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

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

Incorporating machine reliability issue and backlogging into the EMQ model - Part II: Random breakdown occurring in inventory piling time

This paper presents the second part of a research which is concerned with incorporating machine reliability issues and backlogging into the economic manufacturing quantity (EMQ) model. It may be noted that in a production system when back-ordering is permitted, a random machine failure can take place in either backorder filling stage or in on-hand inventory piling time. The first part of the research investigates the effect of a machine failure occurring in backorder filling stage on the optimal lot-size; while this paper (the second part of the research) studies the effect of random breakdown happening in inventory piling time on the optimal batch size for such an imperfect EMQ model. The objective is to determine the optimal replenishment lot-size that minimizes the overall productioninventory costs. Mathematical modelling is used and the renewal reward theorem is employed to cope with the variable cycle length. Hessian matrix equations are utilized to prove convexity of the cost function. Then, the optimal lot size for such a real-life imperfect manufacturing system is derived. Practitioners and managers in the field can adopt these replenishment policies to establish their own robust production plan accordingly

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

Finite production rate model with backlogging, service level constraint, rework, and random breakdown

In most real-life production systems, both random machine breakdown and the production of nonconforming items are inevitable, and adopting a backlogging policy with a predetermined minimum acceptable service level can sometimes be an effective strategy to help the management reduce operating cost or smoothen the production schedule. With the aim of addressing the aforementioned practical situations in production, this study explores the optimal production runtime for the finite production rate (FPR) model with allowable backlogging and service level constraint, rework of defective products, and random machine breakdown. Mathematical modelling is employed along with optimization techniques to derive the optimal production runtime that minimizes the long-run average system costs for the proposed FPR model. The joint effects of the allowable backlogging with a planned service level, rework, and random machine breakdown on optimal runtime decision have been carefully investigated through a numerical example and sensitivity analysis. As a result, important insights regarding various system parameters are revealed in order to enable the management to better understand, plan, and control such a practical production system

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

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

Mathematical modelling for multiproduct EPQ problem featuring delayed differentiation, expedited rate, and scrap

The client requirements of present-day markets emphasize product quality, variety, and rapid response. To gain competitive advantages in marketplaces and meet customer needs, manufacturers today seek the most economical and fastest fabrication schemes and strategies to produce their various goods, especially when commonality exists within these multiple end products. Inspired by the above viewpoints, this study uses a mathematical modelling approach for solving a multiproduct economic production quantity (EPQ) problem featuring scrap, delayed differentiation, and expedited rate on the fabrication of the common part. We build a two-stage multiproduct fabrication scheme. Stage one uses an accelerated rate to produce all necessary common parts for multi-item to shorten its uptime, while stage two fabricates finished products sequentially using a rotation cycle rule. Inevitable random scraps produced in both stages are identified and removed to achieve the anticipated quality. We determined the optimal cost-minimization operating cycle length and used a numerical example to show our model’s capability and to explore collective and individual impacts of scrap, expedited-rate, and postponement strategies on various performances of the studied problem (such as uptime of common part, utilization, rotation cycle time, total system cost, and individual cost contributor, etc.) Our model can offer an optimization solution and in-depth managerial insights for fabrication and operations planning in a wide variety of present-day industries, such as automotive, household goods, clothing, etc