Frontiers in Business, Economics and Management 
ISSN: 2766-824X | Vol. 11, No. 3, 2023 

 

225 

Optimization	Model	for	Captive	Breeding	of	Lake	Sheep	
Based	on	Space	Utilization	
Yingliang Gu1, * 

1College of Acupuncture and Rehabilitation, Dazhou Vocational College of Chinese Medicine, Beijing, China 
*Corresponding author: 2977176435@qq.com 

 

Abstract: Power Lake sheep have excellent characteristics such as fast early growth, early sexual maturity, four-season estrus 
and can be kept in captivity, etc. It is of great significance to explore the development of appropriate production plans to improve 
the space utilization of farms in order to increase the annual turnout of lake sheep. In this paper, under the condition of continuous 
production, in order to determine the reasonable number of breeding rams and basic ewes in the farm and estimate the range of 
the annualized number of sheep, we used a set of inequalities to depict the breeding cycle of the lake sheep from the spatial and 
temporal dimensions, and established a model to analyze the number of annualized number of sheep under the condition of 
deterministic factors, and based on the analytical model, we solved the gap between the number of standard sheep pens and the 
number of standard sheep pens in the existing one, under the requirement of the farms to have not less than 1500 sheep pens per 
year. gap in the number of standard sheep pens. On the basis of the above model, taking into account the influence of uncertain 
factors such as the success rate of mating, the number of lambs delivered and the mortality rate, an optimization algorithm for 
fluctuating reproduction of lake sheep was adopted to achieve the optimization of the production plan in order to maximize the 
economic benefits. 

Keywords: Captive Breeding of Lake Sheep, Continuous Production, Fluctuating Reproductive Cycles, Spatio-Temporal 
Analysis. 

 

1. Introduction 
As a precious lamb breed in China, the lake sheep has a 

long history with excellent characteristics such as fast early 
growth, early sexual maturity, four seasons of estrus and can 
be raised in captivity. According to historical records, as early 
as 1126 AD, lake sheep began to be bred. In the early days, 
lake sheep were raised by the northern people, but as the 
northern immigrants migrated to the south, in the absence of 
natural pasture, they gradually changed from free-range to 
captive breeding. Under the constant cultivation of artificial 
captivity, the sheep eventually adapted to the southern climate 
and settled around Lake Taihu, forming the current lake sheep. 
Since the 14th Five-Year Plan for the development of animal 
husbandry, large-scale raising has become a trend in the 
breeding industry, through scientific methods can improve the 
survival rate and fertility of sheep, make full use of land for 
penning captive breeding, thus improving land utilization and 
saving more resources. The breeding of lake sheep mainly 
includes two key aspects: reproduction and fattening. Due to 
the high technical requirements of artificial insemination, the 
breeding of lake sheep is mostly carried out by natural mating 
of breeding rams and basic ewes, and the sheep need to be fed 
in different pens at different times. Under the premise of 
safeguarding the economic benefits of farm managers, the 
space utilization rate of the farm is improved by formulating 
the corresponding production plan, and the main production 
plan is used to determine the mating time of how many basic 
ewes can be mated, and by controlling the breeding time, the 

space utilization of the sheep pens is improved in order to 
achieve the maximization of economic benefits [1, 2]. 

2. Model Analysis of The Annualized 
Number of Sheep Slaughtered Under 
Deterministic Conditions 

2.1. Reproductive gestation cycle of lake sheep 
based on temporal analysis 

In the actual production process, without considering the 
uncertainty factors and the elimination and renewal 
conditions of the breeding sheep, statistics found that the 
natural mating period of 20 days within the ewes can be 
conceived, gestation period of 149 days, 2 lambs per litter, the 
lactation period of 40 days, lamb fattening period of 210 days, 
the ewes are empty and resting period of 20 days. Now a lake 
sheep farm existing 112 standard sheep pen, in the realization 
of continuous production conditions, in order to determine the 
reasonable number of farm breeding rams and basic ewes and 
estimate the range of the annualized number of sheep, the 
establishment of 2. Determining factors under the conditions 
of the number of annualized number of sheep model analysis, 
and based on the analytical model for the farm each year, not 
less than 1,500 sheep under the requirements of the estimated 
solution to the existing standard sheep pen The gap in the 
number of sheep. In the process of modeling, the reproductive 
gestation cycle modeling process shown in Figure 1 was 
established by using the method of sequential analysis based 
on different times. 

 



 

226 

Natural mating period (20d)
15 rams

Breeding ram: foundation 
ewe
1:14

Breeding period 
(149d)

Sheep pen not more 
than 8

Non-mating rams
No more than 4 
rams in a pen

Lamb fattening (210d)
Sheep pen not 
exceeding 14

Empty resting foundation 
ewes (20 days)

No more than 14 sheep in 
the pen

Lactation (40d)
Sheep pen not 
more than 6

Non-mating rams
No more than 4 
rams in a pen

Non-mating rams
No more than 4 
rams in a pen

Non-mating rams
No more than 4 
rams in a pen

give birth to newborn 
baby animals

 
Figure 1. Breeding gestation cycle diagram 

 
Let the number of breeding rams be x,the number of 

foundation ewes be 14x, the number of lambs be 2 x 14x = 
28x, and given the number of pens N, we have: 

Modeling the natural mating period: 
 

𝑁                     (1) 
 
(2) Models of pregnancy: 
 

2𝑥 𝑁           (2) 
 
(3) Models of lactation: 
 

𝑁           (3) 

 
(4) Modeling of the lambing and nulliparous conditioning 

period: 
 

𝑁           (4) 

 
Based on the time analysis, it was found that the captive 

breeding of lake sheep was multi-batch, and each batch was 
further divided into two rounds of breeding, and the complete 
breeding process was analyzed by taking the first round of the 
first batch as an example, and the results were obtained as 
shown in Figure. 2. 

The number of pens used in the second lactation period is 
the largest, because the lambs are fattened in the first round, 
and the second round of foundation ewes lactating with the 
non-mating period of the breeder rams will take up the pens, 
so according to Equation 3, we can calculate the breeder rams 
as 24 rams. Because the whole breeding cycle is 229 days, 
deducting a natural mating period there are only 209 days, but 
lamb fattening needs 210 days, so we will delay one day, in 
the second round of the second round of lactation period to 
open the new foundation ewes natural mating, so when the 
first batch of the first round of the nursing period is finished, 
at the same time start the second batch of lake sheep breeding, 
can be calculated to get the cumulative number of days of 
captivity under different breeding period as Table 1. 

 

Natural mating 
period

20 days

gestation period
149 days

Lactation period
40 days

Lamb fattening
210 days

20 days resting 
period for 

foundation ewes

Second natural 
mating period 

20 days

Second 
trimester 149 

days

Second lactation 
period of 40 

days

Non-mating 
rams

Non-mating 
rams

Number of 
sheep

Non-mating 
rams

Second Breeding Process

First breeding process

 
Figure 2. Diagram of the complete breeding process of the first round 

 
Comparison of Table 1 reveals that the cumulative number 

of days corresponding to the number of breeding periods 3 is 
the closest to two years, whereas for leap year (366 days) and 
flat year (365 days) under two years the number of lower lake 
sheep was calculated to be 2,823 and 2,816, respectively. 

 

Table 1. Number of Reproductive Stages 
Number of breeding periods Cumulative days 

1 229 
2 458 
3 687 
4 916 



 

227 

2.2. Solving a model for annualized sheep 
numbers in slaughter 

With the premise of achieving continuous production, the 
first natural mating period of the second cohort ends at the 
end of the lactation period of the first cohort, so finding a 
second cohort to initiate the second cohort for natural mating 
at the stage of the first cohort HO is extremely important to 
maximize the benefits of captive breeding, so the following 
two scenarios were considered for the study [3]: 

(1) The second batch is initiated at the end of the first 
gestation period of the first batch. 

The time elapsed includes a natural mating period of 20 
days and a gestation period of 149 days for the first batch, a 
natural mating period of 20 days for the second batch, a 
gestation period of 149 days for the second batch, a lactation 
period of 40 days for the second batch, and a cumulative time 
elapsed of 378 days to enter the fattening of the lambs, which 
takes 210 days. The first batch went through the first natural 
mating period of 20 days, 149 days of gestation, 40 days of 
lactation, 20 days of rest and recuperation in the empty womb, 
20 days of natural mating period in the second batch, 149 days 
of gestation in the second batch, 40 days of lactation in the 
second batch, with a cumulative elapsed time of 438 days. At 
this time, the first batch of lambs have been reared for 229 
days, the first batch of sheep out of the pen on the 419th day, 

there is a time difference of 41 days, indicating that the second 
batch of lambs have been reared for only 60 days, 419 days 
before the first batch of lambs and the second batch of lambs 
need to be reared in the pen, that is, there is a lack of pens, at 
this point, we can draw two important conclusions: 

Conclusion 1, the natural mating of the second batch of 
foundation ewes took place earlier and earlier, i.e. in the 
second case. 

Conclusion 2, 48 pens kept by lambs after day 419, that is, 
days 420 to 438, were vacant. 

(2) The second batch was started earlier than the end of the 
first pregnancy. 

According to conclusion 2 of the first scenario, the first 
batch of lambs is fattened out on day 419. According to the 
analysis the time required for a complete breeding cycle is 
229 days, with 50 free pens, i.e. the second batch is started on 
day 190 to enter the natural mating period. Starting earlier 
than 190 days will inevitably lead to an insufficient number 
of pens by day 419, i.e. the first case above. Starting the 
second batch on the 190th day is the earliest, with the highest 
number of lambs in the pen and more benefits. 

Therefore, the breeding process of the first batch of the first 
round and the third batch of the second round of lake sheep as 
an example of solving the analysis, to obtain the breeding 
process shown in Table 2 and Table 3. 

 
Table 2. Table of the first round of breeding process for the first batch 

Natural mating season Duration of pregnancy Lactation period 
Lamb fattening and resting period in 

empty wombs 

 Foundation ewes 
require 42 pens 

56 pens of foundation 
ewes 

Lambs need 48 pens 

 Breeding rams require 
6 pens 

Breeding rams require 
6 pens 

24 pens of foundation ewes 

Total columns 24 Total columns 48 62 total columns Breeding rams require 6 pens 
   Total columns 78 

88 remaining sheep 
pens 

64 remaining sheep 
pens 

50 remaining goat pens 34 remaining goat pens 

Cumulative number of 
days 20d 

Cumulative days 169d 
Cumulative number of 

days 209d 
Cumulative days 229d 

24 breeding rams   
672 lambs have been reared for 20 

days 
336 foundation ewes    

 
Table 3. Third second round of breeding process table 

Natural mating season Duration of pregnancy Lactation period 
Basic ewe lamb pens: 4 Basic ewe lamb pens: 6 Basic ewe lamb pens: 8 

Lambing pens: 7 Lambing pens: 7 Lambing pens: 7 
Cumulative number of days 249d, 

harmonized time 439d 
Cumulative days 398d, harmonized 

time 588d 
Cumulative days 438d, harmonized 

time 628d 

96 + 28 lambs reared for 40d 96 + 28 lambs reared for 189d 
96 + 28 lambs have been raised for 229 

days and are out of the pen 
 

Based on the results of solving for all batches, it can be seen 
that for the above two cases of second batch start-up time 
points the annual slaughter of lake sheep and the number of 
sheep pen gaps are calculated as follows: 

(1) The second batch is initiated at the end of the gestation 
period of the first batch. 

The first batch of lambs in the first round of complete 
fattening for the time of 419 days, the base ewes 336, the 
number of lambs out of the pen is 672, the annual output is 
586; the second batch of lambs in the first round of complete 
fattening for the time of 439 days, the base ewes for 14, the 

number of lambs out of the pen is 24, the annual output is 20; 
the third batch of lambs in the first round of complete 
fattening for the time of 609 days. With 62 base ewes and 124 
lambs fledged, the annual number of lambs fledged in the 
third batch is 75; the total annual number of lambs fledged in 
the first round of all three batches is 618. 

The number of pens is calculated when the first round of 
lambs in the third batch is fully fattened, so the first batch of 
lambs will require 48 pens and the foundation ewes 24 pens; 
the second batch of foundation ewes will require 1 pen and 
the lambs 1 pen; the third batch of foundation ewes will 



 

228 

require 8 pens and the lambs 7 pens; so the total number of 
pens required will be 89, and the number of shortfalls in the 
number of pens will be 23. 

(2) The second batch is started earlier than the end of 
pregnancy of the first batch. 

The first batch of the first round of lamb fattening start time 
is 209 days, the base ewes are 336, the number of lambs out 
of 672, then the first batch of the annual output is 1174; the 
second batch of the first round of lamb fattening start time is 
229 days, the base ewes are 14, the number of lambs out of 
24, then the second batch of the annual output is 38; the third 
batch of the first round of lamb fattening time is 399 days The 
first round of lambs for the third batch is 399 days, the 
foundation ewes are 62 and the number of lambs fledged is 
124, so the annual lambing quantity for the third batch is 113; 
so the total annual lambing quantity for the first round of 
lambs for the three batches is 1325; 

The number of sheep pens is calculated when the first 
round of lambs of the third batch are fattened, the first batch 
of lambs need 48 pens, the foundation ewes need 42 pens, the 

breeder rams need 6 pens; the second batch of foundation 
ewes need 1 pen, the lambs need 1 pen; the third batch of 
foundation ewes need 5 pens, the lambs need 9 pens; so the 
total number of pens needed is 112, the number of shortfalls 
in the number of sheep pens is zero. 

3. Optimization of A Fluctuating 
Reproduction Model for Lake Sheep 
Under Uncertainty Considerations 

3.1. Analysis of fluctuating reproductive cycles 
of lake sheep under uncertainty 

In the actual breeding process of lake sheep, there are still 
uncertain factors such as the success rate of breeding, the 
number of lambs delivered and the mortality rate, etc. 
Therefore, the fluctuating breeding cycle of lake sheep in 
captivity under the consideration of uncertain factors is shown 
in Figure 3. 

 

Natural mating period (20d)
Sheep pen not exceeding 14
Breeding rams: foundation 

ewes

Maternity (147-150d)
Sheep pen no more than 

8

Non-mating rams
No more than 4 rams 

in a pen

Lamb fattening 
(210d)

Sheep pen not 
exceeding 14

Empty resting 
foundation ewes (20 

days)
No more than 14 sheep 

in the pen

Lactation benchmark 
(40d) 35-45d

Sheep pen no more 
than 6

Average 2.2 per litter
Survival rate 97

Non-mating rams
No more than 4 rams 

in a pen

Non-mating rams
No more than 4 rams 

in a pen

Non-mating rams
No more than 4 
rams in a pen

give birth to 
newborn baby

Observation period 
(30d)

Conception rate 85%
Infertility rate 15%

 
Figure 3. Fluctuating breeding cycles under uncertainty 

 
Natural mating phase: The ram mates naturally with the 

foundation ewe for 20 days. 
Gestation phase: the basic ewe's conception fluctuates from 

147 to 150 days during the gestation phase, we need to 
generate multiple random integer values within this specified 
range and store the basic ewe's state records in the matrix, 
after the gestation period is over, the basic ewe enters into 
lactation state and gives birth to lambs. 

Observation period: Since successful pregnancy can only 
be recognized 30 days after the end of the base ewe's mating 
period, the conception rate of the base ewe's natural mating is 
85%. 

Lactation period benchmark: the lactation period is based 
on 40 days, and the lactation period is controlled to be within 
35 to 45 days, for every 1 day decrease in lactation period, the 
fattening period of the lambs decreases by 2 days, and vice 
versa for every 1 day increase in lactation period, the fattening 
period of the lambs increases by 2 days, and we need to 
generate multiple random integer values within this specified 
range of the status record of the breeding ram, the status 
record of the foundation ewe, the status record of the lambs, 

the status record of the foundation ewe and the status record 
of the lambs. The status record of the breeding ram, the status 
record of the foundation ewe, the status record of the lamb, 
and the use of the sheep pen are put into the matrix variable 
for storage. At the end of the lactation period, the foundation 
ewes return to breeding status and the lambs enter the 
fattening period. Typically, foundation ewes produce an 
average of 2.2 lambs per litter with a survival rate of 97%. 

Empty Rest Period: Foundation ewes require a minimum 
of 18 days in the empty rest period before they can be flexibly 
conditioned. 

In addition, without taking other factors into consideration, 
we need to pen lactating ewes and lambs born with a 
difference of no more than 7 days between delivery dates, as 
well as allow fattening lambs with a difference of no more 
than 7 days between weaning dates to be penned together, and 
base ewes during the resting period to be penned together. 



 

229 

3.2. Optimization algorithm for fluctuating 
reproduction in lake sheep under 
consideration of uncertainties 

To minimize the expected loss of the production plan under 
the consideration of uncertainty, there are the following two 
options: 

(1) When the sheep pens are vacant, the value of loss per 
pen per day is 1, and want to minimize the expected loss value 
to each pen occupied, in each pen among the sheep pens in 
order to put one to the farm to occupy the pens. 

(2) When the sheep pen is full, we need to rent an additional 
sheep pen, the rental fee is 3, as far as possible in the 
feasibility of the number of annualized number of sheep 
slaughtered under the conditions of the sheep pen full of farms, 
reduce the cost of renting the sheep pen, so that the loss of the 
expected value of the minimum. 

Then the flow chart of the optimization algorithm used is 
shown in Fig. 4. 

The constraints of the algorithm are as follows: 
Sheep pen capacity: ensure that the capacity limit of each 

pen is not exceeded, including the number of sheep at 
different stages. 

Availability of leased space: if leased, it is necessary to 
ensure that the leased space is available and has sufficient 
holding capacity. 

Total farm space: Consider the total available space within 
the farm. 

Rental cost: If renting, consider the cost of renting the 

space to ensure it is within budget. 
Movement and deployment restrictions between different 

sheep stages within the farm: consider the rules and 
restrictions on movement of sheep between different farming 
stages. 

The specific implementation process of the algorithm is as 
follows: 

Divide a number of sheep pens from the total number of 
pens and mark the use in these pens as 'natural mating' and the 
rest as 'idle'. The pens were divided into natural mating and 
idle pens, and the status of the breeding rams and foundation 
ewes, as well as the status of the lambs, were recorded, and 
all the pens on the farm were placed in breeding rams, 
resulting in a loss of 1 per pen per day. 

The second step was to calculate the values of conceived 
and foundation ewes at 147-150 days of gestation, 30 days of 
observation, 85% conception rate and 15% non conception 
rate, arrange the pens for the breeding rams, put the non 
conceived ewes in the natural mating phase, number the 
breeding rams and arrange them in separate pens, and the 
remaining ones were arranged according to four. Observe 
whether the monitoring time is reached or not, if it has been 
reached, then output the result, if it has not been reached, we 
need to test and modify the foundation ewes in the natural 
mating period, observation period and lactation period, as 
well as checking and modifying the lambs in the fattening 
period and the foundation ewes in the empty resting period 
until the monitored time is reached and output the result [4, 
5]. 

 

Gestation: 30 days of 
observation

Calculation of conceived and 
unfertilized foundation ewes

Natural mating period:
Allocation of pens Breeding rams 

Foundation ewes, calculated losses

Detection and 
modification of 

lambs in the 
fattening period

Detection or 
modification of 

foundation ewes in 
natural mating periods

Began

close
Is the testing time 

met?

Detection or 
modification of 

foundation ewes in the 
observation period

Checking and 
modifying 

foundation ewes in 
the empty rest period

Detection or 
modification of 

foundation ewes in 
lactation

Detection or 
modification of 
foundation ewes 
during pregnancy

not have 
reached

 
Figure 4. Flowchart of the fluctuating reproduction algorithm 



 

230 

 

4. Conclusions 
In this paper, under the condition of realizing continuous 

production, in order to determine the reasonable number of 
breeding rams and basic ewes in the farm and estimate the 
range of the annualized number of sheep, the inequality group 
was used to portray the breeding cycle of lake sheep from the 
spatial and temporal dimensions, and an analysis of the model 
of the number of annualized number of sheep under the 
condition of determining the factors was set up, and based on 
the analytical model, the existing number of standardized 
sheep pens was solved for the requirement that the farms 
should produce no fewer than 1,500 sheep annually under the 
estimate. The shortfall in the number of standard sheep pens 
was estimated based on this analytical model. The results of 
the calculations showed that the number of standard pens used 
by the farm was 110 when there were 24 breeding rams and 
336 foundation ewes in 112 standard pens. The farm has the 
highest number of lambs and benefits from adding an 
additional batch of foundation ewes in the second round, i.e. 
starting the second batch on day 190 to enter the natural 
mating period. 

Finally, considering the effects of uncertain factors such as 
the success of mating, the number of lambs delivered and the 
mortality rate, an optimization algorithm for fluctuating 

reproduction in lake sheep was used to achieve the 
optimization of the production plan in order to maximize the 
economic benefits. 

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