a scheduling based reorder point system with variable lead time in a multi-item production...

________________________________________

† : Corresponding Author

A scheduling based reorder point system with variable

lead time in a multi-item production environment

Jeongja Jeong† 1and Takashi Irohara2 Sophia University

7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan Email:[email protected]

[email protected]

Abstract. The reorder point system(RPS) in a multi-item production environment is addressed in this paper. The RPS-based inventory control system is the most common procedure for order-release planning in a dis-crete parts environment. Generally, manufacturers need to be more responsive to dynamic markets in order to save a cost while keeping the competitiveness. However the traditional reorder point has been calculated with fixed lead time which is estimated by using historical data or past experiences. And also, in an actual invento-ry control, the lead time does change and assuming fixed lead time is quite unrealistic. However, a large change in lead time may result in excessive stock or shortage of supply. And this assumption is outdated be-cause of its inability to accurately capture the dynamics of the production environment. In reality, lead times may be dependent upon dynamic conditions: amount of load in the shop, capacity of the shop, the order prior-ity, resource schedules, part routings, order lot size, and shop rules and constraints. To support the planning function for an agile manufacturing environment, a planning system must consider real-world capacities so that the order-release plan generated is feasible. In this paper, a new approach to the RPS, in which the lead time is estimated by scheduling, is proposed. The RPS planning approach in the flow shop and the job shop production environment are considered. New approach to the RPS is compared against the traditional reorder point system. The numerical experiment shows the effectiveness of the proposed approach. Keywords: Reorder point, inventory, dynamic, lead-time, multi-item, scheduling.

1. INTRODUCTION

Recent years, manufacturers need to be more respon-sive to dynamic market demands and reduce costs and deal with short product life cycle(Jimmie browne et el. 1996) in order to be competitive. At this point, inventory plays a major role in deciding the overall manufacturing costs, and a good scheduling system should balance the on-time deli-very of products low inventory. But, typical mass produc-tion environment that is characterized by large on-hand inventory, economic order quantity (Hillier et al. 2001)(K.L. HOU et al. 2006) are now being transformed into multipro-duct production environments characterized by zero inven-tory, short production lead time. So, in an advanced plan-ning and scheduling environment, manufacturers are ob-liged to estimate due dates for customer orders and satisfy

their requirements (M. Kuroda et al. 2008). Michael masin et al. presents an auction-based algorithm for simultaneous scheduling to minimize both the due date and inventory cost (Michael et al. 2007). G.Z Mincsovics et al. estimate due date by workload-dependent capacity control (G.Z Minsovics et al. 2009). Joanna jozefowska propose produc-tion planning and control philosophy that seeks to eliminate waste of waiting time, overproduction and inventory (Joan-na jozefowska et al. 2007). And, Ota takako Study on the Optimum Ordering Quantity for Products with a Short De-terioration Time (Ota takako et al. 1974).

In this paper, we propose a reorder point system aim-ing to keep proper inventory under dynamically changing situation. And, in this study, we propose the RPS which decreases the over stock and stock out. Conventional RPS is no longer acceptable because it can’t respond to dynamic

APIEMS2009 Dec. 14-16, Kitakyushu

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demands. The Conventional RPS has assumed production lead

time as fixed value which is calculated by past experience, historical data. But if load excess the capacity of system, there would be waiting queue in front and behind produc-tion facilities and lead time will be long. In addition, if the systems handle various products, lead time varies according to producing products. Therefore it is inadequate to set unique reorder point because production lead time changes time to time according to status of system. In this paper, we propose a RPS approach considering variable lead time and multi-item manufacturing environment. 2. CONVENTIONAL MODEL 2.1 Conventional RPS(1)

Conventional RPS(1) is one of the most common me-

thods of stock control. If stock of item fall below preset reorder point, RPS makes order to replenish stock (Hillier et al. 2001)(G. Hadley et al. 1963).

In practice, there is lead time between order placement and arrival of item. Therefore RPS sets reorder point consi-dering the lead time and safety stock. The safety stock is a term used by inventory specialists to describe a level of extra stock that is maintained below the cycle stock to buf-fer against stock out. Conventional RPS(1) is calculated as below.

Reorder

point

L

Inve

ntor

y lev

el

L Ltime

Figure 1 : Reorder point and lead time

K (1) K : reorder point L : production Lead time : safety stock coefficient σ : standard deviation of demand per period d : average demand per period

The production lead time L of the conventional R

PS(1) is estimated as fixed value which be calculated by preset lead time. Hear, the preset lead time is calculated as sum of processing time and expected waiting time.

2.2 Conventional RPS(2)

Conventional RPS assumes production lead time is set

as fixed value and it is calculated as sum of processing time and room time. In conventional RPS(2), production lead time is figured out by discrete random variable as figure 2 (Suguro Takao et al. 2004).

Conventional reorder point is calculated as below. K = (2) (3) Li : i th production lead time in ascending order P(Li): probability of Li s : standard deviation of demand per period ro : stock out allowable ratio (for example, ro=5% => Lm=6, ro=25%=> Lm=5) Lm : lead time that satisfies the ro (m is calculated according to the formula (3). ) The conventional RPS(2) considered to reduce inven-

tory level through improvements in production lead time of conventional RPS(1). Even so, it does not correspond with practical situation in which lead time is not fixed value and varies according to load status of the system.

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10

event probability

Lead time

Figure 2: Event probability of production lead time 3. PROPOSED MODEL 3.1 Proposed RPS

Considering the above, this paper proposes RPS that

LLd

mm LsLd

m

iio

m

ii LPrLP

1

1

1

)()1()(


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takes into account both production scheduling and process load status for calculation of variable lead time.

The conventional RPS(1) has assumed production lead time as fixed value. But if load excess the capacity of sys-tem, there would be waiting queue in front and behind pro-duction facilities and lead time will be long as figure 3,4. Figure 3: In the absence of waiting time Figure 4 : In the presence of waiting time

Incidentally, this study considers change of the waiting time of production. Thus production lead time of pro-posed RPS is variable by load status of production process. The proposed RPS model approach is to take planned lead times as input to the process scheduling. Specifically, a waiting time occurs at the factory doing multi-item produc-tion environment.

For example, we consider in the case of after a change of production lead time as figure 5.

Conventional RPS : L1=L2=L3 =L(L=preset lead time) Proposed RPS : L1≠L2≠L3 (Lt=lead time of period t) As in the example given above, proposed RPS form an

accurate estimate of production lead time for period t.

Reorder

point

L1

Inve

ntor

y lev

el

L2 L3time

Figure 5: Lead time of period t 3.2 Algorithm

We propose the following algorithm to decide the reorder point for product i=1,2,…,N of the period t=1,2,…,T.

Initial state isｔ＝1. 1) Scheduling in period t, and decide lead time

Li, t 2) Set up reorder point based on the lead time

Li, t (2) 3) Quantity demanded di,t 4) Changes on the upgraded inventory level

(Ii,t+1=Ii,t - di,t) 5) If the inventory is below the reorder point

level, production order is released. If the order of multiple products occur at the same

time, set an order of priority (LPT : Longest processing time). (Ii,t < Ki,t),

6) t=t+1. 7) go back to 1)

K i, t : Reorder point of period t for product i di,t : Demand of period t for product i L i, t : Lead time of period t for product i I i, t : Initial inventory of period t for product i

id : Average demand of product i i : Standard deviation of demand for product i

N : Number of product 3.3 Structure of the proposed model

The proposed system consists of 2 warehouses and m production machineries. The material warehouse is located at the start of system and finished product warehouse is located at the end of the system. The n products are loaded from material house and processed by the system. After p manufacturing processes, products arrive at finished prod-

Process3

Process2

Proces1

JOB2

JOB2

JOB2

L

tiitiiti LLdK ,,,

Process3

Process2

Proces1 JOB1

JOB1

JOB1 JOB2

JOB2

JOB2 Waiting Time

L

time

time


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uct warehouse as completed products. The inventory of finished goods is checked every day.

The firm places the order when the inventory is below the reorder point.

As figure 6, we assume that the process is hybrid flow shop which is combined a flow shop and a parallel machine. Or, as figure 7, a completely job shop is treated in which all routines differ from each other.

If every part route is ordered in a consistent unidirec-tional manner, the shop is considered a pure flow shop. On the other hand, if every item in the shop is ordered in a non-unidirectional fashion, the shop is considered a pure job shop. If it’s different from each process of 3 of hybrid flow shop (figure 6) and production sequence by a processing target thing, or it’s shared with two ways by a job shop(figure 7) and an experiment is made.

Specifically, from the production lead time L by the above-mentioned system it’s possible to find an appropriate reorder point according to the congestion situation of the production place by production reorder point K(1).

If there is order over two products at the same time, it is decided about the order of priority by dispatching rules. (LPT : Longest processing time) (K. Yosimoto et al. 1999)

m : number of machine n : number of product p : number of process

Machine

1Machine

4 Machine

7

Machine2

Machine 5

Machine 8

Machine3

Machine6

Machine 9

material

product

Process 1

Process 2

Process 3

Figure 6: Hybrid flow shop

material

Machine

Machine

Machine

Process 1

Process 2

Process3 product

Our evaluation criteria are frequency of stock out and inventory quantity as figure 8.

Ｌ：Long

Ｌ：Short

Inve

ntor

y lev

el

Frequency of stock out

Figure 8 : Evaluation criteria 4. THE NUMERICAL EXPERIMENT 4.1 Parameter settings

The best way of presenting the approach is through

comparison with the conventional RPS model(1)(2). Ex-perimental input information fixes machining time, lot size, beginning inventory and customer demands like table 1, and handles 5 kinds of product, simulates for 1000 days as table 1 and table 2. A conventional technique and pro-posed technique was compared and checked stock and frequency of stock out were checked like table 4 against 9 types of the safety stock coefficient like table 4.

A conventional technique (1), (2) fixes and handles a production lead time like table 3. But by a proposed tech-nique, a reorder point is changing because a production lead time is changing.

We investigate the relationship between average inventory and frequency of stock out for each technique.

In summary, the simulation planning pass consists of determining order-release dates from production lead time based on actual component and assembly lead times. Production lead time is a function of shop load, order priority, resources schedules, part routings and constraints. But, conventional RPS uses fixed component lead times as table 3.

Figure 7 : Job shop


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Table 1: Input data of hybrid-flow shop

Demand quantity (normal distribution) ： N(120,102)

Table 2: Input data of job shop

product Processing time machining

sequence

Lot sizeInitial inventory

1 7 10 18 M1→M2→M3 800 1000

2 2 17 12 M1→M3→M2 700 900

3 15 10 5 M3→M1→M2 600 800

4 10 7 10 M3→M2→M1 500 700

5 5 10 10 M2→M3→M1 400 600 Table 3 : Job Shop Ratio

Table 4: Safety stock coefficient α

product Processing time Lot size

Initial inventory1 2 3

1 30 40 72 800 1000

2 10 70 50 700 900

3 60 40 20 600 800

4 40 30 40 500 700

5 20 40 40 400 600

JSR

Product 100% 75% 50% 25% 0%

1 M1→M2→M3 M1→M2→M3 M1→M2→M3 M1→M2→M3 M1→M2→M3





α 3.090 2.576 2.326 1.960 1.645 1.282 0.842 0.524 0.253

P(%) 0.001 0.005 0.010 0.025 0.05 0.10 0.20 0.30 0.40


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Table 5: Comparison of the proposed with conventional model

4.2 Hybrid flow shop

Result of the frequency of stock out for each safety factor in the hybrid flow shop test is shown in table 7.And result of the average inventory for each safety stock coeffi-cient in the hybrid flow shop test is shown in table 7 Table 6: Frequency of stock out for each safety stock coefficient (hybrid flow shop)

Safety

stock coef-

ficient

frequency of stock out

conventional proposed

(1) (2)

0.253 0 88.2 18.0

0.524 0 72.8 10.6

0.842 0 69.2 9.0

1.282 0 53.6 6.4

1.645 0 40.8 3.4

1.960 0 39.6 3.6

2.326 0 29.8 4.0

2.576 0 27.0 2.2

3.090 0 19.2 1.6

average 0 47.9 6.5

Table 7 : Average inventory for each safety stock coeffi-

cient(hybrid flop shop)

Safety

stock coef-

ficient

average inventory


(1) (2)

0.253 5929.7 2512.3 2265.6

0.524 5993.9 2541.8 2297.2

0.842 6037.7 2581.0 2328.9

1.282 6105.5 2625.6 2373.4

1.645 6143.5 2663.0 2408.0

1.960 6223.7 2701.2 2447.3

2.326 6270.0 2740.8 2482.7

2.576 6307.7 2791.7 2507.4

3.090 6394.3 2824.6 2562.5

proposed model：L= variable lead time

conventional model(1)：L= maximum of lead time

conventional model(2)：L= expected value of lead time

Describing table 6,7 with graph, it is equal to

figure 9,10

conventional

proposed (1) (2)

Processing time constant constant constant

Waiting time variable variable variable

Production

Lead time constant expected value variable

Reorder point Fixed Fixed variable


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0

10

20

30

40

50

60

70

80

90

100

0.25 0.52 0.84 1.28 1.65 1.96 2.33 2.58 3.09

freq

uenc

y of

stoc

k ou

t

safety stock coefficient

conventionaltechnique (1)conventional technique (2)proposedtechnique .

Figure 9: Frequency of stock out for each safety stock

coefficient (hybrid flow shop)

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

0.25 0.52 0.84 1.28 1.65 1.96 2.33 2.58 3.09

inve

ntor

y le

vel



Figure 10 : Average inventory for each safety stock

coefficient (hybrid flow shop)

Making graph from the relationship between inventory and frequency of stock out for each technique, it becomes to be like figure 11. According to figure 11, we can see that our evaluation criteria is satisfied.

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

0 20 40 60 80 100

inve

ntor

y le

vel



Figure 11 : Relationship between average inventory

and frequency of stock out for each technique. (hybrid flow shop)

4.3 Job shop 4.3.1 Job Shop Ratio =100%

Result of the frequency of stock out for each safety

factor in the hybrid flow shop test is shown in table 8. And result of the average inventory for each safety stock coeffi-cient in the hybrid flow shop test is shown in table 9. Table8 : frequency of occurrence of stock out(job shop)

Safety

stock coef-

ficient

frequency of occurrence


(1) (2)

0.253 0 143 129

0.524 0 132 120

0.842 0 104 89

1.282 0 77 71

1.645 0 70 55

1.960 0 51 45

2.326 0 37 33

2.576 0 30 28

3.090 0 27 22

Table 9: volume of average inventory (job shop)

Safety

stock coef-

ficient

volume of average inventory


(1) (2)

0.253 5213.5 3371.9 1892.7

0.524 5250.0 3391.0 1912.3

0.842 5276.4 3396.7 1926.1

1.282 5345.6 3405.2 1957.5

1.645 5380.9 3448.6 1969.3

1.960 5408.8 3446.1 1971.0

2.326 5441.8 3463.4 2009.6

2.576 5492.8 3490.6 2022.6

3.090 5528.1 3502.4 2050.9

Describing table 8, 9 with graph, it is equal to


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figure12,13.

0

20

40

60

80

100

120

140

160

0.25 0.52 0.84 1.28 1.65 1.96 2.33 2.58 3.09

freq

uenc

y of

stoc

k ou

t

safety stock coefficinet


Figure 12: Frequency of stock out for each safety

stock coefficient (job shop)

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

0.25 0.52 0.84 1.28 1.65 1.96 2.33 2.58 3.09

inve

ntor

y le

vel



Figure 13: Average inventory for each safety stock

coefficient (job shop) Making graph from the relationship between inventory

and frequency of stock out for each technique, it becomes to be like figure 14. According to figure 14, we can see that our evaluation criteria is satisfied.

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

0 50 100 150 200

inve

ntor

y le

vel


conventionaltechnique(1)conventionaltechnique(2)proposedtechnique

Figure 14: Relationship between average inventory

and frequency of stock out for each technique.(job shop)

4.3.1 Job Shop Ratio is variable

The shop characteristics are parameterized such that

the randomness of routines used to manufacture products, called the job-shop ratio. Accordingly, a job shop ra-tio(JSR) of 0% indicates that a flow shop is treated in which all routines are identical, whereas all routines differ from each other. The Job Shop Ratio is calculated as below formula (4) (M. Kuroda et al. 2002).

(4)

Figure 15 : Job Shop Ratio

We graph from relationship frequency of stock out for each job shop ratio figure 16.

0

50

100

150

200

250

300

350

400

450

100% 75% 50% 25% 0%

freq

uenc

y of

stoc

k ou

t

Job shop ratio

proposedtechniqueconventionaltechnique(2)

Figure 16 : Frequency of stock out for each job shop

ratio

4.4 Analysis of the results

Production lead time of conventional technique (1) is calculated by the maximum value of simulation for 1000days . In this case, the inventory has excess compared with a proposed technique. Thus conventional tech-nique(1) keeps a large inventory despite the extremely short lead time. But, proposed technique is always depending on

1001

jobsofnumbertotal

routineidential

thewithjobsofnumber

JSR　　　

　

　　　　


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state of process. Accordingly we could say that the pro-posed RPS is to help limit the inventory level by variable lead time. On the other hand, a way using the lead time on the basis of stock out allowable ratio (conventional tech-nique (2)) has increased average inventory and stock out substantially compared with the proposed technique. Con-ventional RPS(2) takes a new approach to improve the in-ventory problem of conventional RPS(1). Thus convention-al RPS(2) focus on to reduce of inventory. As a result of the above, we found out that inventory or frequency of stock out can be improved by setting lead time appropriate to each product based on a result of the production lead time L scheduling. Results of the experiments show that the proposed technique performs better than conventional tech-nique (1)(2).

In case of a Job Shop Ratio, when the job shop ratio of process is 100%, fixed lead times may be reasonable. On the other hand, when the job shop ratio of process is 0%, proposed technique showed a remarkable improvement over conventional technique because waiting time get long-er as we approach the job shop ratio 0%. The conventional technique does not deal with change of production lead time.

5. CONCLUSION

In this paper, we consider about the production lead time by scheduling when calculating an reorder point, to improve both the over stock level and frequency of stock out which are a problem of conventional reorder point sys-tem. And lead time of the proposed RPS is variable and are based on realistic. The congestion situation of the produc-tion process was considered and the new reorder point sys-tem which considered the changing lead time was proposed. And the validity of the proposed system was inspected by a numerical experiment. Proposed RPS was fairly com-pared with a conventional RPS. In a multi-item production, the proposed RPS will consistently result in better overall system performance relative to a conventional RPS. In summary, this experimentation indicates that proposed RPS may be a superior for a multi-item production environment.

REFERENCES

G. Hadley, T.M. Whitin (1963), Analysis of inventory system: Prentice Hall, Englewood Cliffs, NJ.

Hillier and Lieverman(2001), Introduction to opera-tions research: Mcgraw-hill international, pp.941.

Jimmie Browne, John Harhen, James Shivnan(1996), Production management systems : Addison-wesley, pp. 7.

Joanna Jozefowska (1996), Just-In-Time scheduling , Models and algorithms for computer and manufacturing systems: Springer.

Kazuho Yoshimoto and Takashi Irohara (1999) Pro-duction and operations management : Japanese standards association, pp. 195.

K.L.HOU and L.C.LIN (2006), An EOQ model for deteriorating items with price and stock-dependent selling rates under inflation and time value of money: International journal of systems science, pp. 1131-1139.

Masin M, pasaogullari MO, Joshi S (2007), Dynamic scheduling of production-assembly networks in a distri-buted environment: IIE Transactions, Vol. 39, pp. 395-409.

M.Kuroda and H. Mihira (2008), Strategic inventory holding to allow the estimation of earlier due dates in make-to-order production : International journal of produc-tion research, Vol. 46, No. 2, pp. 495-508.

M.Kuroda and H. Shin and A. Zinnohara (2002), Ro-bust scheduling in an advanced planning and scheduling environment: International Journal of production research, Vol.406, No.15, pp.3655-3668.

Mincsovics GZ, Dellaert NP(2009), Workload-dependent capacity control in production-to-order systems : IIE Transactions, Vol. 41, No. 10, pp. 853-865.

OTA Takako , TAGAWA Shinichi , TAKEOKA Ka-zushige(1974) A Study on the Optimum Ordering Quantity for Products with a Short Deterioration Time: Journal of Japan Industrial Management Association ，pp．11-20.

SUGURO Takao and KURODA Mitsuru(2004)Safety Stock and Reorder Point for Reordering Point System with Variable Lead Times : Journal of Japan Industrial Man-agement Association, Vol.55，No．2，pp.90-94.

SUGURO Takao (2003) Process of finding the proper inventory : Business line, pp.90-94.

Takashi Irohara : “Lagrangian relaxation algorithms for hybrid flow-shop scheduling problems with limited buffers”, International Journal of Biomedical Soft Compu-ting and Human Sciences, Vol.14, No.2,(2009) to appear

Virginia Lo, Jens Mache, and Kurt Windisch(1998), A comparative study of real workload traces and synthetic workload models for parallel job scheduling : Springer-verlag berlin heidelberg, pp. 25-46.


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Yugo HIMOTO, Atsushi SAITO and Masayuki MATSUI (2004): A case study on setting of safety stocks in SCM/MRP : J Jpn Ind Manage Assoc 55, pp．51-58

AUTHOR BIOGRAPHIES

Jeongja Jeong is a graduate student at Sophia University in the Department of Information. She received her B.S in Industrial engineering from Kangwon University, Korea. Her email address is <[email protected]> Takashi Irohara is an Associate Professor at Sophia Uni-versity in the Department of Information and Communica-tion Sciences. He received his B.S, M.S., and Ph.D. in In-dustrial and Management Systems Engineering from Wa-seda University, Tokyo. His research interests include facil-ity layout and material handling, manufacturing scheduling and logistics optimization. His papers have been published in Computers and Industrial Engineering, Journal of Japan Industrial Management Associations, Transactions of the Japan Society of Mechanical Engineers, Journal of the So-ciety of Plant Engineers Japan, Journal of Japan Society of Logistics Systems and others. He is a member of the Japan Industrial Engineering Association, the Japan Society of Mechanical Engineers, the Operations Research Society of Japan and the Scheduling Society of Japan. His email ad-dress is <[email protected]>


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a scheduling based reorder point system with variable lead time in a multi-item production...

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