributor, one manufacturer and one retailer for a single-product and general demand functions under three scenarios is developed. We assume that during the production process, both healthy and defective items are generated. As the first scenario, we develop the first model, in which the defective items are not reworked and all considered as scrape, while in the second model, we assume that the defective items are reworked and are sold as perfect item. In the second scenario, we assume that defective item can be sold with lower price than the selling price. Moreover, raw materials with imperfect quality are sent back from a distributor to outside supplier under a lower price. Determining the order quantity of the distributor and the selling prices of the distributor and the manufacturer as well as the retailer was the goal of this article such that the total profit of each member is maximised. In order to solve the models, the Stackelberg approach is employed between the members, and the concavity of the profit functions is proved using several theorems. Then, closed form solutions are derived for the decision variables and a solution algorithm is proposed to determine the optimal solutions. Finally, a numerical example is presented to illustrate the applicability of the model. [ABSTRACT FROM AUTHOR]

This research proposes an innovative modification to the Economic Production Quantity (EPQ) model that integrates Weibull-distributed gradual deterioration, holding costs, variable demands, and effective strategies for managing shortages and backlogs. The extended EPQ model is designed to optimize production quantity and total variable cost for items with slow degradation, particularly perishable goods. The model is based on modeling inventory with three different demand patterns and utilizing analytical methods for solving ordinary differential equations by incorporating Maclurin series approximation techniques. The proposed model addresses deterministic time-varying demand and defective items during the production processes. The research contributes valuable insights into inventory management and production planning, it has significant implications for decision-making in specifically perishable goods production industries. Two examples are given to validate the model's applicability, finally sensitivity analysis and the scope of possible future research are suggested. • EPQ model integrates Weibull deterioration for perishable goods, optimizing production. • Model addresses demand variations and defective items, aiding decision-making. • Validation examples confirm model applicability for perishable goods industries. • Sensitivity analysis highlights parameter impact on production optimization. • Insights aid decision-makers in optimizing inventory management for efficiency. [ABSTRACT FROM AUTHOR]

Economic Order and Production Quantity models are the most used stock control models. These simple and useful models are inadequate to respond to the real-life problems as they contain rigid assumptions. Therefore, new models are developed by adding additional assumptions to the assumptions in these models or by loosening existing assumptions. Among the rigid assumptions contained in these models are incoming order does not contain any defective items, delay in payments and backordering (shortages) of the demand are not allowed. Many new models have been developed by loosening these assumptions. In this study, unlike the previous studies, an economic order quantity model is proposed which deals with the fact partial backordering of the demand postponed to meet in the next period, defective items and permissible delay in payments. The study consists of three parts. In the first part, partial shortage and backordering of the demand, the defective item in the incoming order and the permissible delay in payments are explained and the studies in this area are summarized. In the second chapter, a new model of economic order quantity is proposed, which allows partial shortage and backordering of the demand. In this section, obtaining the mathematical model, previous studies are the special case of the proposed model and numerical examples related to the model are given. In the third chapter, the effects of the model parameters on the optimal values are examined by sensitivity analysis. Especially changes of optimal order quantity and total profit are discussed in detail. According to the results of the analysis, it was observed that the backordering rate of demand did not have a significant effect on optimal order quantity and total profit. On the other hand, as the permissible delay time in payments increased and the rate of defective products decreased, the optimal order quantity decreased and the total profit increased

This paper develops a two-layer supply chain for defective and non-defective types of items produced by manufacturer and retailer. The manufacturer produces items on the condition that the faulty items will return to the manufacturer after inspecting without financial liability of the retailer. The damaged items are identified subsequent of the conducted full screening process and reworked by the retailer. Hence, the retailer sells the non-defective item to the customer. Effect of deterioration estimates to make this research a more realistic one for the manufacturer and retailer. This study discusses the impact of several business strategies such as production rate, unit production cost, idle time costs in a collaborative marketing system, reliability dependent demand, etc., to determine the optimum average profit of the integrated model. The selling price of the retailer, the purchase cost of the manufacturer, and holding cost has the potential to be a significant breakthrough. Numerical and managerial implications are demonstrated along with sensitivity analysis of different parameters for the proposed two-layer supply chain. [ABSTRACT FROM AUTHOR]

• An (r, Q) inventory policy with deterministic demand and defective items is proposed. • Two formulations for the percentage of defective items are considered. • Full inspection of lot size is conducted • The non-planned shortages during inspection period are modeled. • Properties are established for the optimal problem solution. 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. [ABSTRACT FROM AUTHOR]

For an imperfect production system, to reduce quality-related costs, a manager may consider investing capital in quality improvement. In general, the investment expense in reducing the defective rate of items is often paid by the vendor. On the other hand, the buyer may inspect the product quality as the order is received which implies it incurs an inspection cost. In a supply chain integrated system, to accomplish global optimisation, the vendor and buyer can agree to jointly invest capital to improve the imperfect production processes, and the buyer can remove the inspection programme as the defective rate reaches a certain low-level. Hence, this paper investigates the impacts of collaborative investment and inspection policies on an integrated inventory model with defective items. The objective of this study is to seek the optimal order quantity, shipping times from the vendor to the buyer per production run, and the defective rate that minimise the joint total cost per unit time. An algorithm is developed to find the optimal solution. Several numerical examples are presented to demonstrate the proposed model and solution procedure, and then several management insights are obtained from the numerical examples. [ABSTRACT FROM PUBLISHER]

Şirketler, değişen market koşullarına adapte olmak ve rekabet içinde kalmak için lojistik faaliyetlerini de üretim faaliyetleri gibi iyi bir şekilde yönetmelidir. Lojistik maliyetleri, şirket maliyetlerinin büyük bir kısmını oluşturduğu için etkin bir şekilde yönetilmelidir. Etkin bir lojistik ağı oluşturulurken üretim birimlerinin yerleşimi ve bu birimlerden talep noktalarına giden araç rotalarına karar verilmesi karşılaşılan ve çözülmesi gereken optimizasyon problemleridir. Bu çalışma kapsamında envanter, yerleşim ve rotalama ile ilgili kararlar göz önüne alınarak bir yerleşim-envanter-rotalama probleminin çözülmesi amaçlanmıştır. Yerleşim-rotalama problemi ile karşılanamayan talepli ekonomik üretim miktarı modeli entegre edilerek problemin formülasyonu oluşturulmuştur. Problemin çözümü için bir büyük komşuluk arama algoritması geliştirilmiştir. Algoritmanın değerlendirilmesi, literatürden alınan yerleşim-rotalama problem veri setleri ile yapılan karşılaştırmalı sonuçlar aracılığıyla gerçekleştirilmiştir. Literatürde her iki problem veri seti için de bilinen en iyi sonuçlar bulunmuş veya en iyi sonuçlara yaklaşılmıştır.

We consider a manufacturing system which receives raw material from a supplier, processes it, and delivers it to the customer periodically. The system considered is imperfect and produces defectives at a constant rate. The finished product can only be delivered if the whole lot is quality-certified. Hence, defectives have to be reworked, and the whole lot quality-checked within the cycle. Three different scenarios are considered, viz. (a) a single lot of raw material for multiple lot of finished product and delivery of the product in multiple instalments, (b) a single lot of raw material for a multiple lot of finished product and delivery of the product in a single instalment, and (c) lot-for-lot and delivery of finished product in single instalment. A total cost equation is developed for each model, and the optimal ordering quantities are evaluated. The results found are encouraging and quite simple to use for practical purposes. [ABSTRACT FROM AUTHOR]

An imprecise demand rate creates problems in profit optimization in business scenarios. The aim is to nullify the imprecise nature of the demand rate with the help of the cloudy fuzzy method. Traditionally, all items in an ordered lot are presumed to be of good quality. However, the delivered lot may contain some defective items, which may occur during production or maintenance. Inspection of an ordered lot is indispensable in most organizations and can be treated as a type of learning. The learning demonstration, a statistical development expressing declining cost, is necessary to achieve any cyclical process. Further, defective items are sold immediately after the screening process as a single lot at a discounted price, and the fraction of defective items follows an S-shaped learning curve. The trade-credit policy is adequate for suppliers and retailers to maximize their profit during business. In this paper, an inventory model is developed with learning and trade-credit policy under the cloudy fuzzy environment where the demand rate is treated as a cloudy fuzzy number. Finally, the retailer’s total profit is maximized with respect to order quantity. Sensitivity analysis is presented to estimate the robustness of the model.

This study develops a mathematical model for an integrated vendor-buyer inventory with defective items and inspection error. We assume an imperfect production process with a known of defective probability. The buyer conducts a 100% quality check. However, due to an imperfect inspection process, the quality-controller may reject a good item as a defective item (Type I error), or accept a defective item as a non-defective item (Type II error). Different from existing research, we incorporate freight cost (a function of shipping weight and the distance between the vendor and the buyer) and stochastic lead time (a function of production and transportation times). The objective of the study is to minimize the joint total expected cost. This study indicates that Type I error is more significant when it is compared with Type II error. We also illustrate how the joint total expected cost and the lead time are significantly influenced by the distance of the buyer and the vendor. The results of our analysis also show that lead time can be shortened if the vendors speed up their shipment with smaller production lot sizes. Numerical examples and sensitivity analysis are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Modern business environment focuses on improving the operational efficiency of supplier, retailer and customers through integrating their inventory. Although a smoothly running integrated inventory system is ideal, the reality is to deal with imperfectness in transportation. In actual production environments, inventory items are not perfect and defectiveness occurs in a random process. In this paper, we propose an integrated supplier-retailer inventory model in which both supplier and retailer have adopted trade credit policies, and the retailer receives an arriving lot which contains random defectiveness in quality of items. This paper proposes a mathematical model considering two situations such as (a) risk neutral and (b) risk-averse case and the solution procedures are described with computational algorithm. The optimization procedures are discussed for determining optimal cycle time and the optimal number of shipments by minimizing the expected joint total cost in the integrated inventory system. Numerical examples are provided to illustrate the theoretical results, and sensitivity analysis is made for major inventory parameters. [ABSTRACT FROM AUTHOR]

As businesses operating in the production sector often need to deal with defective products and machine breakdowns, they may experience challenges in their production planning and control processes. This study addresses random machine breakdown to derive an optimal production time for defective products under an abort/resume inventory control policy. Machine time-to-breakdown involves two scenarios: namely, the occurrence of machine breakdown during production and its occurrence after production. Total expected cost functions are derived for each scenario. The average cost is then assessed in terms of expense over time. This study explores assumptions such as the performance of the inspection both during and after production, defective products being discarded, and others being sold at discount prices, and the time-to-breakdown being exponentially distributed. Numerical examples are given to explain the model which has been developed, and the convex formation of the expected function of total cost is illustrated in a graph. Optimal values are investigated for production time and total cost based on two different values of the number of machine breakdowns per unit time. Sensitivity analyses are used to show how a number of system parameters affect the optimal solution. These analyses demonstrate that the demand rate and the number of machine breakdowns have a significant effect on the optimal solutions. [ABSTRACT FROM AUTHOR]

The objective of this article is to maximize the joint profit for supplier and retailer by constructing a combined supplier-retailer inventory model wherein supplier and retailer both have implemented trade credit policies, and some defective items are received by the retailer. The customer's demand is expressed as a function of time, price and credit period, which is appropriate for the products for which demand increases initially and after some time it starts to decrease. Production, directly proportional to the customer's demand rate, is considered as one of the decision variables for the purpose of reducing the holding cost of the supplier. The article estimates the optimum replenishment cycle, customer's credit period, retail selling price by a classical optimization technique. For validation of the derived model, various numerical examples are demonstrated. Finally, implementing sensitivity analysis on the decision variables by varying the inventory parameters, effective managerial insights are generated which is beneficial for the players of supply chain by practically gaining the maximum joint profit through advising to opt for the case N ≤ M ≤ T, i.e., when the customer's trade credit period offered by the retailer is lesser than the retailer's trade credit period offered by the supplier and which is lesser than the replenishment cycle length. [ABSTRACT FROM AUTHOR]

This article investigates an integrated single-vendor single-buyer production-inventory model with imperfect production under a mixed environment where fuzziness and randomness appear simultaneously. The paper focuses on representing the annual customer demand as a triangular fuzzy number together with an associated probability. A further assumption is that the production process is not perfect and goes 'out-of-control' with a certain probability. This causes the vendor, in particular, and the supply chain, in general, to incur an additional warranty cost and also leads to the production of larger batch sizes to compensate the imperfection. In order to avoid these extra costs, the vendor makes an investment to improve the production process quality and hence reduce the number of defective items produced. The expected annual integrated total cost is derived with these assumptions under the n-shipment policy. A methodology is proposed to minimize crisp equivalent of the expected annual integrated total cost so as to obtain the optimal values of the number of shipments, the shipment lot-size, the safety stock factor and the 'out-of-control' probability. A numerical example is given to illustrate this proposed methodology and to highlight the advantage of investing in reducing the probability of the production process going 'out-of-control'. [ABSTRACT FROM AUTHOR]

A control chart of monitoring the number of failures is proposed with a moving average scheme, when the life of an item follows a Weibull distribution. A specified number of items are put on a time truncated life test and the number of failures is observed. The proposed control chart has been evaluated by the average run lengths (ARLs) under different parameter settings. The control constant and the test time multiplier are to be determined by considering the in-control ARL. It is observed that the proposed control chart is more efficient in detecting a shift in the process as compared with the existing time truncated control chart.

Inventory models must consider the probability of sub-optimal manufacturing and careless shipping to prevent the delivery of defective products to retailers. Retailers seeking to preserve a reputation of quality must also perform inspections of all items prior to sale. Inventory models that include sub-lot sampling inspections provide reasonable conditions by which to establish a lower bound and a pair of upper bounds in terms of order quantity. This should make it possible to determine the conditions of an optimal solution, which includes a unique interior solution to the problem of an order quantity satisfying the first partial derivative. The approach proposed in this paper can be used to solve the boundary. These study findings provide the analytical foundation for an inventory model that accounts for defective items and sub-lot sampling inspections. The numerical examples presented in a previous paper are used to demonstrate the derivation of an optimal solution. A counter-example is constructed to illustrate how existing iterative methods do not necessarily converge to the optimal solution.

In this study, an integrated economic order quantity inventory model for deteriorating items with inventory level and ware house capacity constraint is considered. The proposed model allows shortage for buyer and demand to be considered as a constant. By using Lagrange's multiplier technique optimum order quantity is obtained to minimize the system cost while satisfies the both constraints. Numerical examples are also illustrating developed model. [ABSTRACT FROM AUTHOR]

Abstract This study develops a green production quantity model with random imperfect quality items, trade-credit policy, service level constraints, and failure in reworking. In practice, not only the green production system but also the rework process is often imperfect. The presence of a single machine in the production system outcomes in limits for green production and partial backordering. It is assumed that the green production system may generate defective items, which are reworked. As a reworked process is also not perfect, it produces a random number of scrapped items. On the other hand, the supplier also offers a trade-credit policy to the buyer, which minimizes the on-hand stock and boosts sales. In fact, this model will help organizations to minimize their costs through cooperation in industry. The objective of this research is to find the optimal green production and backordered quantity of each item such that the projected total cost is minimized. The optimal solution for each scenario is derived and the application of the designed model is validated with two numerical examples. The results for both examples show that the minimum total cost for the green production system is achieved when the permissible delay-period given by the supplier to the buyer is greater than or equal to the aggregated sum of production uptime, reworking time, production downtime, and allowed shortage time. Sensitivity analysis of the proposed study is also done to illustrate the realistic usage. [ABSTRACT FROM AUTHOR]

This paper proposes a multi-product Vendor-Managed Inventory (VMI) supply chain inventory model by considering defective items. In this way, vendor purchases different kinds of products from a remote supplier and then delivers them to the retailer. On the other hand, delivered batches have some imperfect items while there is no access to the supplier to replace them. Consequently, there are two options to face imperfect items in order to satisfy the demand, buy or repair. Additionally, some innovations and contributions in this paper are Join Replenishment Policy, all units discount, different alternatives for shipping goods and backlogging shortage. They result in a novel VMI model that is as close as possible to the real world. Since, the proposed problem was NP-complete, two evolutionary algorithms are developed to solve it, Genetic Algorithm (GA) and imperialist Competitive Algorithm (ICA). After calibrating their main parameters by Taguchi approach, their performance is compared by solving numerical examples. [ABSTRACT FROM AUTHOR]

During production process, we may encounter with some defective items in spite of all preventive measures. The defective items are either rejected immediately at the time of production or reworked and sold as good ones or customers are given enough refund to maintain the goodwill of the company. Here, no. of defective items is assumed to be some fraction of production. Moreover higher price of a product trims down the demand and vice versa. Also in the market, it is seen that large stacks of consumer goods display lead the customers to buy more. The article considers this realistic situation and involves price-sensitive stock dependent demand. As production advances, we have defective items as a part of outcome. So production rate is one of the key features for the present model while calculating the profit. Cycle time and retail price are the very obvious factors affecting the profit. Hence, maximum total profit is achieved with respect to production rate, cycle time and retail price. When there are no defective items in production process, the optimal results are obtained as well. [ABSTRACT FROM AUTHOR]

In this paper, we develop an integrated inventory model in a single-vendor single-buyer supply chain under an unknown demand distribution at the buyer. It is assumed that each lot delivered to the buyer contains a random fraction of defective items, and lead time can be reduced at an extra crashing cost. We also consider that the unmet demand at the buyer during the stockout period is partially backordered. A model is formulated to minimize the total expected relevant costs of the system considering an exact expression of the service level constraint to ensure that a certain percentage of customer orders are filled by the buyer. Closed-form expressions are derived for the optimal order quantity and safety factor for given lead time and shipment frequency, and then an algorithm is proposed to find the global optimal solution. Finally, a numerical example is presented to illustrate the solution procedure and sensitivity analysis is carried out to analyze the proposed model. [ABSTRACT FROM AUTHOR]

This paper proposes single-supplier single-retailer production and inventory models for maximizing the supplier and retailer's profits in non-cooperative and cooperative environments. The effect of defective items and inspection errors are considered in the proposed models. In addition, we consider that the supplier offers the retailer a quantity threshold to absorb transportation costs for promoting the economies of scale of transport. Mathematical analyses are conducted, and optimal equilibrium production and replenishment strategies for the supplier and retailer are derived under non-cooperative and cooperative situations. Subsequently, we establish two algorithms to explain the optimal equilibrium solutions for these cases. Finally, several numerical examples and a sensitivity analysis with respect to major parameters are presented to demonstrate the theoretical results, compare the distinct solutions, and derive managerial insights. [ABSTRACT FROM AUTHOR]

In this study, an optimal integrated vendor-buyer inventory model with defective items is proposed. Most researches on defective items assumed that an inspection process was carried out by the buyer. We consider that the vendor conducts the inspection process and disposes defective items in multiple batches. We prove that the function of annual cost is convex, and obtain closed-form expressions. A solution procedure is used to derive the optimal order quantity, the number of shipments, and the number of defective item disposals. Numerical examples are provided to illustrate our model. Setting the fraction of defective items to zero, the numerical examples indicate that the proposed model can result in the solutions to the existing models without considering defective items. Moreover, a sensitivity analysis is used to reveal the effects of cost parameters on the optimal solution. We show that when the disposal cost is relatively low, a multiple disposals strategy may perform better than a single disposal strategy. [ABSTRACT FROM AUTHOR]

This research aims to study the factors affecting the crooked fabric cutting and to present the new cutting procedure that complies with the factors affecting the crooked fabric cutting of a case study. The defect in fabric cutting process was crooked fabric making nonconforming product. The cause and effect diagram was utilized to analyze and suggest related factors leading to the problem. It was showed that the number of times of knife sharpening and the number of layers in fabric paving would affect the crooked fabric cutting the design of experiment was applied to determine appropriate the level of these factors. The main factor significantly affected the crooked fabric cutting (p < 0.05) was the number of times of knife sharpening, but the number of layers in fabric paving and interaction between both factors would not significantly affect the crooked fabric cutting. The number of times of knife sharpening in the level 4 had been sharpened twenty times in each cutting round. The least average defective proportion was 0.0173. Then the new cutting procedure would significantly reduce average defective proportion. It could reduce the average number of defective items as 5.74 pieces in each cutting round or 70.52 percents.

An integrated single vendor, single buyer inventory policy is incorporated in a continuous review model with stochastic demand. The production process is assumed to involve deterioration process and produces a certain number of defective items. In the real life production environment, it can often be observed that defective items are produced due to imperfect production processes. A situation in which the vendor invests money to improve the quality of the production process and thereby reduce the defective percentage is considered. 100% error-free screening process adopted by the buyer to separate the defective and non defective items is incorporated to focus on the varying inventory holding costs. Shortages are allowed and completely backlogged. In realistic circumstance, when an order is placed, the products will be delivered only after a certain period of time (lead time). But, owing to tough competition, the buyer may pay an extra amount in order to reduce the lead time. This realistic assumption is incorporated in this model. The model is formulated by allowing the lead time demand to be distributions free and follow a normal distribution. The prime motive is to determine the optimal policies regarding optimal order quantity, reorder point, lead time and the number of lots delivered in a production run by minimizing the expected total cost of the system considered. [ABSTRACT FROM AUTHOR]

Coping with the severe competition in the global markets and the limited in-house capacity, the management of presentday manufacturing firm always pursues possible alternatives to level production schedule by shortening production uptime, assure product quality, and reduce overall system cost in order to stay competitive. Inspired by this concept, the present study attempts to derive the optimal batch size for a fabrication system with outsourcing policy and rework. Mathematical modeling and optimization techniques are employed to explore the problem. As a result, a closed-form optimal batch size for the proposed system is determined. Besides, various critical system performance indicators, such as the breakeven points of unit outsourcing cost, in- house defective rate, and unit reworking cost, etc., can now be revealed for assisting diverse managerial decision-makings. [ABSTRACT FROM AUTHOR]

Recently, Moussawi-Haidar et al. (2016) considered a production process with random supply and an inspection process performed during and at the end of production. Two economic production quantity models with defective items were developed, in which Model 1 assumes that defective items are sold at a discounted price at the end of inspection process, and Model 2 assumes that defective items are reworked at a cost at the end of inspection process. In the paper, there are some mathematical expressions which are to be corrected. We first present the mathematical expressions corrected and establish the necessary conditions for which there is an optimal solution. We next provide the correct solutions to the numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we consider an integrated vendor-buyer inventory policy for a continuous review model with a random number of defective items and screening process gradually at a fixed screening rate in buyer’s arriving order lot. We assume that shortages are allowed and partially backlogged on the buyer’s side, and that the lead time demand distribution is unknown, except its first two moments. The objective is to apply the minmax distribution free approach to determine the optimal order quantity, reorder point, lead time and the number of lots delivered in one production run simultaneously so that the expected total system cost is minimized. Numerical experiments along with sensitivity analysis were performed to illustrate the effects of parameters on the decision and the total system cost.