We develop an economic production quantity (EPQ) model with random defective items and failure in repair. The existence of only one machine results with limited production capacity and shortages. The aim of this research is to derive the optimal cycle length, the optimal production quantity and the optimal back ordered quantity for each product so as to minimize the total expected cost (holding, shortage, production, setup, defective items and repair costs). The convexity of the model is derived and the objective function is proved convex. Two numerical examples illustrate the practical usage of the proposed method.
One of the critical factors in any production process is material. The management of material concerns the regulation of the flow of materials to, within, and from the organization. The efficiency of the material flow can substantially influence costs as well as revenue generation capabilities [1]. The management of material involves a balance between the shortages and excesses of stock in an uncertain environment. With the globalization of business in recent years, firms are sourcing and distributing raw materials, components, and finished goods across the globe. Customers want to receive their quality products quickly. As a result, efficient inventory management, production planning and scheduling to achieve flexibility and quick response has become a core competitive advantage. To achieve operation strategies goals, the company must be able to effectively utilize resources and minimize costs. In manufacturing companies, when items are internally produced instead of being obtained from an outside supplier, the economic production quantity (EPQ) model is often employed to determine the optimal production lot size that minimizes the overall production/inventory costs. The classic EPQ model assumes that during a production run a manufacturing facility functions perfectly. However, due to process deterioration or some other factors, imperfect quality items are inevitable. Some examples of the rework processes are: printed circuit board assembly in the PCBA manufacturing, metal components, and plastic injection molding. A considerable amount of research has been carried out by Cheng [2], Chiu et al. [3], Chung [4], Lee and Rosenblatt [5], and Rosenblatt and Lee [6] to address the imperfect quality EPQ problem. They assumed that at some random time, the process might shift from an in-control to an out-of-control state. Hayek and Salameh [7] derived an optimal operating policy for finite production (EPQ) model with rework and imperfect quality items. They assumed that all defective items were repairable and that backorders were allowed. Numerous studies have been carried out to address the problems of imperfect quality EPQ model with rework (see, for example, [7], [8], [9], [10], [11] and [12]). Chan et al. [13] presented a new EPQ model with increasing lower pricing, rework and reject situations. Teng et al. [14] studied optimal ordering decisions with returns and excess inventory. Islam and Roy [15] formulated an EPQ model considering flexible and reliable production process and with fuzzy demand-dependent-unit production cost. Bayindir et al. [16] considered the EPQ model with general inventory cost rate function and piecewise linear concave production costs, and proposed an effective solution procedure for deriving the economic order quantity. Hou [17] studied an EPQ model with setup cost and process quality as a function of capital expenditure and developed an efficient procedure to derive the optimal production run time, setup cost, and process quality. Chiu et al. [10] investigated an EPQ model with scrap, rework, and stochastic machine breakdowns to determine the optimal run time and production quantity. Chiu [18] later showed that the same problem can be derived without derivatives. Li et al. [19] developed an EPQ-based model with planned backorders to evaluate the impact of the postponement strategy on a manufacturer in a supply chain. Pentico et al. [20] extended the EPQ model with partial back ordering where the decision variables were production quantity and period length. Teng and Chung [21] considered the EPQ model under two levels of trade credit policy to optimize the production quantity and period length. Chiu et al. [22] considered the effects of random defective rate and imperfect rework process on economic production quantity model. Wee et al. [23] developed an inventory model for items with imperfect quality and shortage backordering. Taleizadeh et al. [24] developed an EPQ model under limited production capacity and scraped items production. Taleizadeh et al. [25] developed an EPQ model with stochastic scraped production rate, partial back ordering and service level constraint. From our literature search, none of the above has so far developed an economic production quantity (EPQ) model with random defective items and failure in repair with capacity constraint. In the case of multi product-single machine systems, Haji et al. [26] studied an imperfect manufacturing process with rework where several products are manufactured on a unique machine. Recently, Widyadana and Wee [27] studied the optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time.
This study develops an EPQ model with production capacity limitation and random defective production rate and failure during repair. Our objective is to determine the optimal period lengths, backordered quantities, and order quantities. The objective function of the proposed numerical model is proved to be convex. Two numerical examples and sensitivity analysis using uniform and normal distribution functions for Xj and θiθi are used to illustrate the practical applications of the proposed methodology. The study provides managerial insights for practitioners in designing an EPQ model with random defective items and failure in repair. Future research should focus on multi-product multi-constraint problems in an uncertain environment.
