Optimal production cycle time for multi-item FPR model with rework and multi-shipment policy

This paper determines the optimal common production cycle time for a multi-item finite production rate (FPR) model with rework and multi-shipment policy. The classic FPR model considers production planning for a single product with perfect quality production and a continuous issuing policy. However, in real life production environments, vendors often plan to produce multiple products in turn on a single machine in order to maximize the machine utilization. Also, due to various uncontrollable factors, generation of nonconforming items in any given production run is inevitable. It is also common for vendors to adopt multiple/periodic delivery policy for distributing their finished goods to customers. In this study, it is assumed that all nonconforming items can be reworked and repaired in the same cycle when regular production ends at additional cost per each reworked item. Our objective is to determine the optimal common production cycle time that minimizes the long-run average cost per unit time and to study the effect of rework on the optimal common cycle time for such a specific multi-item FPR model with rework and multi-shipment policy. Mathematical modeling is used, and the expected system cost for the proposed model is derived and proved to be convex. Finally, a closed-form optimal cycle time is obtained. A numerical example and sensitivity analysis is provided to show the practical use of our obtained results

Optimal production cycle time for multi-item FPR model with rework and multi-shipment policy

This paper determines the optimal common production cycle time for a multi-item finite production rate (FPR) model with rework and multi-shipment policy. The classic FPR model considers production planning for a single product with perfect quality production and a continuous issuing policy. However, in real life production environments, vendors often plan to produce multiple products in turn on a single machine in order to maximize the machine utilization. Also, due to various uncontrollable factors, generation of nonconforming items in any given production run is inevitable. It is also common for vendors to adopt multiple/periodic delivery policy for distributing their finished goods to customers. In this study, it is assumed that all nonconforming items can be reworked and repaired in the same cycle when regular production ends at additional cost per each reworked item. Our objective is to determine the optimal common production cycle time that minimizes the long-run average cost per unit time and to study the effect of rework on the optimal common cycle time for such a specific multi-item FPR model with rework and multi-shipment policy. Mathematical modeling is used, and the expected system cost for the proposed model is derived and proved to be convex. Finally, a closed-form optimal cycle time is obtained. A numerical example and sensitivity analysis is provided to show the practical use of our obtained results

Economic lot sizing with imperfect rework derived without derivatives

This paper presents an algebraic method for solving economic production quantity (EPQ) model with imperfect rework. Conventional method for deriving optimal lot size is by using differential calculus on the cost function with the need to prove optimality first. Recent articles proposed algebraic approach to the solution of classic economic order quantity (EOQ) and EPQ model without reference to the use of derivatives. This note extends them to an EPQ model taking into consideration an imperfect rework of defective items. We demonstrate that the optimal lot size and the expected production-inventory cost for such a realistic EPQ model can be derived without derivatives

Integrating a cost-reduction shipment plan into a single-producer multi-retailer system with rework

This study integrates a cost-reduction shipment plan into a single-producer, multi-retailer system with rework process. In a recent article, Chiu et al. [1] have examined a single-producer, multi-retailer integrated inventory model with a rework process. For the purpose of reducing the inventory holding cost, this study combines an alternative n+1 product distribution policy into their model. Under the proposed shipment plan, an extra (initial) delivery of finished items takes place during the production uptime to meet the retailers’ product demands for the periods of the producer’s uptime and reworking time. Upon the completion of rework, multiple shipments will be delivered synchronously to m different retailers. The objectives are to find an optimal production-shipment policy that minimizes the expected system cost for such a supply chain system, and to demonstrate that the result of this study gives significant holding cost savings in comparison with Chiu et al.’s model [1]. With the help of mathematical modelling and Hessian matrix equations, the optimal operating policy for the proposed model is derived. Through a numerical example, we demonstrate our model gives significant savings in stock holding cost for both the producer and retailers

Economic lot sizing with imperfect rework derived without derivatives

This paper presents an algebraic method for solving economic production quantity (EPQ) model with imperfect rework. Conventional method for deriving optimal lot size is by using differential calculus on the cost function with the need to prove optimality first. Recent articles proposed algebraic approach to the solution of classic economic order quantity (EOQ) and EPQ model without reference to the use of derivatives. This note extends them to an EPQ model taking into consideration an imperfect rework of defective items. We demonstrate that the optimal lot size and the expected production-inventory cost for such a realistic EPQ model can be derived without derivatives

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

Reexamining a single-producer multi-retailer integrated inventory model with rework using algebraic method

In this study, a single-producer multi-retailer integrated inventory model with rework is reexamined using mathematical modeling and an algebraic method. It is assumed that a product is manufactured through an imperfect production process, and the reworking of random defective items is done right after the regular process in each cycle. After the entire lot is quality assured, multiple shipments will be delivered synchronously to m different retailers in each production cycle. 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. The conventional approach uses differential calculus on system cost function to derive the optimal production- shipment policy (Chiu et al. [1]); in contrast, the proposed algebraic approach is a straightforward method that enables practitioners who may not have sufficient knowledge of calculus to understand and manage real-world systems more effectively

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

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

Revisiting "integrating a cost-reduction shipment plan into a single-producer multi-retailer system with rework" using an alternative approach

This study revisits a single-producer multi-retailer system with a cost-reduction shipment plan and rework [1] using an alternative approach. Unlike the conventional method that uses differential calculus on system cost function to prove its convexity and derive the optimal production-shipment policy, we proposed an algebraic solution procedure to the problem. Such a straightforward approach may enable the practitioners with little knowledge of calculus to understand real supply chain systems more easily