A two-level replenishment frequency model for TOC supply chain replenishment systems under capacity constraint

Computers & Industrial Engineering 72 (2014) 152–159

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A two-level replenishment frequency model for TOC supply chain replenishment systems under capacity constraint q Horng-Huei Wu a, Amy H.I. Lee b,c,⇑, Tai-Ping Tsai c a

Department of Business Administration, Chung Hua University, Hsinchu 300, Taiwan, ROC Department of Technology Management, Chung Hua University, Hsinchu 300, Taiwan, ROC c Department of Industrial Management, Chung Hua University, Hsinchu 300, Taiwan, ROC b

a r t i c l e

i n f o

Article history: Received 31 May 2012 Received in revised form 20 December 2013 Accepted 3 March 2014 Available online 12 March 2014 Keywords: Supply chain management Inventory replenishment Theory of constraints (TOC) TOC supply chain replenishment system Two-level replenishment frequency

a b s t r a c t Good production planning and replenishment management are important for a firm to keep competitive in the market. The theory of constraints-supply chain replenishment system (TOC-SCRS) is a replenishment method under the TOC philosophy. In the application of the TOC-SCRS in a node of a supply chain, the replenishment frequency (RF) and the reliable replenishment time (RRT) are required parameters. Generally, the RF of a node depends on the public transportation schedule such as ship schedules or its private conveyor schedule. If this node is a plant, however, the RF depends on the setup frequency in this plant, and a higher RF (i.e., once a day) is preferred by the TOC because of lower inventory and quick response to different market requirement. Basically, the RF in a plant is determined by its sales or production quantity. When sales increase significantly, the RF in a plant requires to be elongated from higher frequency (i.e., once a day) to lower frequency (i.e., once every two or more days) due to the limited capacity. Therefore, a two-level replenishment frequency model for the TOC-SCRS under capacity constraint is proposed. This model is especially suitable to a plant in which different products have a large sales volume variation. Numerical examples are utilized to evaluate the application of the proposed method. Employing this proposed methodology will facilitate a plant or a central warehouse to implement an effective TOC-SCRS successfully. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The major objective of inventory management is to satisfy the needs of customers promptly. Because the cost of inventory shortage is usually higher than the cost of excess inventory, traditional inventory management tends to put a larger amount of inventory near customers, i.e. the selling points. Even though the products can be obtained by the customers in a shorter time, excess inventory in the selling points or their upstream warehouses is resulted, and a higher capital is required to maintain the inventory (Belvedere & Grando, 2005; Patnode, 1999; Simchi-Levi, Kamindky, & Simchi-Levi, 2009). To avoid these from happening, another type of inventory management is to put the inventory at the beginning of the supply chain, i.e. the factory, to reduce the loss caused by market changes. However, a higher risk of customers not being able to receive the products on time is resulted. q

This manuscript was processed by Area Editor Iris F.A. Vis.

⇑ Corresponding author at: Department of Technology Management, Chung Hua University, Hsinchu 300, Taiwan, ROC. Tel.: +886 3 5186582; fax: +886 3 518 6575. E-mail address: [email protected] (A.H.I. Lee). http://dx.doi.org/10.1016/j.cie.2014.03.006 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved.

Many of today’s markets are facing short product life cycle, demand fluctuation and customization; thus, fast response to customers’ demand has become the aim of firms. The theory of constraints (TOC) on the supply chain, called the TOC supply chain solution, was first proposed by Goldratt (1994) to avoid the bullwhip effect Lee, Padmanbhan, and Whang (1997) caused by unstable demand and variation in demand forecast and at the same time to maintain a lower inventory level and a higher customer service satisfaction. The replenishment method of the TOC supply chain solution is called the TOC supply chain replenishment system (TOC-SCRS) (Cole & Jacob, 2002; Goldratt, 1994). The TOC-SCRS is being implemented by a growing number of companies these days. In the application of the TOC-SCRS in a node of a supply chain, the replenishment quantity (RQ) is a function of the time to reliably replenish (TRR) and the quantity sold during the TRR. The TRR composes of two parameters, the replenishment frequency (RF) and the reliable replenishment time (RRT) (Cole & Jacob, 2002; Wu, Chen, Tsai, & Tsai, 2010). Generally, the RF of a node depends on the public transportation schedule such as a ship schedule or a private conveyor schedule, and the RRT is the required transportation time from an upstream node to this node.

H.-H. Wu et al. / Computers & Industrial Engineering 72 (2014) 152–159

For example, if the ship schedule is once a week and the RRT from an upstream node to this node is also a week, then the TRR is two weeks. Based on the two weeks of the TRR, this node replenishes only what is sold during the two weeks. That is, the RQ is a function of the TRR (or the RF and the RRT) and what is sold during the TRR. Therefore, the determination of the TRR is one key factor to successfully apply the TOC-SCRS in a node. The detailed model of the TOC-SCRS was described by Wu et al. (2010). When the TOC-SCRS is applied in a plant, however, the determination of the TRR will encounter a conflict with the RQ (Wu et al., 2010). This is because the RF depends on the set up frequency in the plant and the RRT depends on the production lead time. As we know, the set up frequency and the production lead time in the plant depend on the production quantity (i.e., the RQ). It means that the RF and the RRT must depend on the known RQ, especially under the constraint of finite plant capacity (Wu et al., 2010). However, in the TOC-SCRS, the RQ depends on the known parameters of the RF and the RRT. Therefore, this is a big conflict and an issue that needs to be tackled in the TOC-SCRS in a plant. A detailed description of the problem will be presented in the third section. Although the conflict of the RF and the RQ in the application of the TOC-SCRS in a plant has been studied by some authors (Cole & Jacob, 2002; Wu et al., 2010), the RF is the same for all products under each method. However, when different products have rather different sales volumes in a plant, an equal RF for all products is unreasonable. For example, the average daily consumptions of product A and B are 10 K units and 10 units respectively, and the setup time is one hour for each product. In this case, the same RF for both products is unreasonable. Basically, the RF of product A should be lower than that of product B. Therefore, a two-level RF model for the TOC-SCRS under a capacity constraint is proposed in this paper. Such a model is especially suitable for a plant whose products have very different sales volumes. The rest of the paper is organized as follows. In section two, works of the theory of constraints (TOC) and the supply chain replenishment system (SCRS) using the TOC are reviewed. In section three, replenishment frequency model is reviewed. A two-level replenishment frequency model is constructed in section four. An algorithm for the two-level replenishment frequency model is proposed in section five. A numerical example and a case study are demonstrated in section six. Some conclusion remarks and future research directions are made in the last section. 2. Literature review The theory of constraints (TOC) is a global managerial methodology that helps managers to concentrate on the most critical issues. It has drawn a wide range of responses from practitioners and academics (Gupta, 2003; Rahman, 1998). Especially in the academic field, the TOC has been a popular research topic since The Goal first appeared in print (Goldratt & Cox, 1986). For example, a comprehensive review of the publications regarding the philosophy and application of the TOC has been done by Rahman (1998). In addition, Blackstone (2001) overviewed the applications of the TOC in different areas, and Gupta (2003) presented some papers with advanced TOC works and identified some potential research issues. In recent years, there are also many TOC related publications. For example, Finch and Luebbe (2000) compared and contrasted linear programming with the TOC, and stated that the objectives of the TOC go beyond the scope of the objectives of linear programming because the TOC can provide an effective way for continuous improvement. Bhattacharya, Vasant, Sarkar, and Mukherjee (2008) proposed a human–machine intelligent approach using fuzzy linear programming to solve the TOC productmix problem with the consideration of decision maker’s level of satisfaction. Bhardwa, Gupta, and Kanda (2010) explained the steps to implement the drum-buffer-rope (DBR) concept of the TOC for

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planning and controlling the production in some practical manufacturing situations. Gonzalez-R, Framinan, and Ruiz-Usano (2010) performed a multi-objective comparison of dispatching rules in a DBR production control system, and the conflicting objectives include average tardiness, maximum tardiness and average work in process. Kasemset and Kachitvichyanukul (2010) proposed a bi-level multi-objective mathematical model for job-shop scheduling based on the concept of the TOC. On the first level, the initial schedule is generated by minimizing idle time of the bottleneck. On the second level, a multiple objective technique is applied to improve additional performance measurements. Lee, Chang, Tsai, and Li (2010) studied three special job shop environments and suggested some enhancement solutions for the TOC simplified DBR system (SDBR). Gupta and Andersen (2012) used a small company simultation to show how throughput dollar-days and inventory dollar-days can be used for improving the process in a DBR system. Tsou (2013) used three time-series data-mining techniques, sequential probability ratio test, CUSUM chart and autoregression test, to detect the timing of market demand change and proposed a systematic methodology to adjust the target inventory level dynamically for supply chain collaboration from the perspective of the TOC. Although the TOC was applied only to the production in early stage, it has been applied to a wide range of fields including operation (production), finance and measures, project, distribution and supply chain, marketing, sales, managing people, and strategy and tactics (Blackstone, 2001; Kendall, 2006). As stated in Section 1, the TOC-SCRS is a replenishment method for solving the distribution and supply chain problem (Cole & Jacob, 2002; Goldratt, 1994). The goal of the TOC-SCRS is to solve the conflict problem in the inventory management in the supply chain so that a win–win strategy can be devised to maintain a lower inventory level and a prompt delivery and to satisfy customer needs (Blackstone, 2001; Goldratt, 1994). The performance improvement of the companies implemented TOC-SCRS includes the reduction of inventory level, lead-time and transportation costs and the increase in forecast accuracy and customer service levels (Hoffman & Cardarelli, 2002; Novotny, 1997; Sharma, 1997; Waite, Gupta, & Hill, 1998; Watson & Polito, 2003; Wu et al., 2010). Because a plant or a central warehouse acts as a supplying source in the supply chain, its replenishment strategy has a great influence on the entire supply chain. Basically, the TOC-SCRS concerns two strategies to decouple the excess inventory in each node, for example, a plant, a central warehouse or a retailer, and maintain the inventory availability to consumers (next nodes). First, each node holds enough stock to cover the demand during the TRR. Second, each node replenishes only what is sold, i.e., the RQ (Goldratt, 1994; Holt, 1999; Perez, 1997; Simatupang, Wright, & Sridharan, 2004; Smith, 2001; Yuan, Chang, & Li, 2003). In order to respond to unforeseeable events, a control mechanism called buffer management is used to monitor whether the inventory of any item is too low (Cole & Jacob, 2002; Schragenheim & Ronen, 1991; Yuan et al., 2003). When the inventory is too low, replenishment will proceed immediately so that material shortage can be prevented even when demand increases abruptly. Two recent works of the TOC-SCRS are as follows. Wu, Tsai, Tsai, Tsai, and Liao (2011) proposed the adoption of the TOC-SCRS in TFT-LCD plants, and demonstrated the significance and feasibility of the proposed system. Wu, Huang, and Jen (2012) proposed an evaluation and enhancement model to release the inventory shortage occurrence during the replenishment frequency conversion periods in order to successfully implement an effective the TOC-SCRS. The conflict in the determination of the TRR with the RQ, as explained in Section 1, was first being solved by Cole and Jacob (2002) with a method to determine the RF and the RQ simultaneously in a plant. The daily remaining (or unused) capacity is calculated first based the production capacity required by the average daily sales.

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Then the RF is determined by the remaining capacity. If the remaining capacity is enough to replenish (or set up) all products once a day, the RF is once a day. Otherwise, the RF is elongated to be once every two days, and the remaining capacity is checked for sufficiency. The process repeats until the remaining capacity is enough to fulfill the replenishment or setup requirement. Basically, the RF determination model of Cole and Jacob (2002) is a trial and error method and can handle only one machine in a plant or in a bottleneck station. An enhanced RF model for the TOC-SCRS under a capacity constraint is then provided by Wu et al. (2010). Although the conflict of the RF and the RQ in the application of TOC-SCRS in a plant is solved by these authors, the RF is the same for all products under each method. However, when a plant needs to produce different products with very different sales volumes, it is simply unreasonable and unrealistic to have an equal RF for all products. Thus, this paper proposes a two-level RF model for the TOC-SCRS under a capacity constraint for a plant whose products have very different sales volumes. Such a model will be more suitable in real practice. 3. Review of the replenishment frequency model Nomenclature I i Oj fi li e w m C Si pi Di Li ti

vi ri F R k

total product types product index, i = 1, 2, . . ., I different level of products, j = I or II RF for product i, i.e., the time period between deliveries, such as two days, initial fi = F for all i manufacturing lead time for product i, li = one day (i = 1, 2, . . . , I) in this paper working days in a period, e = one day/period in this paper working hours in a working day, w = 24 h/day in this paper number of machines in the bottleneck station plant capacity in a period setup time required for product i in the bottleneck station production output per hour for product i in a bottleneck machine average daily demand (consumption) of product i in a period required capacity for product i in a period the number of times of setup required for product i in a period saved setup time for the RF of product i to be elongated additional required setup time for the RF of product i to be shortened frequency of replenishment for all products proposed by Wu et al. (2010) total saved capacity for the RF of each product in level I to be elongated a parameter used to allocate products into level I and level II

According to the TOC, the output of a plant is determined by its bottleneck station. Therefore, the capacity of the bottleneck station is only considered in the following model. The RF model proposed by Wu et al. (2010) is developed on the basis of minimum capacity loss (i.e., maximum throughput) and minimum replenishment quantity (i.e., minimum inventory). The model is briefly shown as follows: I I X X Si  t i Min C  Li þ fi i¼1 i¼1

Subject to : C P

!

I I X X Si  t i Li þ fi i¼1 i¼1

fi P 1 and f i is an integer

ð1Þ ð2Þ ð3Þ

where C = m  e  w and Li = Di/pi, and t i ¼

l

m

Li eh

is utilized to deter-

mine the minimum number of setup times required for product i in a period. If Li > e  h, product i requires two or more bottleneck machines to produce simultaneously in a period (Wu et al., 2010). Because the number of setup times must be an integer and greater than Li the value of eh , the notation dxe is utilized to represent an integer which is equal to or larger than x. For example, d1:1e; d1:9e and d2e all have the value of 2. Therefore, Si  ti is the setup time required by product i in a period. If the capacity is not sufficient, the setup frequency of product i must be elongated, for example from once i every period to once every two or more periods. Therefore, Si t is fi a normalized setup time for product i in a period when the setup frequency of product i is prolonged to be several periods (Wu et al., 2010). Objective function (1) is to minimize the excess capacity, which is the total available capacity minus the total used capacity (Wu et al., 2010). The total used capacity composes two parts: total production capacity which is used to manufacture products    PI PI Si ti  i:e:; . The longer seti¼1 Li and total setup time i:e:; i¼1 fi up frequency, the shorter total setup time and the larger production quantity are in a period. Therefore, the feasible condition in Eq. (2) requires that the total used capacity must not be greater than the total capacity. On the other hand, objective function (1) requires an optimal setup frequency or production quantity for P all products. If C < Ii¼1 Li , it means that the plant capacity is not enough to produce all products. We cannot deal with it by elongating the setup frequency, and it is a product-mix decision problem (Bhattacharya & Vasant, 2007; Bhattacharya et al., 2008; Luebbe & P P i Finch, 1992). On the other hand, if C P Ii¼1 Li þ Ii¼1 Si t , it means fi that the capacity is enough to produce all products and to setup once for each product i in a period (Wu et al., 2010). Therefore, the minimum excess capacity in objective function (1) is obtained P P when all fi = 1. However, if Ii¼1 Li < C < Ii¼1 ðLi þ Si  ti Þ, it means that the capacity is enough to produce all products but not enough to setup once for each product i in a period. Therefore, we must elongate the setup frequency to reduce the total setup time. There are multiple combinations of feasible setup frequency of different product i to satisfy Eq. (2). The optimal combination is the one that minimizes the excess capacity. The optimal setup frequency (F) in the condition that all fi are equal is shown in Eq. (4) (Wu et al., 2010).

&P I F¼

i¼1 ðSi

C

PI

 ti Þ

’

i¼1 Li

ð4Þ

However, when products have a large sales volume variation, the same RF for all products is unreasonable. Therefore, a two-level RF model, which can obtain different replenishment frequencies for different products (fi), is proposed in this paper. 4. A two-level replenishment frequency model In this model, the same RF for all products proposed by Wu et al. (2010) is utilized to be the initial solution, and then the variant RF is re-evaluated for products in different level via a two-level RF concept. With the consideration of demanded capacity, excess capacity and setup time, the concept of decreasing the RF of higher demanded products and increasing the RF of lower demanded products is applied. That is, for those products with lower required capacity, i.e. products belonging to level I, their RF will increase so that excess capacity can be obtained for processing products with higher required capacity, i.e. products belonging to level II, with a shorter RF, as shown in Fig. 1.

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4.1. Two levels of products Based on the required capacity of product orders (Li), the products are separated into two levels, where level I contains the product orders which require lower capacity, and vice versa. The division of levels I and II is determined by the relationship between product demand and the capacity required for setup. That is, products, which require lower production capacity, belong to level I, as shown in Eq. (5).

OI ¼ fijLi 6 k  Si  t i ; i ¼ 1; 2; . . . Ig:

If the RF shortens to 2 days, the average setup time per day is 1 (=(1/2)  2  1) hour. Therefore, under a 4-day capacity evaluation,     2 ¼ 4  12  14  2  1 hours of additional setup time is resulted. 4.3. The saving setup time per period due to elongating RF When product i is replenished (setup) once in F periods, the average total setup time in the bottleneck station for a period is 1  Si  t i . If F is to elongate into once in fi periods, using F as a basis, F the saving setup time is shown in Eq. (8).

ð5Þ ri ¼ F 

where k is a parameter for evaluating the ratio of setup time and demand capacity. Then products, which do not belong to level I, belong to level II, as shown in Eq. (6).

OII ¼ f1; 2; . . . Ig  OI :

ð6Þ

The allocation of products into two levels can be shown in Fig. 2.

  1 1  Si  t i  F fi

ð8Þ

For example, if a product is originally replenished once every 4 days and the setup time in the bottleneck station is 2 h and ti = 1, the average setup time per day is 1/2 (=(1/4)  2  1) hours. If the RF elongate to 6 days, the average setup time per day is 1/3 (=(1/6) under a 4-day capacity evaluation,   2 1) hour. Therefore,  2=3 ¼ 4  14  16  2  1 hour of setup time is saved. 4.4. The variant RF for products in level I or II

4.2. The additional required setup time per period due to shortening RF When we want to increase or decrease the replenishment (setup) frequency of a product, we need to consider the total setup time for a product in a period in order to calculate the total setup time saved or needed in that period. When product i is replenished (setup) once in F periods, the average setup time in the bottleneck station for a period is 1  Si  ti . If F is to shorten into once in fi periods, using F as a basis, F the additional required setup time is as shown in Eq. (7).

vi ¼ F 

  1 1  Si  t i  fi F

ð7Þ

For instance, if a product is originally replenished once every 4 days and the setup time in the bottleneck station is 2 h and ti = 1, the average setup time per day is 1/2 (=(1/4)  2  1) hours.

Required capacity (Li)

II

I

Product i Shorten replenishment frequency

Elongate replenishment frequency

Increase total setup time

Decrease total setup time

I

II

λ × Si ×

Li j ×H Product i

Fig. 2. The allocation of products to two levels.

fi ¼ Maxfdk  Si  t i =Li e; Fg;

i 2 OI :

ð9Þ

Even though an increase of the RF of a lower capacitydemanded product can decrease its required setup time, the excess time may not be sufficient for the extra setup time needed when increasing the RF of a higher capacity-demanded product. For example, let a plant produce product A and B, there is one bottleneck station, and the setup time in the bottleneck station for each product is 2 h. The original plan is to have the same RF for the two products, that is, once every three periods. If the RF of product A, a lower capacity-demanded product, elongates to once every four     periods, the setup time saved is 1=2 ¼ 3  13  14  2  1 hour based on Eq. (8). In addition, the RF of product B, a higher capacity-demanded product, decreases to  once every  two periods, the additional required setup time is 1 ¼ 3  12  13  2 hour based on Eq. (7), which is longer than the setup time saved from product A. Therefore, an algorithm is required to generate the RF for each product (especially for products in level II) when there are multiple products. 5. An algorithm for the two-level replenishment frequency model In this section, an algorithm for the two-level replenishment frequency model is presented. The flow chart of the algorithm is depicted in Fig. 3. Basically, the saving capacity from the products in level I is contributed to the products in level II in a descending order of the demanded capacity loading, and the RF is shortened by one period a time. The procedure for calculating the RF is as follows:

Fig. 1. The evaluation of RF.

Required capacity (L i)

The upper bound RF for those products in level I depends on the value of dk  Si  ti =Li e, but not less than the initial RF (i.e., F). Therefore, the elongated RF for products in level I is as shown in Eq. (9).

Step 1. Calculate the initial RF (i.e., F) for all products using Eq. (4). Step 2. Separate the products into two levels based on Eqs. (5) and (6). Step 3. Calculate the revised RF and the total saving capacity for each product in level I. Step 3.1. Let O = OI and the initial R be the residual capacity which is the total  capacity minus the used capacity, i.e., P P R ¼ F  C  Ij¼1 Lj  Ij¼1 ðSj  t j Þ .

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Start Start

Step 1

Calculate the initial replenishment frequency (F) for all products

Step 2

Separate the products into Level I and II based on the size of required capacity.

Step 3

Step 3.1

Calculate the excess capacity of the plant and set it as the initial remaining capacity.

Step 3.2

Modify the replenishment frequency of a product in Level I.

Step 3.3

Evaluate the saved capacity of the product in Level I and add it to the remaining capacity.

Step 3.4

Yes

Is there any more product in Level 1? No Copy set O from Level II.

Step 4.1

Step 4

Step 4.2

Select a product with the highest demanded capacity from set O.

Step 4.3

Is the RF of this product equal to one?

Yes

No Is the remaining capacity sufficient to decrease the replenishment frequency of this product?

No

Yes Step 4.4

Shorten the replenishment frequency of this product and recalculate the remaining capacity.

Step 4.5

Is there any more remaining capacity?

Step 4.6

Delete this product from set O.

No

Yes

No Is set O empty? No

Yes Step 4.7 Is the RF of each product in Level II one or is the total remaining capacity insufficient for the extra setup time required when shortening the RF of each product in OII? Yes End

Fig. 3. The flow chart of the two-level RF model.

Step 3.2. Select a product i from OI and evaluate its elongated RF (i.e., fi) by utilizing Eq. (9). Step 3.3. Evaluate the saved capacity of product iby utiliz ing Eq. (8) and add it to R, i.e., R ¼ R þ F  1F  f1  Si  ti . i

Step 3.4. O = O  {i}. If O – /, go to Step 3.2. Step 4. Calculate the revised RF for each product in level II. Step 4.1. Let O = OII and constant x be 0. Step 4.2. Select product i with the highest demanded capacity from OII, that is, i = Max{j|Lj, j e OII}. If fi = 1, go to Step 4.6. Step 4.3. Based on Eq. (7), evaluate whether the total remaining capacity (R) is sufficient for the extra setup time required when shortening the RF of product i. That is

      1 1 if R < F  fi 1  1F  Si  t i  F f1i  1F  Si  t i or R < F  fi 1  f1i  Si  ti , go to Step 4.6. Step 4.4. Shorten the RF of product i and calculate the remaining   1 capacity (R), i.e., R ¼ R  F  fi 1  f1i  Si  ti , fi = fi  1, and x = 1. Step 4.5. If the total remaining capacity is empty, stop the process. That is if R = 0, let x = 0 and go to Step 4.7. Step 4.6. Delete product i from O, i.e. O = O  {i}. If O – /, go to Step 4.2.

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Step 4.7. If the RF of each product in OII is 1 or the total remaining capacity is insufficient for the extra setup time required when shortening the RF of each product in OII, then stop. That is, if x = 0, then stop. Otherwise, go to Step 4.1. 6. Numerical example and demonstration

Table 2 Revised RF for the numerical example. Product index Product level Referenced RFa(day) Revised RF (day) a

1 II 3 2

2 II 3 2

3 II 3 2

4 I 3 6

5 II 3 2

6 I 3 3

7 I 3 4

8 II 3 1

Proposed by Wu et al. (2010).

To effectively demonstrate the proposed algorithm and its applications, a numerical example is first presented. A prototype system based on this algorithm is developed and its application to a machinery factory is employed for further demonstration. 6.1. A numerical example Consider a plant which has three machines in its bottleneck station and produces eight different products. The average daily demand (Di), the production output per hour on the bottleneck station (pi), and the setup time required on the bottleneck station (Si), for each product are as shown in Table 1. The operation is 24 h a day, and a day is treated as a period. In addition, k is set to be 2. Based on Eq. (4) and the model proposed by Wu et al. (2010), the RF of each product is once every three days. Furthermore, based on the TOC-SCRS, the buffer level (Cole & Jacob, 2002; Wu et al., 2010), i.e. the maximum inventory, for product 1 is 2400 pieces, and the total buffer is 23,800 pieces for these eight products. The two-level RF model is then utilized to evaluate the reasonable RF of each product to reduce the total buffer. Based on the proposed algorithm in the above section, the stepwise computation is presented in Appendix A. These products are first categorized into two levels in step 2, i.e., OI = {4, 6, 7} and OII = {1, 2, 3, 5, 8}. Then the RFs of the products in OI are elongated as long as possible in step 3 in order to save the setup time for these products. The final RF of each product in OI is revised to be once every six days, once every three days and once every four days for product 4, 6 and 7 respectively, as shown in Table 2. As result, 6.5 h are saved. These saved hours are then utilized to reduce the RF of each product in OII in step 4. The final RF of product 8 in OII is reduced to be once a day, and those for the other products in OII are once every two days, as shown in Table 2. After this refined process for the RF, the total buffer is 18,430 pieces, with a reduction of 5370 pieces (or 22.6%).

Fig. 4. Production data for the case in the prototype system.

6.2. A prototype and demonstration A prototype system based on the proposed algorithm was further developed to demonstrate the feasibility of this model. The major input items include the number of machines in the bottleneck station, the number of days in a period, the number of working hours in a day, the number of different products, manufacturing leadtime, and k, the parameter for evaluating the ratio of setup time and demand capacity, as shown in Fig. 4. After inputting the average daily demand, production output and setup time for each product, the system can calculate the RF of each product, as shown in Fig. 5.

Table 1 Production data for the numerical example. Product index (i)

1

2

3

4

5

6

7

8

Average daily demand (Di) Production output per hour (Pi) Setup time (Si)

600

2500

800

200

450

320

180

900

100

100

100

100

90

80

60

90

2

2

2

6

1

3

6

1

Fig. 5. The revised RF for the case via the prototype system.

A case study is carried out in a machinery factory. The data, as shown in Fig. 4, is collected from the factory, and is used as an example. The RF based on Wu et al. (2010) is once every four days. The total inventory is 52,100 pieces. The manager of this factory wondered whether the buffer level was too high and would like to have a better buffer management. Therefore, using the prototype system, the products are separated into two levels, as shown in Fig. 5. The products in level I are product 1, 6, 8, 9 and 10. The revised RF is once every eight days, once every four days, once

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every six days, once every eight days and once every six days for each product, respectively. The products in level II are product 2, 3, 4, 5 and 7. The revised RF is once a day for product 2 and once every two days for the others. The buffer level of each product is also provided by this system. The buffer of each product in level II is significantly reduced due to its shortened RF. Although the buffer of each product in level I is increased, the total buffer in the system is reduced from 52,100 pieces to 37,940 pieces. 7. Conclusions and suggestions Although the TOC-SCRS can be an effective inventory replenishment method in practice, the determination of the RF is a difficult task when it is deployed in a plant or a central warehouse with capacity constraints. Some RF determination models have been proposed to solve this problem; however, the RF must be the same for all products under each of the models. When different products have rather different sales volumes in a plant, an equal RF for all products is unreasonable. Therefore, a two-level RF model for the TOC-SCRS under a capacity constraint is proposed in this paper. The concept of this model is to replenish higher demanded products more frequently. Products are separated into two levels, and the products with lower demanded capacity loadings belong to level I, and vice versa. The RF of each product under level I is increased first to obtain extra setup time for the products in level II, and the RF of each product under level II can be decreased as a result. A numerical example is used to examine the feasibility and performance of the approach. A prototype system is further constructed to demonstrate the feasibility and effectiveness of the proposed model. The results show that the proposed approach and the prototype system can bring significant economic benefits in a supply chain replenishment system. Thus, the proposed approach can help managers determine the RF of each product and facilitate production scheduling. Future work will focus on the development of a diverse RF Model for TOC-SCRS with capacity constraints to further improve the buffer level (Jiang, Wu & Tsai, 2013). Moreover, the application the TOC-SCRS in a large-scale factory usually has several hundred or thousand products in practice. The deployment of intelligent algorithms, such as particle swarm optimization or genetic algorithms, to optimize the RF for improving system performance and efficiency is suggested. Acknowledgement This work was supported in part by the National Science Council in Taiwan under Grant NSC 96-2628-E-216-001-MY3. Appendix A The stepwise computation for the numerical case in Section 6.1 is as follows: Step 1. When the replenishment (setup) frequency is the same for all products, it is calculated as follows:

3   tj Þ 2þ4þ2þ6þ1þ3þ6þ1 7 fi ¼ F ¼ 6 P8 7 ¼ 6 3  1  24  63 6 C  j¼1 Lj 7  25 ¼ ¼ 3; i ¼ 1; 2; . . . 8: 9 2P

8 j¼1 ðSj

Step 2. Separate the eight products (O0) into two levels, OI and OII. Because k is 2, OI and OII are:

OI ¼ fijLi 6 k  Si  t i ; i ¼ 1; 2; . . . 8g ¼ f4; 6; 7g

OII ¼ f1; 2; 3 . . . ; 8g  OI ¼ f1; 2; 3; 5; 8g: Step 3. Calculate the revised RF and the total saving capacity for each product in level I. 6, 7} and initial Step 3.1. Let O = OI = {4, R = 3  (3  1  24  63)  25

= 2. 

Step 3.2. i = 4 and f4 ¼ Max 2  62 ; 3 ¼ 6.   Step 3.3. R ¼ 2 þ 3  13  16  6  1 ¼ 5. Step 3.4. O = O  {4} = {6, 7} and go back to Step 3.2.

 Step 3.2. i = 6 and f6 ¼ 2  34 ; 3 ¼ 3. 1Max  Step 3.3. R ¼ 5 þ 3  3  13  6  1 ¼ 5. Step 3.4. O = O  {6} = {7} and go back  to Step 3.2. Step 3.2. i = 7 and f7 ¼ Max 2  63 ; 3 ¼ 4.   Step 3.3. R ¼ 5 þ 3  13  14  6  1 ¼ 13 . 2 Step 3.4. O = O  {7} = /. Step 4. Calculate the revised RF for each product in level II. Step 4.1. Let O = OII = {1, 2, 3, 5, 8}and x = 0. Step 4.2. i ¼ MaxfjjLj ; j 2 OII g ¼ 2 and f2 = 3.    1  Step 4.3. R ¼ 13  13  2  2ð¼ 2Þ. > 3  31 2 13 1 1 Step 4.4. R ¼ 2  3  31  3  2  2 ¼ 92, f2 = 3  1 = 2, and x = 1. Step 4.5. R(=9/2) > 0. Step 4.6. O = O  {2} = {1, 3, 5, 8} and go back to Step 4.2. Step 4.2. i ¼ MaxfjjLj ; j 2 OII g ¼ 8 and f8 = 3.    1    Step 4.3. R ¼ 92 > 3  31  13  1  1 ¼ 12 . 9 1 1 Step 4.4. R ¼ 2  3  31  3  1  1 ¼ 4, f8 = 3  1 = 2, and x = 1. Step 4.5. R(=4) > 0. Step 4.6. O = O  {8} = {1, 3, 5} and go back to Step 4.2. Step 4.2. i ¼ MaxfjjLj ; j 2 OII g ¼ 3 and f3 = 3.  1  Step 4.3. Rð¼ 4Þ > 3  31  13  2  1ð¼ 1Þ. 1 Step 4.4. R ¼ 4  3  31  13  2  1 ¼ 3, f3 = 3  1 = 2, and x = 1. Step 4.5. R(=3) > 0. Step 4.6. O = O  {3} = {1, 5} and go back to Step 4.2. Step 4.2. i ¼ MaxfjjLj ; j 2 OII g ¼ 1 and f1 = 3.  1  Step 4.3. Rð¼ 3Þ > 3  31  13  2  1ð¼ 1Þ. 1 1 Step 4.4. R ¼ 3  3  31  3  2  1 ¼ 2, f1 = 3  1 = 2, and x = 1. Step 4.5. R(=2) > 0. Step 4.6. O = O  {1} = {5} and go back to Step 4.2. Step 4.2. i ¼ MaxfjjLj ; j 2 OII g ¼ 5 and f5 = 3.  1    Step 4.3. Rð¼ 2Þ > 3  31  13  1  1 ¼ 12 . 1 1 Step 4.4. R ¼ 2  3  31  3  1  1 ¼ 32, f5 = 3  1 = 2, and x = 1. Step 4.5. R(=3/2) > 0. Step 4.6. O = O  {5} = /. Step 4.7. Because x = 1, go back to Step 4.1. Step 4.1. Let O = OII = {1, 2, 3, 5, 8}and x = 0. Step 4.2. i ¼ MaxfjjLj; j 2 OII g ¼ 2 and f2 = 2. 1 Step 4.3. Because R ¼ 32 < 3  21  12  2  2ð¼ 6Þ, go to Step 4.6. Step 4.6. O = O  {2} = {1, 3, 5, 8} and go back to Step 4.2. Step 4.2. i ¼ j ; j 2 OII g ¼ 8 and f8 = 2.  MaxfjjL   1    Step 4.3. R ¼ 32 ¼ 3  21  12  1  1 ¼ 32 .  1  3 1 Step 4.4. R ¼ 2  3  21  2  1  1 ¼ 0, f8 = 2  1 = 1, and x = 1. Step 4.5. Because R = 0, let x = 0 and go to Step 4.7. Step 4.7. Because x = 0, stop. This is the end of the steps.

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