- Email: [email protected]
Study on CODP Position of Process Industry Implemented Mass Customization
Systems Engineering - Theory & Practice Volume 27, Issue 12, December 2007 Online English edition of the Chinese language journal Cite this article as: SETP, 2007, 27(12): 151–157
Study on CODP Position of Process Industry Implemented Mass Customization JI Jian-hua, Qi Li-li, GU Qiao-lun Antai College of Economics & Management, Shanghai Jiaotong University, Shanghai 200052, China
Abstract: As the core strategy of MC (Mass Customization), CODP (Customer Order Decoupling Point) positioning of the process industry implemented mass customization is the focus of this article. The article builds a CODP positioning model with the delivery lead time constraint and capacity constraint, aiming at total cost minimization. Based on the analytical and numerical comparisons conducted by MATLAB, this article presents 4 inferences and 4 observations and gains insights on how various factors affect the CODP position. Key Words: process industry; mass customization; CODP; postponement strategy
1 Introduction At present, to implement mass customization in the manufacturing, whether it is a continuous process or discrete product manufacturing, we need to determine the Customer Order Decoupling Point (CODP). CODP is the breaking point between productions for stock based on forecast and customization that respond to customer demand. It is also the breaking point between MTS and MTO, namely, activities before CODP are driven by forecast while activities after CODP are driven by real customer order demand. As to the process industry, there are two characteristics in choosing CODP: firstly, the production process of the process industry is very complex and the quantities of product differentiation points are very limited. In the discrete industry, product differentiation points may appear in each stage, but in the process industry, they are usually related to the production technology and often occur in the storage link, raw material adding link, and so on. Secondly, more factors should be considered. The setup time in the process industry is long and the requirements for equipment production load are rather strict. In addition, the raw materials, work in process, and the finished products are often perishable goods; therefore, the storage conditions are very demanding. Thus, there are several considerations in choosing CODP in the process industry, such as production technology cost, customer service level, production utilization rate, and the requirements of work in process for storage conditions and time. To implement mass customization, we usually partly change products or production technology, which is named as re-manufacturing. In the literatures abroad on CODP, Aviv[1] analyzed the influence of the postponement strategy on multi-product inventory system with production capacity constraints, but he did not suggest a position model of CODP. Su[2] built models based on the Queuing Theory and offered a comparative analysis between form postponement and time post-
ponement according to the measure of the two performance indexes: total cost and customer waiting time. Diwakar[3] studied the costs and incomes caused by the postponement strategy based on the Queuing Theory, put forward the optimal position model of CODP, and constructed an approximate solution. Lee[4] considered that the position of CODP was related to not only inventory cost, but also to the disposal cost, the investment cost, and so on, through model results analysis. The main consideration of his model was cost but the lead time constrains were not involved. On the basis of literature [1], Gara[5] conducted a research on multidifferentiation points with decentralized inventory control strategy and centralized inventory control strategy, but his model only considered inventory cost and supposed that there was no limit on the inventory. Lead time constrains were also not involved in his article. Aviv[6] analyzed the benefits brought about by the postponement strategy with uncertain distribution of demand and made quantitative analysis on the benefits brought by the postponement strategy with different order costs. However, he did not consider the production capacity constraints and lead time constrains. All the literatures above focus on the discrete industry. In the literatures on CODP in China, Qi[7] presented that CODP should be moved to downstream as far as possible so as to reduce the costs in design, manufacture, assembly, etc., which were caused by meeting customers’ special requirements. Liu[8] considered that the position of CODP was related to the scale of postponement activity, postponement type, and customization degree; besides, enterprise operation features will also affect the position of CODP. Shao[9] built models with the following considerations such as lead time constrains, decentralized inventory control strategy, and centralized inventory control strategy. He presented that it was not better to move CODP to downstream as far as possible and the position of CODP was impacted by demand, production cycle of each stage, and inventory cost. These
Received date: September 26, 2007 ∗ Corresponding author: Tel: +86-21-62932696; E-mail: [email protected] Foundation item: Supported by the National Nature Science Foundation (No.70472030) c 2007, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
Figure 1. Position of CODP
literatures focus on the discrete industry as well. The relevant researches and models on CODP positioning of the process industry are rather rare. Lian[10] took it as the study direction of her doctorial paper and conducted the primary research. Thus, this article built a CODP positioning model with the delivery lead time constraints and capacity constraints, aiming at the minimization of the total cost, according to the characteristics of the process industry, and on this basis, the impact factors were analyzed and some relevant suggestions were put forward.
2 Problem description and model assumptions The entire production logistics system of the process industry in this article is shown in Figure 1. There are K independent stages in the production system. Each stage can process only one unit at a time. Also, there are no setup costs or setup times. No inventories of finished products exist in the system. The products will be delivered to customers as soon as possible after completion. As to work in process, the basic inventory strategy was adopted, namely, as long as the inventory level is not met, replenishment will be ongoing to achieve the determined inventory level. Suppose K as an alternative point, where production can be interrupted. In the discrete industry, k can be any integer between 1 to K, but in the process industry, because of non-splittability of conditions of work in process, and technology constrains, there will not be any interrupt point between some stages. As a result, CODP cannot be set up under this condition. Therefore, only available interrupt points can be involved in the model of this article. Model assumptions: Assumption 1 The quality of finished products is independent of the CODP position. Assumption 2 The delivery lead time and unit inventory cost of work in process in each stage are independent of the CODP position. Assumption 3 The work in the process inventory in stage j is independent of the CODP position[11] . Assumption 4 The delivery lead time is the same for each customer order and all customer orders are handled in accordance with the “first come, first served” principle. Assumption 5 There are no setup costs or setup times. Assumption 6 No inventories of finished products exist in the system. The products will be delivered to customers as soon as possible after completion. Assumption 7 The buffer inventory of work in process is set up in CODP, and before CODP, there are no buffer inventories. Model formulation and notation: i: Types of finished products, i = 1, 2, · · · , M , the demand of product i is Poisson distributed; λi : The arrive rate;
Λ: The demand rate of all the products; J: The stage set, J = {1, 2, · · · , K}; K: Alternative CODP; µ: The service rate at stage j; Tj : The unit production times at each stage, which are exponentially distributed and the average E(Tj ) = µ1j ; ρ: Capacity utilization rate, and the capacity utilization rate of stage j is ρj = ΛE(Tj ), ρj ≤ 1; b: The basestock level; Z(k, b): The minimum cost to implement mass customization when the basestock level of work in process is b and CODP is set up in stage k; h(k): The inventory cost of unit work in process in unit time, when CODP is set up in stage k; ¯ b): The average inventory of work in process when I(k, the inventory level is b and CODP is set up in stage k; c(k): The re-manufacturing cost of products/process, when CODP is set up in stage k; F (k, b): The postponing time for order delivery when the basestock level is and CODP is set up in stage k. F¯ (k, b) is the average delivery postponing time; α: A determined parameter, α > 0; r: The quantity of work in process in the CODP k.
3 3.1
CODP position model Model
The service objective of the production material system in the process industry is to minimize system cost under certain service level. The model is as follows: ¯ b) + c(k) min Z(k, b) = h(k)I(k, s.t. F¯ (k, b) ≤ α k≤K k, b ≥ 0
(1) (2) (3) (4)
The objective function (1) represents achieving the goal of minimum cost to implement mass customization when the basestock level of work in process is b and CODP is set up in stage k; Both k and b are independent variables. Under certain service level, the costs of implementing mass customization include the inventory cost and the prod¯ b) + uct/process re-manufacturing cost, Z(k, b) = h(k)I(k, ¯ c(k). h(k)I(k, b) represents the inventory cost of work in process, when the basestock level is and CODP is b set up in stage k. In constraint condition (2), F¯ (k, b) ≤ α denotes that the average delivery postponing time must be not more than α, when the basestock level is b and CODP is set up after stage k. α indexes the constraint on the postponing time, which is the measure standard of the customer service level. The lesser the α, the higher will be the customer service level, namely, the enterprise can deliver the customized products to the customers very fast and achieve high customer satisfaction. Constraints (3) and (4) indicate the value ranges of parameters. ¯ b) is very difficult. In the model above, calculating I(k, To solve this problem, we introduce other intermediate variables. Literature [11] summarized the classic tandem-queue
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
models and designed an inventory model, which could be applied in MTSMTOMTS-MTO production processes. In this model, intermediate and finished goods could be produced and stored in advance of demand, and demand was a Poisson process. The unite production times were exponentially distributed. The orders were handled in accordance with the “first come, first served” principle. Each stage was considered as an M/M/1 queuing system. The assumptions above are consistent with those in this article. Literature [11] proposed a tractable approximation scheme and conducted computer simulation to test the results. In the following, we will refer to this approximate solution method. In the literature [11], the formula for calculating the average inventory of work in process is Nk = b − Ik + Gk . Nk represents the quantity of work in process in stage k; Ik is the output-buffer inventory after stage k; Gk indexes the outstanding backorders at stage k. Nk = b − Ik + Gk is the core of the model. We change the equation as follows: Nk + Ik − Gk = b, which represents that the quantity of work in process at stage k. Nk plus out put-buffer inventory Ik minus Gk , which is outstanding backorders at stage k is the basestock level b. It indicates that in unit time, the buffer basestock level in the CODP remains unchanged. To calculate the value of Ik , we adopted an approximate solution method as follows: Define the j × j matrices Cj and Pj ⎞ ⎛ −ν1 ν1 ⎟ ⎜ −ν2 ν2 ⎟ ⎜ ⎟ (5) ⎜ · · · Cj = ⎜ ⎟ ⎝ −νj−1 νj−1 ⎠ νj Hereinto, νj = µj − λj = µj (1 − ρj ), Pj = Λ(ΛI − Cj )−1 , and I denotes an identity matrix. Define row vector γ = (1, 0, · · · , 0), j = 1, · · · , k − b b , (1 − γk−1 Pk−1 e)], j = k, where, e is 1; γk = [γk−1 Pk−1 unit column vector; γj+1 = [γj , (1 − γj e)], j = k, · · · , K − 1. Set πj = γj Pj in literature [11]; Gj is the outstanding backorders at stage k. When positioning CODP in stage k with the basestock level of work in process b, the mean backorder is given by: j = k πj (1 − Pj )−1 e ¯j = (6) G πk Pkb (1 − Pk )−1 e j = k The average postponing time can be considered as F¯j ≈ G¯j /Λ The mean output-buffer inventory is 0 j = k I¯j = ¯k j = k ¯k + G b−N N¯k = πk (1 − Pk )−1 e
(7)
(8) (9)
Based on the formulas above, we can obtain the optimal result, but the solving process is extremely complicated. Literature [3] put forward that in the MTS production process, the quantity of work in process followed the general
binomial distribution. In that case, the calculation of mean backorder, the average postponing time, and the mean output-buffer inventory can be simplified. Therefore, to simplify the calculation, suppose that the quantity of work in process follows the general binomial distribution in the MTS production process. The mean backorder, the average postponing time, and the mean out-putbuffer inventory can be denoted as the following formulas: ¯ b) = I(k,
b (b − r)P r(Nk = r)
(10)
r=0
P r(Nk = r) represents the possibility of the quantity of work in process being r in stage k while positioning CODP in stage k.
Pr (Nk = r) =
⎧ r + k − 1 ⎪ ⎪ (1 − ρ)k ρ, ⎪ ⎪ k ⎨
ρj = ρ
k ρr+k−1 Πki=1 (1 − ρi ) ⎪ j ⎪ ⎪ , ⎪ ⎩ Πi=j (ρj − ρi )
others
j=1
¯ b) ≈ I(k, ¯ b)+ Since Nk +Ik −Gk = b, G(k, K
j=1
ρj 1−ρj
−
¯ b)/Λ, the model can be transformed as follows: b, F¯j ≈ G(k, min Z(k, b) = h(k)
b k ρr+k−1 j (b − r) r=0
j=1
k
i=1 (1
− ρi ) + c(k) i=j (ρj − ρi )
(11)
s.t. ⎡ ⎤ k b k K ρr+k−1 (1−ρ ) ρ j j j i=1 ⎣ (b−r) −b⎦ Λ ≤ α + 1−ρj i=j (ρj−ρi ) r=0 j=1 j=1 (12) k≤K
(13)
k, b ≥ 0
(14)
3.2
Approach According to the formulas above, if the value of k, unit inventory cost, and re-manufacturing cost are determined, the optimal value of b can be obtained, which will lead to the minimum cost with the service constrains. It can be denoted as b∗ . The approach to get the optimal value of k (k ∗ ) is given below: • Choose the available value of k and begin with the smallest one; then apply it into the model Owing to the characteristics of the process industry, some stages cannot be chosen as CODP. Take Oil Refinery as an example; the production cannot be interrupted in some links. Once a certain link is interrupted, the entire production line will be stopped and the losses may be up to one million RMB. • Calculate the optimal b(k) and z(k, b(k)). If k = K, turn to step 4 • k = k + 1, (k = 1, 2, · · · , K); if k is an available point, then turn to step 2, otherwise repeat step 3
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
• Find the value of k, which leads to the minimum value of z(k, b(k)), and denote it as k ∗ , which is the optimal position of CODP.
4 Numerical results In this section, we use MATLAB to simulate the effects of each factor on the optimal position of CODP k ∗ and the optimal basestock level b∗ . Suppose there are 10 stages and the capacity utilization rate of each stage is 0.8. Based on the simulation results, we obtain the following corollaries: Corollary 4.1 The optimal CODP k ∗ and the optimal basestock level b∗ decrease with increasing α (Figure 2). Table 1 provides the specific data of some typical points in Figure 2. From Table 1, it is obvious that when the value of is small, the postponing time to complete the order is controlled strictly. Therefore, the CODP has to move to the downstream and a large number of stocks of work in process are indispensable, which results in high cost. When the value of increases gradually, the CODP moves upstream and the optimal basestock level declines. The total cost decreases sharply. Therefore, the customer service level influences the optimal position of CODP and the optimal basestock level at CODP directly. Corollary 4.2 With the increase of unit inventory cost in CODP and the position of CODP moving downstream, the optimal basestock level tends to become smaller. (Figure 3)
Owing to the restrictions on technology and other factors, different enterprises implement mass customization with different growth rates of cost. The enterprise with flexible production and advanced technology costs far lesser to achieve mass customization. As the production lines are widely used in the process industry and the production processes are relatively rigid, to achieve mass customization, the enterprises need to rebuild or even renew equipments or re-develop formulations. Therefore, the impact of remanufacturing cost on the position of CODP in the process industry is larger than that in discrete industry. Corollary 4.3 shows a strong evidence of the relationship between the position of CODP and the re-manufacturing cost. Corollary 4.4 In the MC system, the utilization rate of equipment in the MTS production phase, which is before CODP should be relatively high, while that in the MTO production phase after CODP should not be very high. In terms of production layout, the equipments with high fix investment or those that need high utilization rate or full load running should be arranged before CODP. The specific analysis is shown below: As seen in Figure 2, there are five stages. When k ∗ = 2, the first two stages are operated in the MTS phase and the other three stages are operated in the MTO phase. Table 2 reports the simulation results. Comparing the results of different conditions and analyzing the impact of utilization rate on the optimal basestock level and total cost, we can make the following observations:
Corollary 4.3 With the increase of the product/process re-manufacturing cost, the optimal position of CODP moves upstream and the optimal basestock level tends to become larger. (Figure 4)
Table 1. Relationship of optimal position of CODP and optimal basestock level with service level
α 0.2 0.4 0.5 1 2 10
k∗ 8 5 3 3 1 1
b∗ 69 57 46 43 40 27
Z∗ 726 560 97 81 6 4
Figure 2. Relationship of optimal position of CODP and
Figure 3. Relationship of optimal position of CODP and optimal basestock level with unit inventory cost (K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
Figure 4. Relationship of optimal position of CODP and optimal
optimal basestock level with service level
basestock level with product/process re-manufacturing
(K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
cost (K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
Table 2. Relationship among the optimal basestock level, minimum cost, and production utilization rate
Condition 1 2 3 4 5 6 7 8 9 10 11 12 13
(ρ1 , ρ2 , ρ3 , ρ4 , ρ5 ) (0.9,0.8,0.7,0.6,0.5) (0.9,0.7,0.8,0.6,0.5) (0.9,0.6,0.8,0.7,0,5) (0.9,0.5,0.8,0.7,0.6) (0.8,0.9,0.7,0.6,0.5) (0.7,0.6,0.9,0.8,0.7) (0.5,0.6,0.9,0.8,0.7) (0.5,0.6,0.8,0.7,0.9) (0.5,0.6,0.8,0.7,0.9) (0.5,0.6,0.7,0.8,0.9) (0.5,0.6,0.55,0.45,0.35) (0.5,0.6,0.95,0.45,0.35) (0.5,0.6,0.95,0.9,0.45)
b∗ 2 2 2 2 2 2 17 17 17 17 2 23 31
Z∗ 3.3465 3.3465 3.426 3.9505 3.3465 3.585 48.848 22.3562 22.3562 9.2419 5.4536 69.8346 84.4971
Observation 1 with a decrease of utilization in the MTS production phase, the optimal basestock level in CODP tends to become larger and the cost tends to become higher. (See Table 2 conditions 1 to 7) This observation is reasonable. If the scale production can be achieved in the MTS production phase, the cost of products can be reduced. If the utilization rate in MTS production phase declines, there is no doubt that the total cost will increase. Observation 2 Changing the sequence of utilization rate among stages in MTS production phase will not affect the optimal basestock level in CODP and the total cost. (See Table 2 conditions 1 and 2) Observation 3 When the utilization rates of stages in the MTS production phase remain unchanged and those in the MTO production phase increase, both the optimal basestock level and the total cost tend to become lager. (See Table 2 conditions 11 to 13) Observation 4 Changing the sequence of utilization rate among stages in the MTO production phase will not influence the optimal basestock level in CODP but will impact the total cost. The closer to CODP the equipment with the maximum utilization rate is, the greater will be the increase in the total cost. (See Table 2 conditions 1 to 10). Based on these four observations, the article puts forward Corollary 4.4. These observations and Corollary 4.4 have huge operational significance for the enterprises to implement mass customization. In the process industry, various types of equipments have to be run with full load or high load because of their high fix investment or other considerations such as security and technology. When implementing mass customization, these equipments with high utilization rate should be arranged after CODP, namely, CODP cannot be positioned before the bottleneck of the production system.
In addition, the utilization rate of equipments arranged after CODP cannot be very high.
5
Conclusions
Implementing mass customization is one approach for enterprises in the process industry to improve their competitive capacity. Taking the process industry implementing mass customization as a background, this article presented the model definitions and relevant assumptions, and then established a CODP positioning model with the delivery lead time constraint and capacity constraint, aiming at the minimization of the total cost. At last, software Matlab was used to get numerical results and obtain some valuable corollaries. In the future research, the CODP positioning model can be extended in the following aspects: customization level influences not only price but also demand; by-products exist in the production process; there are buffer inventories in the interrupt point after CODP.
References [1] Aviv Y, Federgruen A. Capacitated multi-item inventory systems with random and seasonally fluctuating demands: Implications for postponement strategies. Management Science, 2001, 47(4): 512–532. [2] Su J C P, Chang Y L, Ferguson M. Evaluation of postponement structures to accommodate mass customization. Journal of Operations Management, 2004, 23: 305–318. [3] Diwakar G, Benjaafar S. Make-to-order, make-to-stock, or delay product differentiation? A common framework for modeling and analysis. IIE Transactions, 2004, 36: 529–546. [4] Lee H L, Tang C S. Modeling the costs and benefits of delayed product differentiation. Management Science, 1997, 1: 40–53. [5] Gara A, Tang C. On postponement strategies for product families with multiple points of differentiation. IIE Transaction, 1997, 29(8): 641–650. [6] Aviv Y. Design for postponement: A comprehensive characterization of its benefits under unknown demand distributions. Operations Research, 2001, 49(4): 578–598. [7] Qi G N, Gu X J, Li R W. Study on mass customization and its models. Computer Integrated Manufacturing Systems, 2000, 10(2): 41–45. [8] Liu J. Study on the positioning and form selection of decoupling point in postponed manufacturing. Jiangsu Commercial Forum, 2004, 1: 69–70. [9] Shao X F, Wang R, Huang P Q, Ji J H. Research on postponement in mass customization with multiple product differentiation points. Systems Engineering — Theory Methodology Application, 2001, 10(4): 332–336. [10] Lian H J. Mass customization oriented research on production planning of process industry. Doctoral Thesis of Shanghai Jiao Tong University, 2006. [11] Lee Y J, Zipkin P. Tandem queues with planned inventories. Operations Research, 1992, 40(5): 936–947.
Study on CODP Position of Process Industry Implemented Mass Customization JI Jian-hua, Qi Li-li, GU Qiao-lun Antai College of Economics & Management, Shanghai Jiaotong University, Shanghai 200052, China
Abstract: As the core strategy of MC (Mass Customization), CODP (Customer Order Decoupling Point) positioning of the process industry implemented mass customization is the focus of this article. The article builds a CODP positioning model with the delivery lead time constraint and capacity constraint, aiming at total cost minimization. Based on the analytical and numerical comparisons conducted by MATLAB, this article presents 4 inferences and 4 observations and gains insights on how various factors affect the CODP position. Key Words: process industry; mass customization; CODP; postponement strategy
1 Introduction At present, to implement mass customization in the manufacturing, whether it is a continuous process or discrete product manufacturing, we need to determine the Customer Order Decoupling Point (CODP). CODP is the breaking point between productions for stock based on forecast and customization that respond to customer demand. It is also the breaking point between MTS and MTO, namely, activities before CODP are driven by forecast while activities after CODP are driven by real customer order demand. As to the process industry, there are two characteristics in choosing CODP: firstly, the production process of the process industry is very complex and the quantities of product differentiation points are very limited. In the discrete industry, product differentiation points may appear in each stage, but in the process industry, they are usually related to the production technology and often occur in the storage link, raw material adding link, and so on. Secondly, more factors should be considered. The setup time in the process industry is long and the requirements for equipment production load are rather strict. In addition, the raw materials, work in process, and the finished products are often perishable goods; therefore, the storage conditions are very demanding. Thus, there are several considerations in choosing CODP in the process industry, such as production technology cost, customer service level, production utilization rate, and the requirements of work in process for storage conditions and time. To implement mass customization, we usually partly change products or production technology, which is named as re-manufacturing. In the literatures abroad on CODP, Aviv[1] analyzed the influence of the postponement strategy on multi-product inventory system with production capacity constraints, but he did not suggest a position model of CODP. Su[2] built models based on the Queuing Theory and offered a comparative analysis between form postponement and time post-
ponement according to the measure of the two performance indexes: total cost and customer waiting time. Diwakar[3] studied the costs and incomes caused by the postponement strategy based on the Queuing Theory, put forward the optimal position model of CODP, and constructed an approximate solution. Lee[4] considered that the position of CODP was related to not only inventory cost, but also to the disposal cost, the investment cost, and so on, through model results analysis. The main consideration of his model was cost but the lead time constrains were not involved. On the basis of literature [1], Gara[5] conducted a research on multidifferentiation points with decentralized inventory control strategy and centralized inventory control strategy, but his model only considered inventory cost and supposed that there was no limit on the inventory. Lead time constrains were also not involved in his article. Aviv[6] analyzed the benefits brought about by the postponement strategy with uncertain distribution of demand and made quantitative analysis on the benefits brought by the postponement strategy with different order costs. However, he did not consider the production capacity constraints and lead time constrains. All the literatures above focus on the discrete industry. In the literatures on CODP in China, Qi[7] presented that CODP should be moved to downstream as far as possible so as to reduce the costs in design, manufacture, assembly, etc., which were caused by meeting customers’ special requirements. Liu[8] considered that the position of CODP was related to the scale of postponement activity, postponement type, and customization degree; besides, enterprise operation features will also affect the position of CODP. Shao[9] built models with the following considerations such as lead time constrains, decentralized inventory control strategy, and centralized inventory control strategy. He presented that it was not better to move CODP to downstream as far as possible and the position of CODP was impacted by demand, production cycle of each stage, and inventory cost. These
Received date: September 26, 2007 ∗ Corresponding author: Tel: +86-21-62932696; E-mail: [email protected] Foundation item: Supported by the National Nature Science Foundation (No.70472030) c 2007, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
Figure 1. Position of CODP
literatures focus on the discrete industry as well. The relevant researches and models on CODP positioning of the process industry are rather rare. Lian[10] took it as the study direction of her doctorial paper and conducted the primary research. Thus, this article built a CODP positioning model with the delivery lead time constraints and capacity constraints, aiming at the minimization of the total cost, according to the characteristics of the process industry, and on this basis, the impact factors were analyzed and some relevant suggestions were put forward.
2 Problem description and model assumptions The entire production logistics system of the process industry in this article is shown in Figure 1. There are K independent stages in the production system. Each stage can process only one unit at a time. Also, there are no setup costs or setup times. No inventories of finished products exist in the system. The products will be delivered to customers as soon as possible after completion. As to work in process, the basic inventory strategy was adopted, namely, as long as the inventory level is not met, replenishment will be ongoing to achieve the determined inventory level. Suppose K as an alternative point, where production can be interrupted. In the discrete industry, k can be any integer between 1 to K, but in the process industry, because of non-splittability of conditions of work in process, and technology constrains, there will not be any interrupt point between some stages. As a result, CODP cannot be set up under this condition. Therefore, only available interrupt points can be involved in the model of this article. Model assumptions: Assumption 1 The quality of finished products is independent of the CODP position. Assumption 2 The delivery lead time and unit inventory cost of work in process in each stage are independent of the CODP position. Assumption 3 The work in the process inventory in stage j is independent of the CODP position[11] . Assumption 4 The delivery lead time is the same for each customer order and all customer orders are handled in accordance with the “first come, first served” principle. Assumption 5 There are no setup costs or setup times. Assumption 6 No inventories of finished products exist in the system. The products will be delivered to customers as soon as possible after completion. Assumption 7 The buffer inventory of work in process is set up in CODP, and before CODP, there are no buffer inventories. Model formulation and notation: i: Types of finished products, i = 1, 2, · · · , M , the demand of product i is Poisson distributed; λi : The arrive rate;
Λ: The demand rate of all the products; J: The stage set, J = {1, 2, · · · , K}; K: Alternative CODP; µ: The service rate at stage j; Tj : The unit production times at each stage, which are exponentially distributed and the average E(Tj ) = µ1j ; ρ: Capacity utilization rate, and the capacity utilization rate of stage j is ρj = ΛE(Tj ), ρj ≤ 1; b: The basestock level; Z(k, b): The minimum cost to implement mass customization when the basestock level of work in process is b and CODP is set up in stage k; h(k): The inventory cost of unit work in process in unit time, when CODP is set up in stage k; ¯ b): The average inventory of work in process when I(k, the inventory level is b and CODP is set up in stage k; c(k): The re-manufacturing cost of products/process, when CODP is set up in stage k; F (k, b): The postponing time for order delivery when the basestock level is and CODP is set up in stage k. F¯ (k, b) is the average delivery postponing time; α: A determined parameter, α > 0; r: The quantity of work in process in the CODP k.
3 3.1
CODP position model Model
The service objective of the production material system in the process industry is to minimize system cost under certain service level. The model is as follows: ¯ b) + c(k) min Z(k, b) = h(k)I(k, s.t. F¯ (k, b) ≤ α k≤K k, b ≥ 0
(1) (2) (3) (4)
The objective function (1) represents achieving the goal of minimum cost to implement mass customization when the basestock level of work in process is b and CODP is set up in stage k; Both k and b are independent variables. Under certain service level, the costs of implementing mass customization include the inventory cost and the prod¯ b) + uct/process re-manufacturing cost, Z(k, b) = h(k)I(k, ¯ c(k). h(k)I(k, b) represents the inventory cost of work in process, when the basestock level is and CODP is b set up in stage k. In constraint condition (2), F¯ (k, b) ≤ α denotes that the average delivery postponing time must be not more than α, when the basestock level is b and CODP is set up after stage k. α indexes the constraint on the postponing time, which is the measure standard of the customer service level. The lesser the α, the higher will be the customer service level, namely, the enterprise can deliver the customized products to the customers very fast and achieve high customer satisfaction. Constraints (3) and (4) indicate the value ranges of parameters. ¯ b) is very difficult. In the model above, calculating I(k, To solve this problem, we introduce other intermediate variables. Literature [11] summarized the classic tandem-queue
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
models and designed an inventory model, which could be applied in MTSMTOMTS-MTO production processes. In this model, intermediate and finished goods could be produced and stored in advance of demand, and demand was a Poisson process. The unite production times were exponentially distributed. The orders were handled in accordance with the “first come, first served” principle. Each stage was considered as an M/M/1 queuing system. The assumptions above are consistent with those in this article. Literature [11] proposed a tractable approximation scheme and conducted computer simulation to test the results. In the following, we will refer to this approximate solution method. In the literature [11], the formula for calculating the average inventory of work in process is Nk = b − Ik + Gk . Nk represents the quantity of work in process in stage k; Ik is the output-buffer inventory after stage k; Gk indexes the outstanding backorders at stage k. Nk = b − Ik + Gk is the core of the model. We change the equation as follows: Nk + Ik − Gk = b, which represents that the quantity of work in process at stage k. Nk plus out put-buffer inventory Ik minus Gk , which is outstanding backorders at stage k is the basestock level b. It indicates that in unit time, the buffer basestock level in the CODP remains unchanged. To calculate the value of Ik , we adopted an approximate solution method as follows: Define the j × j matrices Cj and Pj ⎞ ⎛ −ν1 ν1 ⎟ ⎜ −ν2 ν2 ⎟ ⎜ ⎟ (5) ⎜ · · · Cj = ⎜ ⎟ ⎝ −νj−1 νj−1 ⎠ νj Hereinto, νj = µj − λj = µj (1 − ρj ), Pj = Λ(ΛI − Cj )−1 , and I denotes an identity matrix. Define row vector γ = (1, 0, · · · , 0), j = 1, · · · , k − b b , (1 − γk−1 Pk−1 e)], j = k, where, e is 1; γk = [γk−1 Pk−1 unit column vector; γj+1 = [γj , (1 − γj e)], j = k, · · · , K − 1. Set πj = γj Pj in literature [11]; Gj is the outstanding backorders at stage k. When positioning CODP in stage k with the basestock level of work in process b, the mean backorder is given by: j = k πj (1 − Pj )−1 e ¯j = (6) G πk Pkb (1 − Pk )−1 e j = k The average postponing time can be considered as F¯j ≈ G¯j /Λ The mean output-buffer inventory is 0 j = k I¯j = ¯k j = k ¯k + G b−N N¯k = πk (1 − Pk )−1 e
(7)
(8) (9)
Based on the formulas above, we can obtain the optimal result, but the solving process is extremely complicated. Literature [3] put forward that in the MTS production process, the quantity of work in process followed the general
binomial distribution. In that case, the calculation of mean backorder, the average postponing time, and the mean output-buffer inventory can be simplified. Therefore, to simplify the calculation, suppose that the quantity of work in process follows the general binomial distribution in the MTS production process. The mean backorder, the average postponing time, and the mean out-putbuffer inventory can be denoted as the following formulas: ¯ b) = I(k,
b (b − r)P r(Nk = r)
(10)
r=0
P r(Nk = r) represents the possibility of the quantity of work in process being r in stage k while positioning CODP in stage k.
Pr (Nk = r) =
⎧ r + k − 1 ⎪ ⎪ (1 − ρ)k ρ, ⎪ ⎪ k ⎨
ρj = ρ
k ρr+k−1 Πki=1 (1 − ρi ) ⎪ j ⎪ ⎪ , ⎪ ⎩ Πi=j (ρj − ρi )
others
j=1
¯ b) ≈ I(k, ¯ b)+ Since Nk +Ik −Gk = b, G(k, K
j=1
ρj 1−ρj
−
¯ b)/Λ, the model can be transformed as follows: b, F¯j ≈ G(k, min Z(k, b) = h(k)
b k ρr+k−1 j (b − r) r=0
j=1
k
i=1 (1
− ρi ) + c(k) i=j (ρj − ρi )
(11)
s.t. ⎡ ⎤ k b k K ρr+k−1 (1−ρ ) ρ j j j i=1 ⎣ (b−r) −b⎦ Λ ≤ α + 1−ρj i=j (ρj−ρi ) r=0 j=1 j=1 (12) k≤K
(13)
k, b ≥ 0
(14)
3.2
Approach According to the formulas above, if the value of k, unit inventory cost, and re-manufacturing cost are determined, the optimal value of b can be obtained, which will lead to the minimum cost with the service constrains. It can be denoted as b∗ . The approach to get the optimal value of k (k ∗ ) is given below: • Choose the available value of k and begin with the smallest one; then apply it into the model Owing to the characteristics of the process industry, some stages cannot be chosen as CODP. Take Oil Refinery as an example; the production cannot be interrupted in some links. Once a certain link is interrupted, the entire production line will be stopped and the losses may be up to one million RMB. • Calculate the optimal b(k) and z(k, b(k)). If k = K, turn to step 4 • k = k + 1, (k = 1, 2, · · · , K); if k is an available point, then turn to step 2, otherwise repeat step 3
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
• Find the value of k, which leads to the minimum value of z(k, b(k)), and denote it as k ∗ , which is the optimal position of CODP.
4 Numerical results In this section, we use MATLAB to simulate the effects of each factor on the optimal position of CODP k ∗ and the optimal basestock level b∗ . Suppose there are 10 stages and the capacity utilization rate of each stage is 0.8. Based on the simulation results, we obtain the following corollaries: Corollary 4.1 The optimal CODP k ∗ and the optimal basestock level b∗ decrease with increasing α (Figure 2). Table 1 provides the specific data of some typical points in Figure 2. From Table 1, it is obvious that when the value of is small, the postponing time to complete the order is controlled strictly. Therefore, the CODP has to move to the downstream and a large number of stocks of work in process are indispensable, which results in high cost. When the value of increases gradually, the CODP moves upstream and the optimal basestock level declines. The total cost decreases sharply. Therefore, the customer service level influences the optimal position of CODP and the optimal basestock level at CODP directly. Corollary 4.2 With the increase of unit inventory cost in CODP and the position of CODP moving downstream, the optimal basestock level tends to become smaller. (Figure 3)
Owing to the restrictions on technology and other factors, different enterprises implement mass customization with different growth rates of cost. The enterprise with flexible production and advanced technology costs far lesser to achieve mass customization. As the production lines are widely used in the process industry and the production processes are relatively rigid, to achieve mass customization, the enterprises need to rebuild or even renew equipments or re-develop formulations. Therefore, the impact of remanufacturing cost on the position of CODP in the process industry is larger than that in discrete industry. Corollary 4.3 shows a strong evidence of the relationship between the position of CODP and the re-manufacturing cost. Corollary 4.4 In the MC system, the utilization rate of equipment in the MTS production phase, which is before CODP should be relatively high, while that in the MTO production phase after CODP should not be very high. In terms of production layout, the equipments with high fix investment or those that need high utilization rate or full load running should be arranged before CODP. The specific analysis is shown below: As seen in Figure 2, there are five stages. When k ∗ = 2, the first two stages are operated in the MTS phase and the other three stages are operated in the MTO phase. Table 2 reports the simulation results. Comparing the results of different conditions and analyzing the impact of utilization rate on the optimal basestock level and total cost, we can make the following observations:
Corollary 4.3 With the increase of the product/process re-manufacturing cost, the optimal position of CODP moves upstream and the optimal basestock level tends to become larger. (Figure 4)
Table 1. Relationship of optimal position of CODP and optimal basestock level with service level
α 0.2 0.4 0.5 1 2 10
k∗ 8 5 3 3 1 1
b∗ 69 57 46 43 40 27
Z∗ 726 560 97 81 6 4
Figure 2. Relationship of optimal position of CODP and
Figure 3. Relationship of optimal position of CODP and optimal basestock level with unit inventory cost (K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
Figure 4. Relationship of optimal position of CODP and optimal
optimal basestock level with service level
basestock level with product/process re-manufacturing
(K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
cost (K = 10, h(i) = 5i, c(i) = i, ρ = 0.8, Λ = 1)
JI Jian-hua, et al./Systems Engineering – Theory & Practice, 2007, 27(12): 151–157
Table 2. Relationship among the optimal basestock level, minimum cost, and production utilization rate
Condition 1 2 3 4 5 6 7 8 9 10 11 12 13
(ρ1 , ρ2 , ρ3 , ρ4 , ρ5 ) (0.9,0.8,0.7,0.6,0.5) (0.9,0.7,0.8,0.6,0.5) (0.9,0.6,0.8,0.7,0,5) (0.9,0.5,0.8,0.7,0.6) (0.8,0.9,0.7,0.6,0.5) (0.7,0.6,0.9,0.8,0.7) (0.5,0.6,0.9,0.8,0.7) (0.5,0.6,0.8,0.7,0.9) (0.5,0.6,0.8,0.7,0.9) (0.5,0.6,0.7,0.8,0.9) (0.5,0.6,0.55,0.45,0.35) (0.5,0.6,0.95,0.45,0.35) (0.5,0.6,0.95,0.9,0.45)
b∗ 2 2 2 2 2 2 17 17 17 17 2 23 31
Z∗ 3.3465 3.3465 3.426 3.9505 3.3465 3.585 48.848 22.3562 22.3562 9.2419 5.4536 69.8346 84.4971
Observation 1 with a decrease of utilization in the MTS production phase, the optimal basestock level in CODP tends to become larger and the cost tends to become higher. (See Table 2 conditions 1 to 7) This observation is reasonable. If the scale production can be achieved in the MTS production phase, the cost of products can be reduced. If the utilization rate in MTS production phase declines, there is no doubt that the total cost will increase. Observation 2 Changing the sequence of utilization rate among stages in MTS production phase will not affect the optimal basestock level in CODP and the total cost. (See Table 2 conditions 1 and 2) Observation 3 When the utilization rates of stages in the MTS production phase remain unchanged and those in the MTO production phase increase, both the optimal basestock level and the total cost tend to become lager. (See Table 2 conditions 11 to 13) Observation 4 Changing the sequence of utilization rate among stages in the MTO production phase will not influence the optimal basestock level in CODP but will impact the total cost. The closer to CODP the equipment with the maximum utilization rate is, the greater will be the increase in the total cost. (See Table 2 conditions 1 to 10). Based on these four observations, the article puts forward Corollary 4.4. These observations and Corollary 4.4 have huge operational significance for the enterprises to implement mass customization. In the process industry, various types of equipments have to be run with full load or high load because of their high fix investment or other considerations such as security and technology. When implementing mass customization, these equipments with high utilization rate should be arranged after CODP, namely, CODP cannot be positioned before the bottleneck of the production system.
In addition, the utilization rate of equipments arranged after CODP cannot be very high.
5
Conclusions
Implementing mass customization is one approach for enterprises in the process industry to improve their competitive capacity. Taking the process industry implementing mass customization as a background, this article presented the model definitions and relevant assumptions, and then established a CODP positioning model with the delivery lead time constraint and capacity constraint, aiming at the minimization of the total cost. At last, software Matlab was used to get numerical results and obtain some valuable corollaries. In the future research, the CODP positioning model can be extended in the following aspects: customization level influences not only price but also demand; by-products exist in the production process; there are buffer inventories in the interrupt point after CODP.
References [1] Aviv Y, Federgruen A. Capacitated multi-item inventory systems with random and seasonally fluctuating demands: Implications for postponement strategies. Management Science, 2001, 47(4): 512–532. [2] Su J C P, Chang Y L, Ferguson M. Evaluation of postponement structures to accommodate mass customization. Journal of Operations Management, 2004, 23: 305–318. [3] Diwakar G, Benjaafar S. Make-to-order, make-to-stock, or delay product differentiation? A common framework for modeling and analysis. IIE Transactions, 2004, 36: 529–546. [4] Lee H L, Tang C S. Modeling the costs and benefits of delayed product differentiation. Management Science, 1997, 1: 40–53. [5] Gara A, Tang C. On postponement strategies for product families with multiple points of differentiation. IIE Transaction, 1997, 29(8): 641–650. [6] Aviv Y. Design for postponement: A comprehensive characterization of its benefits under unknown demand distributions. Operations Research, 2001, 49(4): 578–598. [7] Qi G N, Gu X J, Li R W. Study on mass customization and its models. Computer Integrated Manufacturing Systems, 2000, 10(2): 41–45. [8] Liu J. Study on the positioning and form selection of decoupling point in postponed manufacturing. Jiangsu Commercial Forum, 2004, 1: 69–70. [9] Shao X F, Wang R, Huang P Q, Ji J H. Research on postponement in mass customization with multiple product differentiation points. Systems Engineering — Theory Methodology Application, 2001, 10(4): 332–336. [10] Lian H J. Mass customization oriented research on production planning of process industry. Doctoral Thesis of Shanghai Jiao Tong University, 2006. [11] Lee Y J, Zipkin P. Tandem queues with planned inventories. Operations Research, 1992, 40(5): 936–947.