Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 957970, 10 pages

http://dx.doi.org/10.1155/2015/957970

## An EPQ Model for Deteriorating Production System and Items with Rework

^{1}Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong^{2}Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 7 December 2014; Accepted 5 February 2015

Academic Editor: Antonio Tornambe

Copyright © 2015 N. Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the economic production quantity model jointly considering product deterioration and a deteriorating production system with rework. In this imperfect deteriorating production system, not only does the machine produce defective product but also the machine is subjected to quality deterioration. To be more specific, the defective rate increases at certain time intervals. The defects produced are stored until the end of normal production process. Then they are reworked with extra cost to restore their quality and regarded as perfect product. The main objective is to minimize the total cost per unit product by determining the optimal combination of production run time and backlog quantity. Numerical experiments are carried out to illustrate the behavior of the inventory and show the impact of different parameters on the model. Discussion and conclusions are made at the end of the paper.

#### 1. Introduction

Inventory control and management are crucial to the modern companies in order to survive in the competitive market. Too much inventory causes high inventory cost and low turnover rate, while having extremely low inventory level puts the company at the risk of losing incoming orders and customers [1]. For manufacturing companies, how many products to produce to meet the demand and keep the cost at low level is always a challenge. Economical production quantity (EPQ) is one of the classical problems in inventory control. The aim of this study is to identify the optimal stocking levels and production quantity to minimize the total cost generated. Nowadays the consideration of different demand type, product deterioration, production system reliability, and other uncertainties makes these classical problems even more complicated. In terms of product deterioration, perishable goods such as food, fruit, and drink and other products like electronic devices and metal processing are the good examples and their quality is heavily influenced by the storing condition and length of storage time after production. If the quality does not meet the standard or requirement anymore, either extra cost is needed to recover the quality or the product will be disposed of [2]. The reliability of production system, on the other hand, addresses the quality of production process such as defective rate, machine breakdown, and production speed. Defects can be generated from processing mistakes, setup mistakes, adjustment mistakes, and tools mistakes [3]. Although management techniques such as quality control and the improvement of manufacturing technology have successfully reduced the defect rate to a relatively low level, it is still a problem for companies, especially when the complexity of production is high. Mobile phone industry can be used as a good example for having extra cost caused by defects [4]. Similar to defective rate, the decrease of production and machine breakdown are also the obstacles to achieving a high productivity in a manufacturing system [5]. Maintenance is an effective method that prevents the serious machine breakdown and restores the production quality. But for deterioration within each production cycle, maintenance cannot solve the problem.

In academic world, the EPQ problem has attracted a large amount of researchers in the past 80 years. Goyal and Giri [6] have provided an in-depth overview of the literatures focusing on the deteriorating products. They have clearly defined the cause of product deterioration and categorized the different models used based on demand type, pricing policy, and so forth; interested reader can refer to it for more information. In 1993 Wee [7] first proposed an EPQ model with partial backlog and product deterioration and later he and Law [8] integrated the problem with the consideration of pricing policy in a finite planning horizon. Goyal and Gunasekaran [9] also combined the pricing problem with EPQ problem on deteriorating product. In addition, advertisement frequency is also considered in their model. The model with time-dependent deteriorating rate was developed by Manna and Chaudhuri [10] as an extension of previous works. In their research, not only is the deteriorating rate time-dependent but also the production rate is proportional to demand rate. All the previously mentioned research work assumed a continuous issuing policy to meet the demand of customers. Discontinuous issuing or delivery policy was taken into consideration by Chiu et al. [11]. Discontinuous issuing policy, as stated in their work, was more practical and common in the industry. In their paper, the delivery of products is divided into installments with a fixed time interval and quantity. Numerical methods were utilized to obtain the optimal solution. Later, they further looked at the discontinuous issuing EPQ problem with partial rework [12]. However in these two papers, the number of shipments is assumed to be known. Cardenas-Barron et al. [13] jointly considered the multidelivery EPQ problem with both lot size and number of deliveries to minimize the inventory cost. Backlog is also an important element in the traditional EPQ problem. Research works such as [14–18] have examined the EPQ problem with full/planned/partial backlog. Interested readers can refer to those papers for more details. Other similar works include Abad [19] and Sarkar [20].

For imperfect manufacturing system, Khouja and Mehrez [21] in 1994 stated that the previous research normally assumed constant production rate and perfect quality. In their work, an EPQ model with variable production rates and imperfect quality was developed where defective rate is the function of production speed. Kim and Hong [22] assumed that the machine changes from in-control to out-control state after a random period and defects are generated in out-control state. Chiu [15] further looked at the problem in which rework of defective products is carried out on the same machine and the impact of reworking on the EPQ model is investigated. Machine breakdown was also taken into consideration. The deterioration of production system was introduced by Lin and Kroll [23] and a time-dependent defective rate was utilized to model the deterioration of machines. In recent studies, random defective rate [24], stochastic breakdown [25], stochastic demand, and maintenance scheduling [26] were also combined with the traditional EPQ problem.

The extant literatures show that majority of the papers have only considered either rework and imperfect system or the product deterioration. The few integrated studies can be found in the work of Teunter and Flapper [27], Inderfurth et al. [28], and Tai [29]. In the first paper, the rework and production were assumed to use the same machine and the defective rate follows a given probability distribution. The production quantity was desired to be determined to maximize the profit obtained. While in the second one, the model proposed in [28] generalized the problem, assuming the demand for good items is limited and the demand will be entirely satisfied. And also closed form results were obtained by utilizing constant and deterministic defective and deteriorating rate, while Tai [29] added inspection errors into the model with deteriorating items and imperfect production. In addition, he also investigated the effect of selling imperfect product to customer. However in all of the three studies mentioned above, the defective rate is regarded as stationary and it would not change along with time. In addition, backlog is not considered in the first two models. Hence as the main contribution of this paper, we model a single machine, single product EPQ problem with the joint consideration of deterioration of product and production systems with rework. Defective rate is assumed to be time-dependent and backlog is allowed in the model. The effects of production deterioration on the system are investigated and analyzed.

The rest of the paper is arranged as follows. Section 2 introduces all the notations and assumptions used in this paper, while the detailed mathematical model is presented and elaborated in Section 3. Numerical experiments are included in Section 4 with sensitivity analysis followed by the conclusions in Section 5.

#### 2. Notations and Assumptions

In this research, a production system with single machine and single product is modelled. In the system, the machine can conduct both production and rework process. The product is subjected to quality deterioration and the machine is assumed to be imperfect and deteriorating in terms of an increasing defective rate. The detailed assumptions made and the notations used in this paper are shown in the following subsections.

##### 2.1. Notations

: the defective rate of time period ; : constant length between each change of defective rate (hour); : deteriorating ratio; : demand rate (unit/time); : production rate (unit/time); : rework rate (unit/time); : the total number of intervals in normal production period; : the total number of intervals in backlog period; : holding cost of perfect products ($/unit/time); : holding cost of imperfect products ($/unit/time); : deteriorating cost ($/unit); : production cost ($/unit); : rework cost ($/unit); : penalty cost for backlog ($/unit/time); : fixed setup cost ($/cycle); : backlog quantity (unit); : economic production quantity (unit); : the inventory level at the end of normal production period (unit); : the inventory level at the end of rework process (unit); : the inventory level of imperfect product (unit).

##### 2.2. Assumptions

(1)Unsatisfied order will be backlogged and the backlog will be fulfilled at the beginning of the cycle.(2)During the production period, the defective items are produced at a constant rate in the time interval , . We assume that the production system is deteriorating in the sense that the defective rate increases at time , . Hence we have . Through this paper, we consider a linear relation , where is a constant.(3)To reduce the complexity of the cost function, the normal production run time and the length of backlog period are assumed to be the integer multiple of since the value of is small.(4)Maintenance is carried out after the whole production period, so at the beginning of each cycle, the defective rate is minimized.(5)The imperfect products are reworked after the normal production process with an extra rework cost and the rework is assumed to be perfect.(6)The deterioration only occurs to perfect products with a constant rate .(7)The deteriorated products are disposed of with cost.(8)Demand rate is known and constant.

#### 3. Mathematical Modelling

According to the assumptions and description of the production system, a mathematical model for the system has been formulated. The behaviour of the inventory level in one production cycle is shown in Figures 1 and 2 for perfect and imperfect products, respectively. As illustrated in Figure 1, the inventory level starts from backlog and the backlogged orders are satisfied first. During the first time interval , the defective rate is maintained as . And at the end of the time interval, the defective rate increases to . After the backlog is made up, the production will be continued until the desired economical production quantity is achieved. All the imperfect products are reworked in and the whole production process is completed. In and , the stocks are consumed by demand and backlog is generated during . Similarly in Figure 2, the total amount of imperfect product piles up in periods and due to the defects produced. The gradient increases along with the rise of defective rate. So in general, the inventory level can be represented with the following equations.