Determining replenishment lot size and shipment policy for an extended EPQ model with delivery and quality assurance issues

AbstractThis paper derives the optimal replenishment lot size and shipment policy for an Economic Production Quantity (EPQ) model with multiple deliveries and rework of random defective items. The classic EPQ model assumes a continuous inventory issuing policy for satisfying demand and perfect quality for all items produced. However, in a real life vendor–buyer integrated system, multi-shipment policy is practically used in lieu of continuous issuing policy and generation of defective items is inevitable. It is assumed that the imperfect quality items fall into two groups: the scrap and the rework-able items. Failure in repair exists, hence additional scrap items generated. The finished items can only be delivered to customers if the whole lot is quality assured at the end of rework. Mathematical modeling is used in this study and the long-run average production–inventory-delivery cost function is derived. Convexity of the cost function is proved by using the Hessian matrix equations. The closed-form optimal replenishment lot size and optimal number of shipments that minimize the long-run average costs for such an EPQ model are derived. Special case is examined, and a numerical example is provided to show its practical usage

Solving finite production rate model with scrap and multiple shipments using algebraic approach

This paper solves a finite production rate (FPR) model with scrap and multiple shipments using an algebraic method. Classic FPR model assumes a continuous inventory issuing policy to satisfy demand and perfect quality production for all items produced. However, in real life vendor-buyer integrated production-inventory system, multiple shipment policy is practically used in lieu of a continuous issuing policy and generation of defective items during production run is inevitable. In this study, it is assumed that all defective items are scrap and the perfect quality items can only be delivered to customers if the whole lot is quality assured at the end of the production run. A conventional approach for solving the FPR model is the use of differential calculus on the long-run average cost function with the need to prove optimality first. This paper demonstrates that optimal lot size and its overall costs for the aforementioned FPR model can be derived without derivatives. As a result, it enables students or practitioners who have little knowledge of calculus to understand and to handle with ease the real-life FPR model

Production lot sizing with rework and fixed quantity deliveries

This paper is concerned with determination of the optimal lot size for an economic production quantity (EPQ) model with the reworking of random defective items and fixed quantity multiple deliveries. Classic EPQ model assumes continuous issuing policy for satisfying product demand and perfect quality production for all items produced. However, in real life vendor-buyer integrated production-inventory system, multi-delivery policy is used practically in lieu of the continuous issuing policy and generation of defective items during production run is inevitable. In this study, all nonconforming items produced are considered to be repairable and are reworked in each cycle when regular production ends. The finished items can only be delivered to customers if the whole lot is quality assured at the end of the rework. Fixed quantity multiple installments of the finished batch are delivered to customers at a fixed interval of time. The long-run average integrated cost function per unit time is derived. A closed-form optimal batch size solution to the problem is obtained. A numerical example demonstrates its practical usage

Effect of variable shipping frequency on production-distribution policy in a vendor-buyer integrated system

This paper investigates the effect of variable shipping frequency on production-distribution policy in a vendor-buyer integrated system. In a recent article Chiu et al. [1] derived the optimal replenishment lot size for an economic production quantity problem with multi-delivery and quality assurance, based on an assumption that the number of shipment is a given constant. However, in a vendor-buyer integrated system in supply chain environment, joint determination of replenishment lot size and number of shipments may help such a system to gain significant competitive advantage in terms of becoming a low-cost producer as well as having tight linkage to customer. For this reason, the present study extends the work of Chiu et al. [1] by considering shipping frequency as one of the decision variables and incorporating customer’s stock holding cost into system cost analysis. Hessian matrix equations are employed to certify the convexity of cost function that contains two decision variables, and the effect of variable shipping frequency on production-distribution policy is investigated. A numerical example is provided to demonstrate practical usage of the research result

A single-producer multi-retailer integrated inventory model with a rework process

This study considers a single-producer multi-retailer integrated inventory model with the reworking of random defective items produced. The objective is to find the optimal production lot size and optimal number of shipments that minimizes total expected costs for such a specific supply chains system. It is assumed that a product is manufactured by a producer. All items are screened for quality purpose and random nonconforming items will be picked up and reworked at the end of regular production in each cycle. After the entire lot is quality assured, multiple shipments will be delivered synchronously to m different retailers in each production cycle. Each retailer has its own annual product demand, unit stock holding cost, and fixed and variable delivery costs. Mathematical modeling and analysis is used to deal with the proposed model and to derive the expected system cost. Hessian matrix equations are employed to prove the convexity of the cost function. As a result, a closed-form optimal replenishment-delivery policy for such a specific single-producer multi-retailer integrated inventory model is obtained. A numerical example is provided to show the practical usage of the proposed model

A note on ‘impacts of random scrap rate on production system in supply chain environment with a specific shipping policy’

This paper employs an alternative approach to reexamine the impacts of random scrap rate on production system in supply chain environment with a specific shipping policy. A straightforward approach in terms of a two-phase algebraic derivation is proposed in this study to replace the conventional method with the need of applying first-order and second-order differentiations to the system cost function for proof of convexity before derivation of the optimal production-shipment policy. The research result of this study is confirmed that is identical to what was obtained by Cheng et al. [1] where they used the conventional method to solve the same problem. The proposed approach is helpful for practitioners, who may not have sufficient knowledge of differential calculus to understand such an integrated production-shipment system in supply chain environment

Replenishment Decision Making with Permissible Shortage, Repairable Nonconforming Products and Random Equipment Failure

Abstract: This study is concerned with replenishment decision making with repairable nonconforming products, backordering and random equipment failure during production uptime. In real world manufacturing systems, due to different factors generation of nonconforming items and unexpected machine breakdown are inevitable. Also, in certain business environments various situations between vendor and buyer, the backordering of shortage stocks sometimes is permissible with extra cost involved. This study incorporates backlogging, random breakdown and rework into a production system, with the objective of determination of the optimal replenishment lot size and optimal level of backordering that minimizes the long-run average system costs. Mathematical modeling along with the renewal reward theorem is employed for deriving system cost function. Hessian matrix equations are used to prove its convexity. Research result can be directly adopted by practitioners in the production planning and control field to assist them in making their own robust production replenishment decision

Joint production, quality control and maintenance policies subject to quality-dependant demand

This thesis is a strive to find a proper solution, using the stochastic optimal control means for an unreliable production system with product quality control and quality-dependent demand. The system consists of a single machine producing a single product type (M1P1) subject to breakdowns and random repairs and must satisfy a non-constant rate of customer demand, which response to the quality of parts received. Since the machine produces with a rate of noncompliant products, an inspection of the products is made to reduce the number of bad parts that would deliver to the customer. It is done continuously and consists of controlling a fraction of the production. Approved products are put back on the production line, while bad products are discarded. The intended objective of this study is to provide optimal quality control and production policy, which maximize the net revenue consisting of the gross revenue, the cost of inventory, the cost of shortage, the cost of the inspection, the cost of maintenance and the cost of no-quality parts. Main decision variables are the sampling rate of the quality control system as well as the threshold of finished product inventory. The demand function reacts to the average outgoing quality level (AOQ) of finished products. In the third chapter of this study, preventive maintenance and dynamic pricing policies are added up to the optimal policy, cited above. To achieve the optimal points of the policy, which maximize our net production revenue, a simulation approach is implemented as an experimental design and its results were used in response surface methodology. To implement the experiment design (simulation approach) which thoroughly reflects model considerations such as its continuous nature and the variety, first, a continuous variable for the probability of defectiveness was introduced, functioning with the age of machine up until its next breakdown maintenance. Second, so as to reflect the effect of quality control process that results in Average Outgoing Quality rather than simple defectiveness possibility, this function (AOQ) was built based on instant behavior of mentioned function above as its independent variable. Third, due to the use of prospect theory assumptions in building a demand function that responds to the level of client delivered defectiveness (AOQ), a responsive continuous function was created for the demand, reacting to the level of product quality by determining it's needed per time amount. Finally. To illustrate the machine’s manufacturing policy based on Hedging Point, finished product inventory variable was introduced in the experiment design. In a nutshell, we have a production system that has been designed in a way that by raising its age (At), leads to more possibility of defectiveness and less demand in time units. This manner continuous up until the next maintenance action of the system, which restores all factors to their initial conditions. By use of the simulation approach of optimization an experiment is designed and implemented to control decision variables of the policy and maximize the objective function of average net revenue (ANR). Decision variables are statistically and practically in the matter of consideration such as finished product inventory threshold (Z), the proportion of inspection (F) and PM thresholds (Mk or Pk)