A Study on the Prediction of Productive Schedule during ...nova.bime.ntu.edu.tw/~dsfon/RiceNursery/linearP.pdf · programming (LP) are widely applied as an analysis tool in operational

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邱 \育 \英文861210

A Study on the Prediction of Productive Schedule during the Period of Paddy-Rice Nursery

Yi-Chich Chiu1, Din-Sue Fon2

1. Associate Professor, Dept. of Agricultural Machinery Engineering, National I-Lan Institute of Agriculture and Technology. 2. Professor, Dept. of Agricultural Machinery Engineering, National Taiwan University.

ABSTRACT

The objective of this study was to develop a linear programming model to analyze the productive schedule during the period of paddy-rice nursery. The model can help the managers of the rice seedling centers to make a more precise decision. To verify the model, a rice seedling center was chosen to analyzed on the first rice crop in 1997. The result shows that the cost needs about $1,789,043 NTD, and obtains a profit of about $754,202 NTD in a supplying of 91,385 boxes. In this productive schedule, the maximum quantities of boxes in the greening field are about 4,600 boxes, and about 95.5 % of seedling can be sold out in 4 days after the grown-up.

The limiting condition on minimum treatment boxes in each batch affects the cost significantly, with the result that it needs a lower cost, about $1,790 thousand NTD, under 4,000 boxes/batch, while it needs a higher cost, about $1,950 thousand NTD, at 6,000 boxes/batch. Consequently, using the model to predict the productive schedule not only reduces the cost but also provides the seedling in the best status for transplanting.

The supplying quantities of the boxes in each day during the period of transplanting season can be expressed by a quadrinomial equation. Hereby, a supplying total quantities from 73,928 to 110,892 boxes was generated by the polynomial to analyze the productive schedule. The result shows that the productive scale could be expanded to about 110,892 boxes under the same hardware. In this status, the profit could be increased to about $920 thousand NTD, and the cost/profit ratio is 2.35. As to the relationship between supplying quantities scale and greening field size, it can be calculated by a regressive formula.

Keywords : Linear programming, Productive Schedule, paddy-rice nursery.

Introduction

Rice is a staple crop in Taiwan, which traditional cultivation is to transplant rice seedlings to the paddy field by rice-transplanter. Hereby, there are about 600 rice seedling centers providing the rice seedlings to the farmers. The seedling is nursed in winter season for the first crop and summer season for the second crop.

The operation of rice nursery is seasonal and time effective. A over-ripe or unripe of seedling will reduce the quantities and quality of grain production in the future harvest.

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Seriously, a over-grown of seedling is not properly transplanted so that the seedling will be relinquished causing a great benefit loss for farmers. Therefore, some factors, such as market supply and demand, productive scale and cost, etc., were considered before the managers of rice seedling centers decide the productive schedule and the quantities of the seedling during the period of rice nursery. Other factors, such as the daily seeding throughput, the capacity of greening field, and the minimum treatment boxes in each batch, also affect the productive schedule. Therefore, it is difficult to decide the productive quantities of the seedling during the period of rice nursery.

For the boxes capacity of greening field, it concerns about the type and size of furrows. Chiu and Fon (1997) analyzed this problem, which results can be referred to transfer the known area of greening field to the boxes capacity.

Hiller and Lieberman (1986) pointed out that Statistic, simulation, and linear programming (LP) are widely applied as an analysis tool in operational research. LP uses the optimum probable method to allocate the limited resources in each relative activity , which model is built by one-order mathematical function.

The objectives of this study were to develop a LP model for productive schedule planning during the period of rice nursery. In the LP model, some important constraint conditions were considered to achieve the optimum production.

Literature review

An analysis on the proper age of seedling for transplanting

Tzeng(1977) analyzed the relationship between the age of seedling and the production rate of grain for the first crop of seedling, with the result that the age of seedling is properly transplanted into the field at the period of 15 - 25 days after seeding stage. In other words, the other age of seedling would easily die after transplanting into the field.

It is necessary that the seeded boxes must be piled up in a conditioning room for 3 days for the seeds easy and rapid sprouting. Then, the seeded boxes will be moved into the greening field for further growth, that is, called input operations. Thus, the useful supply duration of the seedling is 12-22 days averaged 17 days in the greening field. In other words, the seedling will can be provided to farmer to transplant started on the 12th day after the seedling moves into the greening field, which the day is defined the 1st day after the seedling grown-up. The seedling stayed on the 13th day in the greening field is defined the 2nd day after the grown-up, and so on. The seedling will not be properly used to transplant if it stayed over the 22nd day in the greening field. Thus, for the rice seedling center, it will lose the over-grown of seedling.

An analysis on the nursing cost of the seedling

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Tuan(1995) analyzed the nursing cost of seedling for the first rice crop, with a result that the nursing cost was about $ 19.55 NTD per box while the labor cost was 32 % of total cost. However, the labor cost in the aspect of the boxes management in greening field is about 6.1 % of the total labor cost. So, accounting for the stayed average day of boxes in greening field is 17 days, the daily management cost(Sc) in greening field is $ 0.02 NTD, that is:

Sc= × × × =19550320061117

002 -------. . . . ----------------- (1)

The more day the boxes stay in greening field, the costlier the manager of rice seedling center needs. The effective stayed day is 12-22(X) days for the seedling in greening field, that is, the 1st - the 11th (R) day after the grown-up. Thus, the nursing cost (SR)would be calculated as follows.

S XR = − − ×195517 002 -----. ( ) . ------------------------- (2)

So, the nursing cost(S1 - S11) for the 1st- the 11th day after the grown-up can be calculated by formula (2), that is 19.45, 19.47, 19.49, 19.51, 19.53, 19.55, 19.57, 19.59, 19.61, 19.63, and 19.65, respectively. It represents all of the preceding bathes of seedling are successfully sold for transplanting, and the price is $ 27.83 NTD per boxes (Tuan,1993). Oppositely, for those seedling over the 12th day after the grown-up, the lost value is that nursing cost plus the seedling price for rice seedling center. Thus, S12 = 19.65+27.83=47.48 NTD/box。

MODEL DEVELOPMENT

Background

A local rice seedling center in I-Lan county, located in the northeast of Taiwan was chosen to develop the model, which center produces about 90,000 boxes in the first crop in 1997 providing about 383 ha. of rice seedling needed. The supply period for rice seedling was from February 12 to March 25 totaled 42 days. Figure 1 shows the daily supplying quantities of rice seedling during the period of rice seedling transplanting in the first crop. It is obvious that the supplying quantities is fewer both in the early and late stages but larger in the middle stage. A polynomial regression analysis was applied by the software EXCEL 7.0 to analyze the relationship between supplying quantities (Nd) and the date (d) in the period of rice seedling transplanting. The regression equation is as follows, which relative coefficient (R2) is 0.87.

Nd = 0.0152 d4-1.3041 d3+29.645 d2-93.572 d+1440.2 ---------- (3)

Variable definition

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Limiting in both the labor and resource allocations, the rice seedling center do a batch of seeding operation every 4 day. The seeding operation is a multitude of procedures and labor-intensive, which procedures are mainly including seed sorting, sterilizing, soaking, pre-germination, sprouting, and drying of seeds, seeding, stacking, moving into greening field for greening etc.. Each batch of rice seedling must be sold out within 11 days after the grown-up. Otherwise, the rice seedling center will lose the over grown-up seedling. Thus, it is supposed that variable Xij means the supplying quantities for batch number i after the jth day of the grown-up. It means the seedling is useless when the j is 12. So, the j is from 1 to 12. The known supplying duration of the seedling is 42 days, thus the nursery period, that is from seeded boxes moving into the greening field to the end of the supply, is 53 days because the first batch of seedling will start to be sold on the 12th day of seedling moving into the greening field. It is known that the rice seedling center produces a batch of seedling per 4 days, so there are 10 batches of seedlings produced, that is, i from 1 to 10. Through above analysis, Table 1 shows the variables in contrast with each batch of seedling, the supplying date, and nursery date. However, the last two variables, X9,11 and X10,7 , in this Table represent the seedling will lose because of over the whole supplying period.

Objective function

The purpose of the productive schedule is to obtain the lowest productive cost under limited resource conditions. The rice seedling center can thereby get the maximum profit. Therefore, lowering the productive cost was chosen as the objective function in this study. However, the maximum productive profit can be calculated as the seedling price per box pluses the supplying quantities, then minus the productive cost. The objective function is as follows.

min Z = S1(X11+X21+X31+… … +Xn1) +S2(X12+X22+X32+… … +Xn2)

+S3(X13+X23+X33 +… … +Xn3) +… … +S12(X1,12+X2,12+X3,12+… … +Xn,12)

-------- (4)

As the batch number of seedling is 10 (n=10). Formula (4) was hereby abbreviated as

min ------------------------------------- (5)Z S Xjj

iji

== =

∑ ∑1

12

1

10

( ),

Constrained conditions

1. the constraint on the sale of rice seedling

The rice seedling centers mostly predict their daily sale quantities of seedling during the transplanting period based on their experiences in past years, including the prediction of transplanting date and cultivating area of the farmers, so that the rice seedling center can

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further decide the nursery schedule. Therefore, according to the seedling demand in the first crop referred to Figure 1, the constraint on the sale of rice seedling can be described as follows.

the sale of seedling on the 1st day : X11 = 980 ------------------- (6)

the sale of seedling on the 2nd day : X12 = 1250 ------------------ (7)

the sale of seedling on the 3rd day : X13 = 1710 ------------------- (8)

the sale of seedling on the 4th day : X14 = 920 ------------------- (9)

the sale of seedling on the 5th day : X15 +X21 = 2305 --------------- (10)




the sale of seedling on the 9th day : X19 +X25 +X31 = 2650 ----------- (14)

the sale of seedling on the 10th day : X1,10 +X26 +X32 = 1640 -------- (15)


the sale of seedling on the 12th day : X28 +X34 = 2470 -------------- (17)

the sale of seedling on the 13th day : X29 +X35 +X41 = 2250 ---------- (18)



the sale of seedling on the 16th day : X38 +X44 = 2980 -------------- (21)

the sale of seedling on the 17th day : X39 +X45 +X51 = 2880 ---------- (22)

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the sale of seedling on the 18th day : X3,10 +X46 +X52 = 3670 --------- (23)

the sale of seedling on the 19th day : X3,11 +X47 +X53 = 3860 --------- (24)


the sale of seedling on the 21st day : X49 +X55 +X61 = 2840 ----------- (26)

the sale of seedling on the 22nd day : X4,10 +X56 +X62 = 3430 ---------- (27)

the sale of seedling on the 23rd day : X4,11 +X57 +X63 = 3270 ---------- (28)

the sale of seedling on the 24th day : X58 +X64 = 3150 ---------------- (29)

the sale of seedling on the 25th day : X59 +X65 +X71 = 3650 ------------ (30)

the sale of seedling on the 26th day : X5,10 +X66 +X72 = 3140 ----------- (31)

the sale of seedling on the 27th day : X5,11 +X67 +X73 = 3270 ----------- (32)

the sale of seedling on the 28th day : X68 +X74 = 2940 ----------------- (33)

the sale of seedling on the 29th day : X69 +X75 +X81 = 2870 ------------- (34)

the sale of seedling on the 30th day : X6,10 +X76 +X82 = 2920 ------------ (35)

the sale of seedling on the 31st day : X6,11 +X77 +X83 = 2250 ------------ (36)

the sale of seedling on the 32nd day : X78 +X84 = 1620 ------------------ (37)

the sale of seedling on the 33rd day : X79 +X85 +X91 = 1050 -------------- (38)

the sale of seedling on the 34th day : X7,10 +X86 +X92 = 1340 ------------- (39)

the sale of seedling on the 35th day : X7,11 +X87 +X93 = 870 ------------- (40)

the sale of seedling on the 36th day : X88 +X94 = 1270 ------------------ (41)

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the sale of seedling on the 37th day : X89 +X95 +X10,1 = 740 ------------- (42)

the sale of seedling on the 38th day : X8,10 +X96 +X10,2 = 960 ------------ (43)

the sale of seedling on the 39th day : X8,11 +X97 +X10,3 = 670 ------------ (44)

the sale of seedling on the 40th day : X98 +X10,4 = 650 ------------------ (45)

the sale of seedling on the 41st day : X99 +X10,5 = 410 ------------------ (46)

the sale of seedling on the 42nd day : X9,10 +X10,6 = 390 ----------------- (47)

2. the constraint on the capacity of greening field (D)

From Table 1 we can know that there were 3 batches (the 1st , the 2nd, and the 3rd batches) of seedling in the greening field before the sale of seedling. Thus, the sum of the first 3 batches of seedling must less than the capacity of greening field. Following the sale of seedling proceeding, it just constrains that the quantities of next following batch of seedling plus the remainder of seedling quantities in greening field to less than the capacity of greening field. The greening field, analyzed in this study, has a seedling capacity of 97,214, that is, D = 97,214 boxes. So, the constrained equations are

before the sale of seedling : i

ijj

X= =

∑ ∑ ≤1

3

1

12

, D ------------------------------------- (48)

moving the 4th batch of seedling into greening field :

X Xjj i

ijj

13

12

2

4

1

12

= = =∑ ∑ ∑+ ≤, D ------------------------- (49)


X X Xjj

jj i

ijj

17

12

23

12

3

5

1

12

= = = =∑ ∑ ∑ ∑+ + ≤, D -------------------- (50)


X X X Xjj

jj

jj i

ijj

111

12

27

12

33

12

4

6

1

12

= = = = =∑ ∑ ∑ ∑ ∑+ + + ≤, D ------------- (51)

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X X X Xjj

j jjj i

ijj

211

12

3 43

12

7

12

5

7

1

12

+ + + ≤= == = =∑ ∑∑ ∑ ∑ , D ---------- (52)


X X X Xjj

j jjj i

ijj

311

12

4 53

12

7

12

6

8

1

12

+ + + ≤= == = =∑ ∑∑ ∑ ∑ , D --------- (53)


X X X X Xjj

jj

jj i

ij jjj

411

12

57

12

63

12

7

8

91

11

1

12

+ + + + ≤= = = = ==∑ ∑ ∑ ∑ ∑∑ , , D ---- (54)


X X X X X Xjj

j jjj

j j jjjj

511

12

6 73

12

7

12

8 9 101

7

1

11

1

12

+ + + + + ≤= == ===∑ ∑∑ ∑∑∑ , , D -- (55)

3. the constraint on the daily maximum seeding throughput (Q)

All rice seedling centers in Taiwan follow a similar pattern of seeding operation with an automatic seeding system. Through a survey, the daily maximum throughput is 12,592 boxes, counting for working 8 hour per day in this rice seedling center. Thus, the seedling quantities of each batch have to less than or equal to the daily maximum seeding throughput. So, the constrained equation is

X iijj

, ,,,....,≤ ==

∑ Q for ---------------- (56) 123 10 -------1

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4. the constraint on the minimum quantities of each batch of seedling (P)

In the real-world seeding operation, it is usually treated with a batch processing for some relative materials, such as seeds, soil, and liquid pesticides. According as the managers of the rice seedling centers said that it is a reasonable operation for each batch of seedling must be larger than 4,000 boxes. Thus, the constraint is

X iijj

, ,,,....,≥ ==

∑ P for ---------------- (57) 123 10 -------1

12

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5. the constraint on non-negative variables

Xi,j >=0 for i= 1 - 10 ; j= 1 - 12 ----------------(58)

RESULTS AND DISSCUSIONS

The software LINDO version release 5.02 was applied to solve the model for finding the optimum solution. Because the input parameters, such as daily sale quantities of seedling, the capacity of greening field, and the minimum treatment boxes in each batch, were collected from the real operations, the executive results of the model can hereby be compared with the status of the real operations to verify the model. And further, using the model to predict the productive schedule, the profit will be increased under the same constraint conditions.

1. executive results

The results with the model analysis show the total nursing cost is $ 1,798,470 NTD and the profit is $ 744,775 NTD in a supplying of 91,385 boxes, while the actual nursery quantities is 91,650 boxes with a loss of 265 boxes. As contrasted with the model analysis, the experienced predicted operations is 91,565 boxes of total nursery quantities with a loss of 180 boxes and the cost is $ 1,789,043 NTD while the profit is $ 754,202 NTD. Both results are close to each other and the profit can be increased about $ 9,427 dolls for model application. Table 2 shows the analyzed value for each variable, as a result that 24.3% of total nursery seedling were sold out on the 1st day of the grown-up, 24.4% on the 2nd day, 24.9% on the 3rd day, and 21.9% on the 4th day. Therefore, almost all seedlings (95.5%) has effectively been transplanted within 4 days after the grown-up. However, that is a contrast with experienced prediction by the manages of rice seedling centers. With the experienced prediction, it is quite obvious in Figure 2 that only about 24.4% of seedling were sold out within the 4th day after the grown-up, 62.2% of seedling during the 5th - 7th day, and 13% of seedling after the 7th day. As it is hard to keep the best of seedling status for transplanting by pass experience to predict the productive schedule during the period of seedling nursery, so it is worth using the model to predicate the productive schedule. Consequently, the model not only could be applied as an analysis tool to predict the productive schedule but also could increase the profit.

Two factors, the seedling quantities in each batch and the quantitative variations of seedling in greening field during the nursery period, were compared with the experienced operation and model’s prediction. Figure 3 shows the comparative results with seedling quantities in each batch, as a result that the seedling quantities for model predicted was higher than the experienced operation for the 4th - 7th batches, and those batches of seedling quantities were highly reached to the daily maximum seeding throughput, namely,

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the seedling quantities in the 4th - 7th batches were constrained in the daily maximum seeding throughput.

The quantitative variations of seedling in greening field during the nursery period for both predicted and experienced operations were shown in Figure 4, as a result that both follow a similar trend. And both in the first and in the last period of supplying were close to each other, but the model’s prediction was larger than the experienced operation in the middle period. In the experienced operation, the maximum quantities of seedling was only reached to about 40,000 boxes by contrast with 46,000 boxes for model’s prediction. However, it is worth mentioning that the capacity of greening field was 97,214 boxes, so it is obvious that there was a lower utilization rate of greening field. Thus, it is necessary to expand the supplying scale for augmenting the utilization rate of greening field and productive profit.

2. sensitivity analysis

In post-analysis, it is important to change the parameters in the model, and then to analyze the variations of optimum solution in order that the model can acquire the more accurate parameters. Two parameters, namely, the minimum treatment boxes in each batch and the daily maximum seeding throughput, affect the nursery cost significantly.

2-1. An analysis on the minimum treatment boxes in each batch

It was analyzed from 3,200 boxes/batch to 6,000 boxes/batch at an interval of 400 boxes for the minimum treatment boxes in each batch, which individual nursery cost were shown in Figure 5. The nursery cost is lower than in other cases at 4,000boxes/batch, which nursery cost was about $ 1,790,000 NTD. When the minimum treatment boxes is above 4,000 boxes/batch, the nursery cost is linearly increased following the expanding of the minimum treatment boxes in each batch. At the 6,000 boxes/batch, the cost was highly reached to about $ 1,950,000 NTD, so the minimum treatment boxes should be kept at 4,000 boxes/batch for presently productive scale.

2-2. An analysis on the daily maximum seeding throughput

It was analyzed from 85%(10,703 boxes/day) to 120% (15,110 boxes/day) of present seeding throughput (12,592 boxes/day) at an interval of 5% (630 boxes/day) for the daily maximum seeding throughput, with the result shown in Figure 6. The results show that the cost is highest in 10,703 boxes/day, which cost needs about $ 1,791,000 NTD. As the daily maximum seeding throughput enlarges to 13,222 boxes/day, the cost will reduce to about $1,788,000 NTD. Besides, the nursery cost is no significant variation when the daily maximum seeding throughput is greater than 13,222 boxes. Thus, under the currently supply scale, it is enough to provides the seedling for farmers at the seeding throughput of 12,592 boxes/day.

3. The prediction of productive schedule under various seedling supplying scale

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As above-mentioned, the managers of the rice seedling centers can precisely predict the seedling productive schedule by the model. It was known that the daily seedling supplying quantities follows a distribution curve during the period of seedling nursery, which quantities can be calculated by formula 3. The distribution curve provides a reference guide of seedling supplying for the managers making a decision to plan the productive schedule, and the seedling quantities can be increased or decreased based on the variety for their productive policy and customer’s demand. Therefore, this study also analyzed the productive schedule under various seedling supplying quantities based on the prediction of formula 3, which supplying quantities was 92,410 boxes, and a series of analyses were proceeded from a 20% decrease of currently supplying quantities to a 20% increase at a 10% intervals, that is, from 73,928 boxes to 110,892 boxes at 9,241 boxes interval. The cost/profit ratio was also calculated to provide for managers to find a proper productive scale. It could not produce such seedling supplying quantities under presently productive conditions if the supplying scale expands over 20% increase.

The relationships with nursery cost, profit, and the cost/profit ratio under various supplying quantities were shown in Figure 7. The result shows that the nursery cost and the profit is linear increasing with supplying quantities. The cost/profit ratio is the highest, about 2.53, in 73,928 boxes of supplying, while the profit is about $580,000 NTD. At the 110,892 boxes of supplying, the profit is the highest, about $920,000 NTD, while the cost/profit ratio is about 2.35. And there was the lowest cost/profit ratio, about 2.33, which profit is about $770,000 NTD at the 92,410 boxes of supplying. As there is no significant difference for the cost/profit ratio between the supplying in 92,410 and 110,892 boxes, so the supplying quantities are possibly enlarged to 110,892 boxes for obtaining more profit.

For the seedling quantities in each batch, the seedling quantities highly reach to the daily maximum seeding capacity on the 2nd -7th batches at the supplying quantities of 110,892 boxes (Figure 8) whereat the system bottleneck was caused by the daily maximum seeding capacity. In other words, it will cause the variance on the productive schedule when the daily maximum seeding capacity changed. Similarly, the quantities also reach to the daily maximum seeding capacity on the 3rd -7th batches in the 101,654 boxes of supplying, and likewise the 4th - 6th batches reach the daily maximum seeding capacity in the 92,410 boxes of supplying. There was no such situation under 92,410 boxes of supplying, namely, the daily maximum seeding capacity will not cause the system bottleneck.

Figure 9 shows the sale portions in each day after the seedling grown-up. Almost all the seedlings have been sold out within 4 days after the grown-up in the 83,498 boxes of supplying, so it represents the seedlings were perfectly transplanted in the optimum status. There was a bad supplying situations in the 110,892 boxes of supplying, which only 75 % of seedling were sold out within 4 days after the grown-up. And further even 11% of seedlings were sold out until the 11th day after the grown-up that will caused the seedling missing the best status to transplant.

For the seedling quantities in the greening field, it intensely reaches about 54,663 boxes on the 21st day of nursery period in 110,9892 boxes of supplying (Figure 10). In other supplying scale, the maximum seedling quantities in greening field likewise happened on the 21st day of nursery period, that is, the middle of nursery period. The maximum boxes were only about 36,134 boxes in the greening field in 73,928 boxes of supplying. It is

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important for a suitably size of greening field for rice seedling centers because a smaller of greening field will significantly affect the supplying scale and otherwise a larger of greening field will increase the nursery cost for maintenance of greening field. Thus, the regression analysis was applied to analyze the relationship between the maximum seedling quantities in greening field and the seedling supplying scale by using the data in Figure 10, with the result that the maximum seedling quantities in greening field has a inseparably relation (R2 = 0.9996) to the seedling supplying scale. The regression equation is

y = 0.4609 x + 3.143 ------------------------ (59)

where y is the maximum seedling quantities in greening field, and x is the supplying quantities. Both units are 10 thousand boxes.

CONCLUSIONS

A linear programming model has successfully been developed to analyze the productive schedule during the period of paddy-rice nursery, which model can help the managers of rice seedling centers to make a more precise decision about the relationship between the supplying quantities and their productive schedule under some constrained conditions ,namely, market demand, boxes capacity of greening field, the daily productive quantities, the limitation on minimum treatment boxes in each batch, and so on. A local rice seedling center was chosen to analysis/verify the model. A polynomial regression analysis was applied to analyze the daily supplying quantities during the transplanting period.

In the 91,385 boxes of supplying, the analyzed result shows that the total nursery quantities is 91,565 boxes with a loss of 180 boxes and the cost is $ 1,789,043 NTD while the profit is $ 754,202 NTD. Almost all seedlings (95.5%) has effectively been transplanted within 4 days after the grown-up. In comparison with experienced prediction by the manages of rice seedling centers, about only 24.4% of seedling were supplying within the 4th day after the grown-up.

Through the sensitivity analysis, two factors, the minimum treatment boxes in each batch and the daily maximum seeding quantities, affect the nursery cost significantly. The nursery cost is lower than in other cases at 4,000boxes/batch, which nursery cost was about $ 1,790,000 NTD. Hence, the minimum treatment quantities should be kept at 4,000 boxes/batch for presently productive scale.

This study also analysis the productive schedule under various seedling supplying quantities, with a result that the nursery cost and profit is linear increasing with seedling supplying quantities. The cost/profit ratio is the highest, about 2.53, in the 74,665 boxes of supplying, while its profit is about $580,000 NTD. In the 110,892 boxes of supplying, the profit is the highest, about $920,000 NTD, while the cost/profit ratio is about 2.35. Consequently, the supplying quantities are possibly enlarged to 110,892 boxes for obtaining more profit.

ACKNOWLEDGEMENTS

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The authors wish to acknowledge support given to project NSC 86-2313-B-197-003. Thanks are also extended to Dr. Wei Fang , Mr. W. S. Su, and C. N. Chen for their helpful comments.

REFERENCE

1. Annual Report of Agriculture in Taiwan. 1995. Complied and printed by the Department of Agriculture, Taiwan Provincial Government.

2. Chiu, Y. C. and D. S. Fon. 1997. Analysis on the transportation system for rice seedling centers in Taiwan. The 12th T.V.E. conference of R.O.C. p193-202. Taichung: Taiwan.

3. Tuan, C. L., 1994. Analysis on management and development of rice seedling center. Research report of the department of agricultural business management technology, Pintung Technical Institute. Pintung:Taiwan.

4. Tuan, C. L., 1995. Analysis on Economic magnitude and adequate site number of rice seedling center. Research report of the department of agricultural business management technology, Pintung Technical Institute. Pintung:Taiwan.

5. Tzen, M. T. 1977. Effects of low temperature treatments of rice seedlings on the subsequent growth and grain yield of the first rice crop. Journal of Chinese agriculture 97:45-60.

6. Hiller, F. S. and G. J. Liberman. 1986. Introduction to operation research. Fifth edition, Holden-Day Inc, Oakland, California, U.S.

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Table 1﹑the variables contrast with nursery date, batch no., and supplying date

Nursery date 1 2 3 4 5 6 7 8 9 10 Batch no. No.1 No.2 No.3

Nursery date 11 12 13 14 15 16 17 18 19 20 Batch no. No.4 No.5

Supplying date 1 2 3 4 5 6 7 8 9 Variable X11

♣ X12 X13 X14 X15 X16 X17 X18 X19 X21 X22 X23 X24 X25 X31

Nursery date 21 22 23 24 25 26 27 28 29 30 Batch no. No.6 No.7 No.8

Supplying date 10 11 12 13 14 15 16 17 18 19 Variable X1,10 X1,11 X1,12 X29 X2,10 X2,11 X2,12 X39 X3,10 X3,11

X26 X27 X28 X35 X36 X37 X38 X45 X46 X47 X32 X33 X34 X41 X42 X43 X44 X51 X52 X53

Nursery date 31 32 33 34 35 36 37 38 39 40 Batch no. No.9 No.10

Supplying date 20 21 22 23 24 25 26 27 28 29 Variable X3,12 X49 X4,10 X4,11 X4,12 X59 X5,10 X5,11 X5,12 X69

X48 X55 X56 X57 X58 X65 X66 X67 X68 X75 X54 X61 X62 X63 X64 X71 X72 X73 X74 X81

Nursery date 41 42 43 44 45 46 47 48 49 50 Supplying date 30 31 32 33 34 35 36 37 38 39

Variable X6,10 X6,11 X6,12 X79 X7,10 X7,11 X7,12 X89 X8,10 X8,11 X76 X77 X78 X85 X86 X87 X88 X95 X96 X97 X82 X83 X84 X91 X92 X93 X94 X10,1 X10,2 X10,3

Nursery date 51 52 53 54 Supplying date 40 41 42

Variable X8,12 X99 X9,10 X9,11 X98 X10,5 X10,6 X10,7 X10,4

♣Xi,j : the supplying quantities for batch no. i of seedling after the seedling(boxes) was grown-up on the jth day.

15

Table 2﹑the value of the variables by model’s analysis

Xi,j Batch no. (i) Seedling

quantities j=1 j=2 j=3 j=4 j=5 j=6 j=7 j=8 j=9 j=10 j=11 j=12

1 4860 980 1250 1710 920 0 0 0 0 0 0 0 0

2 8935 2305 1800 2110 2720 0 0 0 0 0 0 0 0

3 9212 2650 1640 1950 2470 0 502 0 0 0 0 0 0

4 12592 2250 2168 3330 2980 0 0 0 1864 0 0 0 0

5 12592 2880 3670 3860 1676 0 0 506 0 0 0 0 0

6 12592 2840 3430 2764 3150 0 0 0 408 0 0 0 0

7 12592 3650 3140 3270 2532 0 0 0 0 0 0 0 0

8 9660 2870 2920 2250 1620 0 0 0 0 0 0 0 0

9 4530 1050 1340 870 1270 0 0 0 0 0 0 0 0

10 4000 740 960 670 650 410 390 180 0 0 0 0 0

Total 91565 22215 22318 22784 19988 410 892 686 2272 0 0 0 0 proportions in total

quantities 24.3 24.4 24.9 21.9 0.45 0.98 0.75 2.5 0 0 0 0

16

R2 Nd = 0.0152 d4 -1.3041d3 + 29.645d2 - 93.572d + 1440.2

= 0.87

0

1000

2000

3000

4000

1 6 11 16 21 26 31 36 41 46

Supplying date

Supplying quantities

(Nd ; boxes)

Actual quantities

Polynomial equation

Figure 1. the seedling supplying Situation for the first rice crop.

051015202530

1 2 3 4 5 6 7 8 9 101112The day after the grown-up

Supp

lyin

g pe

rcen

tage

(%) Experienced

Predicted

Figure 2. The daily supplying percentage during the period

after seedlings were grown-up.

17

0

2000

4000

6000

8000

10000

12000

14000

1 2 3 4 5 6 7 8 9 10

Batch no.

Nur

sery

qua

ntiti

es(b

oxes

)

PredictedExperienced

Figure 3. Comparison with the nursery quantities of each batch.

0

1000

2000

3000

4000

5000

1 7 1 1 2 3 3 4 4 5

Nursery date

The

see

dlin

g qu

antit

ies

in g

reen

ing

field

(box

es)

Predicted

Experienced

Figure 4. The variations with the seedling quantities in greening field.

18

1.651.701.751.801.851.901.952.00

32003600400044004800520056006000

The seeding quantities per batch (boxes/batch)

Nur

sery

cos

t(m

illio

ns N

TD)

Figure 5. The sensitivity analysis in the relationship between

the nursery cost and seeding quantities per batch.

1787

1788

1789

1790

1791

1792

1070311333119621259213222138511448115110

The daily maximum seeding throughput (boxes/day)

Nur

sery

cos

t(th

ousa

nds

NT

D)

Figure 6. The sensitivity analysis in the relationship between

nursery cost and daily maximum seeding throughput.

19

0

0.5

1

1.5

2

2.5

73,928 110,892101,65192,41083,169

Supplying quantities (boxes)

Cos

t & P

rofit

(mill

ions

NT

D)

2.22.32.32.42.42.52.52.6

The

cos

t/pro

fit ra

tio

CostProfitThe cost/profit ratio

Figure 7. The relationship with the nursery cost, profit, and the cost/profit ratio

under various supplying scale.

0

200

400

600

800

1000

1200

1400

1 2 3 4 5 6 7 8 9 1

Batch no.

Nursery quantities (boxes)

73928 boxes83169 boxes92410 boxes101651 boxes110892 boxes

Figure 8. The nursery quantities in each batch under various supplying scale.

20

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12

The day after the grown-up

The

per

cent

age

of s

uppl

ying

Supplying total quantites

73928 boxes8316919 boxes92410 boxes101651 boxes110892 boxes

Figure 9. the sale percentage in each day after the seedling grown-up under

various supplying scale.

0

10000

20000

30000

40000

50000

60000

1 7 13 19 25 31 37 43 49 55Nursery date

The

see

dlin

g qu

antit

ies

ingr

eeni

ng fi

eld

(box

es)

73928 boxesSupplying scale

83169 boxes92410 boxes101651 boxes110892 boxes

Figure 10. The seedling quantitative variations in greening field

under various supplying scale

21

y = 0.4609x + 3.143

R2 = 0.9996

30

40

50

60

70

75 83 92 102 111

Supplying quantities ( thousand boxes)

The maximum boxes in greening

field (thousand boxes)

Figure 11. The relationship between the maximum boxes

in greening field and supplying boxes

Documents

A Study on the Prediction of Productive Schedule during ...nova.bime.ntu.edu.tw/~dsfon/RiceNursery/linearP.pdf · programming (LP) are widely applied as an analysis tool in operational