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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]

• 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]

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]

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]

Over the last few decades, inventory management of produced, recovered, and returned items in the closed loop supply chain (CLSC) systems has been significantly taken into consideration. Most of available works in the literature assumed that manufactured and remanufactured items have the same quality. However in practices, some customers perceive remanufactured items as lower quality products rather than newly produced items. Therefore, this paper assumes demands for manufactured items and remanufactured ones are not same. This assumption causes lost sales in which there are shortage periods for both produced and reproduced items. In the real-life production systems, the generation of nonconforming items is inevitable owing to deteriorating process or other uncontrollable factors. In order to take this issue into account, the production and reproduction processes are assumed to be imperfect as it can produce some defectives which would be reworked in the same cycles. Moreover, to make the study realistic, we assume the return rate of used items as a variable function which depends on the minimum acceptable quality level of returns. Furthermore, in this research the quality of returned items is considered as a random variable and based on this assumption, buyback cost, remanufacturing cost and salvage value depend on quality level of returned items. Basically, three different stochastic models, with different probability distribution functions, are mathematically developed for multiple manufacturing and remanufacturing cycles. A solution procedure is introduced and then there numerical studies are provided to show the applicability of the developed models and also to perform a sensitivity analysis. Finally, conclusion and future researches are presented. [ABSTRACT FROM AUTHOR]

This paper considers a multi-product problem with non-identical machines. This manufacturing system consists of various machine types with different production capacities, production costs, setup times, production rates and failure rates. One of the major issues in the planning phase of a manufacturing system is to take the best decision about which machines must be utilized to manufacture which items. As a result, the decision makers face three critical questions: what machines must be purchased, which items should be allocated to each machine, and what is the optimal cycle length. These decisions must be made to minimize system costs including utilization, setup, production, holding and scrap costs. The multi-machine multi-product economic production quantity (EPQ) problem for an imperfect manufacturing system is formulated as a mixed integer non-linear programming (MINLP), where the convexity property of multi-product single machine EPQ model is used to convert the problem into a bi-level decision-making problem. In the first level, decisions about machine utilization and items allocation are made. After, in the second level the optimal cycle length for each machine is determined. To solve the problem at hand, a hybrid genetic algorithm (HGA) is proposed integrating genetic algorithm and derivatives method. In the proposed HGA, the solutions of the first level are obtained randomly and then, for the second level, the derivatives method is applied to obtain optimal cycle length based on solutions of the first level. Finally, the results of HGA method are compared to the results of general algebraic modeling system (GAMS) and it is found that HGA method has better and more efficient results. Also, a numerical experimentation and a sensitivity analysis of the model are done. [ABSTRACT FROM AUTHOR]

In developing countries with many OEMs (Original Equipment Manufacturer), self-producing companies sometimes face a one-time-only discount for the same products offered by the OEMs. Making such a special order from the OEM may or may not be beneficial to the manufacturer who usually produces himself. In this paper, EPQ (economic production quantity) models were developed to evaluate the optimal make-or-buy decisions when such a manufacturing company faces a one-time-only discount offered by an OEM. To reflect real situation, imperfect quality of both self-produced and purchased products was also considered in the models. Conditions under which the manufacturer should place a special order from the OEM and related optimal ordering quantities were obtained. A numerical example was presented to illustrate the managerial insights of the models. [ABSTRACT FROM AUTHOR]

Abstract: We studied inventory models with defective items, crashable lead time and distribution-free approach to provide a complete solution procedure. Our procedure will reveal that the numerical comparison of two upper bounds is unnecessary in Tung et al. (2010). Moreover, our analytical results could explain why the previous incomplete solution of Tung et al. (2010) is accidentally corrected. Our findings can be generalized to several related papers and help researchers to locate the optimal replenishment policy. [Copyright &y& Elsevier]

Abstract: A lot of researchers develop their inventory models under trade credit by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, from the viewpoint of practice, it can often be found that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the specific quantity. Furthermore, the products received usually contain some defective items. This paper establishes the EOQ model with defective items and partially permissible delay in payments linked to order quantity. It also uses the rigorous method of mathematics to derive the solution procedure to locate the optimal solution. Finally, numerical examples are given to illustrate all theoretical results in this paper. [Copyright &y& Elsevier]

Abstract: 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 containing some defective items. The customer’s market demand rate depends on the length of the credit period offered by retailer. Our objective is to determine the retailer’s optimal order cycle length, the order quantity, and the optimal number of shipments per production run from the supplier to the retailer so that the entire supply system has maximum profit. We develop an algorithm to find the optimal solution for the supply chain. Several numerical examples are provided to illustrate the theoretical results, and sensitivity analysis of major parameters including the defective rate in a production batch, the retailer’s trade credit period and the customer’s trade credit period in the model are presented. [Copyright &y& Elsevier]

Abstract: This study discusses a mixture inventory model with back orders and lost sales in which the order quantity, reorder point, lead time and setup cost are decision variables. It is assumed that an arrival order lot may contain some defective items and the number of defective items is a random variable. There are two inventory models proposed in this paper, one with normally distributed demand and another with distribution free demand. Finally we develop two computational algorithms to obtain the optimal ordering policy. A computer code using the software Matlab is developed to derive the optimal solution and present numerical examples to illustrate the models. Additionally, sensitivity analysis is conducted with respect to the various system parameters. [Copyright &y& Elsevier]

Abstract: In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms. [Copyright &y& Elsevier]

Abstract: This paper investigates the finite replenishment inventory models of a single product with imperfect production process. In this process, a certain fraction or a random number of produced items are defective. These non-conforming items are rejected or reworked or if they reached to the customer, refunded. Here, a generalised unit cost function is formulated incorporating the several factors like raw material, labour, replenishment rate and others factors of the manufacturing system. The rate of replenishment is considered to be a variable. The selling price of an unit is determined by a mark-up over the production cost. Optimum production of the product is suggested to have maximum profit using a gradient based mathematical programming technique for optimization. Finally, numerical examples are given to illustrate the results and the significant features of the production system. As a particular case, the result of the perfect system (without defective items) are obtained. Also, the effect of changes in the selling rate, defectiveness, production cost and other parameters on the optimal average profit are graphically presented. Some interesting decisions regarding production policy are established. [Copyright &y& Elsevier]

This paper considers a statistical method of estimating mean shift for a fraction defective of population. One traditional method for this estimation problem has been known as the CUSUM (cumulative sum) method, and it provides a method of estimating the occurrence of a shift in the mean from the observed data. We consider this estimation problem of shift occurrence in a production process. It is assumed that the process has two states, one is good (fraction defective low) and the other bad (fraction defective high), and starts in good state with probability one. We are interested in judging when the state has moved to the bad state by analyzing the observed data.In this paper, we model such a phenomenon as a hidden-Markov model. The states which are unobservable in a hidden-Markov model can be analyzed from the sequence of observed results. Hence, the advantage of this modeling technique is that the unknown parameters which are included in the hidden states can be estimated. We compare the performance of this hidden-Markov model with the CUSUM method based on several simulation data sets. [Copyright &y& Elsevier]