CONOIC PRODUCTION UANTITY ODL WITH CONTINUOU UALITY INPCTION Jia-Chi TOU and Jen-ing CHN Abstract. In this paper, we consider a continuous product quality inspection process in a production line. Three heuristic solutions have been proposed for this problem, and the lot size and total annual cost for each solution have been evaluated. These three solutions are: 1) the maximum inventory method, 2) the safety setup time method, and 3) the economic production quantity method. From our research, the maximum inventory method provides the largest lot size and the safety setup time method gives the smallest. The economic production quantity method can provide the lowest total annual cost. The total annual cost of the safety setup time method and the maximum inventory method depend on other parameters in the production system. 1. Introduction The relationship between quality and the economics of production has been diversely studied over the last decade. Porteus is believed to be the first person who started the study in this field. In his paper, a mathematical model characterizing optimal simultaneous investment in setup cost reduction and quality improvement has been developed [9]. Almost at the same time, Rosenblatt and Lee formulated and analyzed a model similar to that reported by Porteus [10]. The difference is that they do not consider investment in process improvements. In a subsequent paper, Lee and Rosenblatt considered using process inspection during the production run so that the shift to out-of-control state can be detected and restored earlier [8]. In 1987, Tapiero linked the optimal quality inspection policies and the resulting improvements in manufacturing costs [12]. In 1988, Fine used a stochastic dynamic programming model to characterize optimal inspection policies and added the Graduate Institute of Industrial anagement, National Central University, Chung-Li, Taiwan 32054 Graduate Institute of Industrial anagement, National Central University, Chung-Li, Taiwan 32054
quality-based learning effects in the model [5]. In 1989, Fine and Porteus refined the original work of Porteus to allow smaller investments over time with potential process improvement of random magnitude [4]. In 1989, Chand validated Porteus' model when learning effects present in setups and process quality [1]. In a series of papers, Cheng involved the production process reliability into a classic economic order quantity model by 1989 and 1991 [2, 3]. In 1993, Hong et al. established the relationship between process quality and investment [7]. In 2000, alameh and Jaber considered a special production/inventory situation where items, received or produced, were not of perfect quality [11]. In alameh's paper, the quality inspection process is an one time inspection process. In 2001, Ganeshan et al. brought Taguchi s cost of poor quality into the economic production quantity model, and linked lot-size determination to the loss-based quality accounting system [6]. All the papers that we mentioned above focus largely on the linkages between quality improvement cost and production economics such as set-up cost and defect-rate reduction. However, none of them has touched the field of continuous product quality inspection process on the economic of production. In order to fill this gap, this paper focuses on this subject to study the mathematical model of a continuous product quality inspection process. In this paper, we make two significant contributions to the area. Firstly, this is the first paper that studies the operation of a continuous inspection process in a production line. Although some papers like Porteus and Fine are under the assumption of a continuous inspection process, none of them studied the operation of this process [9, 5]. econdly, three heuristic solutions have been proposed for the change in production system. Basing on the classic economic production quantity model, the inventory level of these three solutions has been analyzed. In the next section, the mathematical model of this paper is introduced. Three heuristic solutions are analyzed individually and the comparison of these three solutions is made. Afterward, a numerical example is used to verify the models, and a numerical analysis is made to compare the difference between these three solutions with the conclusion at the end. 2. odel and assumptions Notation P: unit production cost H: holding cost per unit per year C: setup cost per production run. : lot size : economic production quantity in the classic economic production quantity model (quality is perfect) p: production rate r: demand rate R: annual demand in units L: scheduling and production setup time in days q : defect rate
t : time to reach the maximum inventory level in the maximum inventory method : lot size in the maximum inventory method TC : total annual cost in the maximum inventory method : lot size in the safety setup time method TC : total annual cost in the safety setup time method : lot size in the economic production quantity method TC : total annual cost in the economic production quantity method In the beginning, let us consider the classic economic production quantity model without shortages or backorders [13]. There is an inspection station at the end of the production line and the inspection cost has been added into the production cost, P, of each product. H is the holding cost per unit per year, C is the setup cost per production run, is the lot size of each batch, p is the production rate, r is the demand rate for the production line, and L is the scheduling and production setup time in days. The annual demand rate, R, for the product is constant over year. In the classic economic production quantity model, the quality of product is assumed to be perfect and the defect rate, q, is zero. The behavior of the inventory level over time can be drawn as figure 1. Figure 1. Inventory level in the classic P model over time We make the following basic assumptions about the model: 1. No shortage or backorder is allowed. 2. All the variables above are strictly positive. 3. The production rate is larger than the demand rate and the sum of the demand rate and the defect rate. This statement can be written as p r and p ( r pq) The total annual cost, TC (), of the traditional economic production quantity model can be written as [13]: TC( ) CR H( p r) PR (1)
The economic production quantity,, that minimizes the cost of production can be written as [13]: 2CRp H ( p r) (2) After describing the classic P model with perfect product quality, we will make an assumption that some variances occur in the production line. These variances could be the change in raw material quality, change in machine precision or both. These variances affect the quality of product. Defective products are produced and defect rate increases. This statement can be translated to its mathematical form, 1 q 0. We suppose all defective products are detected in the inspection site of the production line. The question is: Facing this change, how does the manufacturer adjust his production line to satisfy demands? In this paper, we propose three heuristic solutions for this problem i.e. 1. aximum inventory method 2. afety setup time method 3. conomic production quantity method 2.1. aximum inventory method When the management chooses the maximum inventory method, the production line will keep producing until the inventory reaches the same level as the highest level of a perfect quality production line. The inventory level of this method has been shown in figure 2. Figure 2. Inventory level in the maximum inventory method ince the defect rate, q, is larger than zero, the increasing rate of inventories will decrease to p-r-pq. The time to reach the inventory level as the highest level of a perfect quality production line can be written as t ( p r) 1 p ( p r pq) The lot size in the maximum inventory method can be expressed as (3)
t p Because r and pq are strictly positive and ( p r) ( p r pq) concluded that 1 and. ( p r) ( p r pq) (4) p r and p ( r pq), it can be The total annual cost in the maximum inventory method can be expressed as TC PR CR( p r pq) H ( p r) (1 q) (1 q)( p r) (5) 2.2. afety setup time method In this method, the manufacturer keeps the lot size in each batch as before and prepares to produce next batch as the inventory to reach the production reorder point. This point is the inventory amount that is enough to support the demand during the time to prepare for the next production. The production reorder point is equal to r L, where L is the production setup time and r is the demand rate. The behaviour of the inventory level in the safety setup time method can be drawn as figure 3. Figure 3. Inventory level in the safety setup time method In this method, the lot size is the same as the lot size in a perfect quality production line. This can be written as The total annual cost in the safety setup time method can be expressed as TC PR CR H ( p r pq) (1 q) (1 q) (7) (6)
2.3. conomic production quantity method In this method, the manufacturer accounts the economic production quantity again in the new production situation. The total annual cost of the economic production quantity method can be written as a function of : PR CR H( p r pq) TC ( ) (1 q) (1 q) 2p (8) To obtain the economic production quantity,, take the first derivative of the total annual cost with respect to the production quantity,, and set it equal to zero: dtc (9) ( ) d CR H( p r pq) (1 q) 2p 2 olving equation (9) for, we can find the minimum cost production order quantity, as CR ( p r) H(1 q)( p r pq) (1 q)( p r pq) (10) ince ( p r) ( p r pq) and 1 q 1 ( p r) 1 (1 q)( p r pq), we get: and ubstituting into equation (8), it can be found that the total annual cost in the economic production quantity method can be expressed as 0 TC PR CR (1 q) ( p r pq) H ( p r)(1 q) 2p ( p r)( p r pq) (1 q) (11) The behavior of the inventory level in the economic production quantity method can be drawn as in figure 4.
Figure 4. Inventory level in the economic production quantity method 2.4. Comparison of the three methods In this section, we compare the lot size and the total annual cost of the three methods. In sections 2.1 and 2.3, it has already been found that and equation (6),. Hence it can be concluded that euqation (4) from equation (10), we get:. From,. ubtracting ( p r) ( ( p r pq) p r p r pq 1 ) 1 q (12) ince values of and and and ( p r) ( p r pq), the value of p r p r pq p r 1 p r pq 1 q >0, and, 1 1 q >1, to compare the should be accounted first. p r 1 rq 0 p r pq 1 q ( p r pq)(1 q) From equations (12) and (13), it can be concluded that,, it can be found that (13). ince (14)
To compare the total annual cost of the three methods, we start from the total annual cost of the economic production quantity method, TC. ince TC is the minimum value of the total annual cost, it can be concluded that TC, TC TC. To compare TC and as TC TC, we subtract TC from TC TC H pq( TC CR ) (1 q). Using equations (5) and (7), it can be calculated PR CR( p r pq) H ( p r) PR CR H ( p r pq) ( ) ( ) (1 q) (1 q)( p r) (1 q) (1 q) H ( CR (1 q) From equation (15), if ) 0, then TC TC. H If ) 0, then ( CR (1 q) (15) TC TC, and TC TC, and TC TC TC. 3. An example In order to verify the method and relationship we proposed, a numerical example is used to demonstrate the usefulness of these three methods. Horsy Industry Co. is a manufacturer of automotive shock absorbers. There is a continuous inspection site integrated to its production line, and the inspection cost has been added into the production cost of each product. The production cost of each shock absorber is $7. The setup cost is $500 per batch. The setup cost includes the cost to change the mold and adjust the material in the production line. The holding cost per absorber is $er year. The production rate is 2000 units per day, and the demand rate is 500 units per day. The annual demand is 100000 units. There are 250 working days in a year. The scheduling and production setup time is 3 days. In the initial stage, we propose that the quality is perfect and no defective products are produced in the production line. From the classic economic production quantity model, the economic production quantity,, can be calculated as 2CRp H( p r) 2500100000 2000 2 (2000 500) 8165 units/batch When the batch size maintains the economic production quantity, the total annual cost in this production line can be calculated as
CR H ( p r) 500100000 28165 (2000 500) TC( ) PR 7 100000 8165 2 2000 $712247 / year Now, we assume that some unknown variances occur in the production line and the defect rate of product increases from 0 to 3%. All defective parts have been detected by the continuous inspection site in the production line. How does the management adjust its production capacity to manage this situation? According to the solution we proposed, there are three different ways. 3.1. aximum inventory method In the maximum inventory method, the management can keep producing until inventories reach the highest level as the defect rate is zero. From equation (4), the lot size in this method can be accounted as ( p r) (2000 500) 8165 8505 ( p r pq) (2000 500 2000 0.03) ch Using equation (5), the total annual cost in this method can be accounted as: 28165 (2000 500) $733833.8/ year 2 2000 units/bat PR CR( p r pq) H ( p r) 7 100000 500100000 (2000 500 2000 0.03) TC (1 q) (1 q)( p r) (1 0.03) 8165 (1 0.03) (2000 500) 3.2. afety setup time method In this method, the manufacturer keeps the lot size in each batch the same as the defect rate is zero and prepares to produce next batch as the inventory reaches the production reorder point r L. In this case, r L 5003 1500 units. ince the lot size is the same as the production line with zero defect rate. The lot size in this method,, is equal to 8165 units/batch. From equation (7), the total annual cost can be found as PR CR H ( p r pq) 7 100000 500100000 TC (1 q) (1 q) (1 0.03) 8165 (1 0.03) 28165 (2000 500 2000 0.03) $733841/ year 2 2000
3.3. conomic production quantity method In the economic production quantity method, the management accounts the economic production quantity again in the new production situation. From equation (10), we can find that the lot size is equal to ( p r) (1 q)( p r pq) 8165 (2000 500) (1 0.03) (2000 500 2000 0.03) 8461 Based on equation (11), the total annual cost is equal to PR CR TC (1 q) ( p r pq) H ( p r)(1 q) ( p r)( p r pq) 7 100000 (1 q) (1 0.03) 500100000 (2000 500 2000 0.03) 28165 8165 (2000 500) (1 0.03) 2 2000 $733833.64 / year (2000 500) (2000 500 2000 0.03) (1 0.03) 4. ummary and Conclusion In this paper, we consider a continuous product quality inspection process in a production line. The inspection site in the production line can detect the quality of defective products. Three heuristic solutions have been proposed for management when defect products were found in the production line. The inventory level and lot size of these three different methods have been studied. From our analysis, the maximum inventory method provided the largest lot size and the safety setup time method gave the smallest. The total annual costs of three different solutions have also been discussed. The economic production quantity method can produce the smallest total annual cost. The total annual cost of the safety setup time method and the maximum inventory method depend on the parameters of the production system. References [1] Chand,, Lot sizes and setup frequency with learning in setups and process quality, uropean Journal of Operations Research, 42, 1989, 190-202. [2] Cheng, T.C.., An economic production quantity model with flexibility and reliability considerations, uropean Journal of Operations Research, 39, 1989, 174-179. [3] Cheng, T.C.., conomic order quantity model with demand-dependent unit production cost and imperfect production processes, II Transactions, 23, 1991, 23-28. [4] Fine, C.H., and Porteus.. L., Dynamic process improvement, Operations Research, 37, 1989, 580-591.
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