426

THE ECONOMIC OPTIMISATION OF SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING

R. B. M. Huirne, Th. H. B. Hendriks, A. A. Dijkhuizen and G. W. J. Giesen*

A stochastic dynamic rograrnming model is designed to determine

This optimal policy maximises the present value of net revenues from sowspresent in the herd and from subsequent replacement gilts over a given planning horizon. The model also calculates the total extra profit to be expected rom trying to retain an individual sow until her optimal lifespan an d not replacing her immediately. This total extra profit is an economic index which makes it possible to rank sows within the herd on future profitability and, therefore, can be used as a management guide in culling decisions. For typical Dutch values the optimal replacement decisions result in an average herd life of 3.43 parities. The maximum economic life of sows of average productivity turns out to be 10parities. All data can easily be adjusted to represent aspecific herd or a different region of the world.

the economic optima P replacement policy in swine breeding herds.

Introduction

The decision whether to keep or to replace sows is a frequent and an important one. Sows are culled because of health problems and failure to conceive, on the one hand, and because of insufficient production, on the other. Especially in the latter case, culling decisions are based on economic considerations. A sow is re laced not because she is no longer able to produce in a biological sense,

1986). Knowledge of the optimal replacement policy and of the influence of changes in production and prices on it may support farmers in taking these management decisions. Most of the ublished work on optimising this type of decision has dealt with dairy cattle &eddies, 1972; Renkema and Stelwagen, 1979; Gartner, 1981; Dijkhuizen et al, 1985; Van Arendonk, 1985; Kristensen, 1987). Although on average 43-65% of the sows in a herd are replaced annually (Dagorn and Aumaitre, 1978; Bisperink, 1979; n o e s and Van Male, 1979; Van der Steen, 1984; Arkes et al, 1986), surprisingly little research has been done on the economics of culling in swine breeding herds. Recently, however, Dijkhuizen et al(l986) developed a spreadsheet model for microcomputers, named Pork-CHOP, in which the benefits of increased lifespan in swine * Department of Farm Management and Department of Mathematics. Address for

correspondence: Wa eningen Agricultural University, De artment of Farm Management, Hollandsewe 1,670fKN Wageningen, The Netherlands. &e authors are indebted to H. A. M. Van der .keen, Department of Animal Breeding, Wageningen, for his useful advice in developing the model.

but 1 ecause a replacement gilt is expected to yield more (Dijkhuizen et al,

SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING 427

breeding herds can be quantified and replacement decisions for sows with poor productive and/or reproductive performance can be optimised using the marginal net revenue approach.

In this paper, a more detailed research model for mainframe computers is presented. In this model, the application of stochastic dynamic programming for the analysis of sow culling decisions is introduced. This technique makes it possible to take into account the biological variation in piglet production of sows and replacement gilts. Influences of changes in production and prices on the optimal policy can also be established. Furthermore, an economic index is calculated that can be used as a culling guide for individual sows within a herd.

Material and Methods

Dynamic Programming Finite state dynamic programming is a mathematical o timisation technique for solving multistage decision problems (Bellman, 195'77. Multistage decision problems involve a sequence of decisions extending over a given period of time. Many farm management decisions, including decisions concerning the replacement of sows, can be formulated as multistage decision problems. Subject to some constraints, the goal in such problems is maximising (or minimising) an objective function, for example the present value of net revenues. The number of time periods over which decisions will be made is referred:to as the planning horizon. The lanning horizon is divided into several time-intervals (weeks, months, years!. An interval is called a stage. At each stage, the condition or state of the process being managed is described by a set of parameters, called state variables, such as age and production level of the sow. Each state variable has only a finite number of distinct values. Given the time and a state variable, the manager has to take a decision, chosen from a set of possible decisions.

More formally, dynamic programming is a method for maximising (or minimising) an objective function, which has the following general form:

I Maximum . I { (Gi (Xi, U,))}

'Ji

where: I Xi Ui Gi (Xi, Ui) = function describing the impact of specific state and control

variables at stage i on the objective function for stage i.

= length of the planning horizon = vector of state variables at stage i (1 < i S I) = vector of control variables at stage i

Optimisation generally starts at the end of the planning horizon and moves backwards in time to the present stage. At each stage the optimal decision is determined for all possible combinations of the state variables. The decision about which state-change option is best is based on the combination of two effects: the effect of the immediate state change plus the previous determined optimal effect of remaining state changes. In effect the decision at each stage collapses into a single stage decision problem. If we define the so-called value

428 R. B. M. HUIRNE,TH. H. B. HENDRIKS, A. A. DIJKHUIZEN AND G. W. J. GIESEN

function Vi, then mathematically this process can be represented by the recurrence relation (Bellman, 1957):

Vi (X,) = Maximum {Gi (X,, Ui) + Vi+l (Ti(&, UJ)} (2) Ui

where: Vi(Xi) = maximal value of the objective function during the remainder of

the planning horizon under optimal decisions given the state Xi Ti (Xi, Ui) = transformation function: given the state vector Xi and control

vector U. at stage i the system moves in stage i + l to state Xi+l = Ti (x, ui)

The other symbols have been previously defined. The decision problem can be either deterministic (with certainty) or

stochastic (Burt and Allison, 1963; Van Beek and Hendriks, 1985). The objective function in the stochastic case has the following general form:

and the recurrence relation:

Vi(X,) = Maximum { Gi(Xi,Ui,Ei) + Vi+,(Ti(Xi,Ui,Ri))} (4) Ui

where: -1 R. Gi(Xi ,Ui ,Ei)

= vector of stochastic variables at stage i = function describing the expected impact of specific state

and control variables at stage i on the objective function = expected value of the objective function during the

remainder of the planning horizon under optimal decisions given state X,

= transformation function: given the state and control level at stage i and realisation Ri of the stochastic vector - R,, this function directs the system from state Xi in stage i to state Xi+l = Ti(X,,Ui,Ri) in stage i + l

Vi(Xi)

Ti(& ,Ui ,si)

The other symbols have been previously defined. This collapsing of the problem so that only the current stage, the current

possible decisions, and the optimal solutions to subsequent stages are considered is a common feature of dynamic programming. The fundamental principle of dynamic programming that must hold to yield an optimal solution is called the principle of optimality, i.e. an optimal decision at any stage must be based on an optimal set of decisions with regard to the state resulting from that decision (Bellman, 1957). For a more detailed description of the technique and possible applications in agriculture, reference is made to Bellman (1957), Dreyfus and Law (1977), Kennedy (1981) and to Van Beek and Hendriks (1985).

Dynamic programming has the advantage that it places no immediate restrictions on the nature of functions used to specify the structure of the system, such as the nonlinear marginal net revenue objective function (Throsby, 1964; Dannenbring and Starr, 1981). Furthermore, it is possible to alter structural characteristics of the process being managed over time (Burt and Allison, 1963).

SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING 429

The Sow Replacement Model The stochastic dynamic programming technique described in the preceding sub-section is used to determine the optimal replacement policy for individual sows. This stochastic formulation makes it possible to account for involuntary culling and for variation in production traits, including litter size. Given the production level in the present stage, the distribution of production levels in the next stage can be defined taking into account repeatability of production (Van Arendonk, 1985).

The economic criterion for sow replacement decisions can be described as follows: a particular sow should be kept in the herd as long as the present value of the marginal cash-flow from keeping her until the next parity is higher than the present value of the lifetime average cash flow from a replacement gilt for that time pegod. This income potential of the gilt cannot be realised as long as the sow is kept in the herd. It can therefore be inte reted as the opportunity

In the model developed for this study, the planning horizon I is divided into 50 sow production-cycles i. Therefore, 0 d i S 50 and I = 50. Parity number j varies between 1 and 15, whereas litter size n varies between 0 and 20 pigs born alive. The decision whether to replace a sow is considered at the time of weanin ,35 days after each parturition. At this time, a particular production- cycle i ?stage i) ends and the next production-cycle i + l (stage i+ l ) starts. At the end of parity j (1 5 j d 14) at stage i, the farmer has to decide whether to keep a particular sow in order to let her farrow in the next parity j + 1 at stage i+ 1, or to replace her by a replacement gilt resulting in farrowing of that replacement gilt (parity 1) at stage i + l . Because the maximum parity number considered is 15, all sows in panty 15 are replaced at the end of that parity. When a sow is mated, it is assumed that she conceives with a probability of 100%.

In the model, the optimal replacement policy is determined, taking into account the risk of premature disposal of retained sows. Involuntary disposal, among others, consists of removal for failure to conceive, leg problems, diseases and accidents, and death of the sow.

Optimisation starts at the end of the planning horizon by setting the values of sows in each possible state equal to the expected value of their piglet production. This is the sum of the expected discounted returns from piglet production over their remaining life (taken from Bisperink (1979) and Dijkhuizen (1986)) and the anticipated slaughter value. This value is used instead of slaughter value alone in order to reduce the length of the planning horizon, by eliminating the possible influence of the starting situation on the final results. The process then continues backwards, stage by stage, until the first stage of the planning horizon is reached. At each stage the present value of cash flow associated with keeping the sow until the next stage and then following an optimal policy during the remainder of the planning horizon is compared with the present value of cash flow when the sow is replaced immediately.

In the model, sows are described by a state vector (Xi) with components consisting of parity number j and production level n in 3 previous parities of the sow. If j < 3 the litter size in only 2 previous parities respectively 1 previous parity is considered. The maximal expected present value of cash flow during the remainder of the planning horizon under optimal replacement policy (Vi(Xi)) given the initial state Xi of the sow at the beginning of stage i (0 d i d 1-1) equals:

cost of postponed replacement (Dijkhuizen et al, 198 ‘g ).

Vi(X,) = Maximum [VRi(Xi), VKi(Xi)] ( 5 )

430 R. B. M. HU1RNE.m. H. B. HENDRIKS, A. A. DIJKHUIZEN AND G. W. J. GIESEN

where:

where: = vector representing the state of the sow at stage i ; Xi = 0 in the

case of a replacement gilt = the maximal expected value of cash flow during the remainder

of the planning horizon under optimal replacement policy given the initial state of the sow at the beginning of stage i

= expected cash flow given the current state of the sow, the decision to keep the sow at stage i and an optimal replacement policy during the remainder of the planning horizon

= expected cash flow given the current state of the sow, the decision to replace the sow by a replacement gilt at stage i and an optimal replacement policy during the remainder of the planning horizon

= marginal probability of involuntary disposal at the beginning of stage i of a sow in state Xi

= probability of n pigs born alive at stage i + l for a sow in state Xi at stage i

= probability of n pigs born alive at stage i + l for a replacement gilt

- n=n = the outcome of the stochastic variable of litter size E is n pigs born alive (n varies between 0 and 20)

Gi+,(Xi,n=n) = net revenues between stage i and stage i + l for a sow kept in state Xi with a litter size of n pigs born alive

Gi+,(O,n=n) = net revenues between stage i and stage i + l for a replacement gilt with a litter size of n pigs born alive

Ti(Xi,g=n) = transformation function: if a sow in state Xi at stage i gives a litter size of n pi s born alive, she comes at stage i + l in state Xi+l = Ti(Xi,;=nT

Xi

Vi(Xi>

VKi(Xi)

VRi(Xi)

PID(Xi)

Pn(Xi)

Pn(0)

S(Xi) R G FL

DF = discount factor The slaughter value (S(Xi)) is calculated per kilogram of live weight. The financial loss (FL) associated with involuntary disposal is built into the model to account for the loss in slaughter value, costs of veterinary treatment prior to the disposal, and idle production factors associated with disposal.

The discount factor (DF) is calculated using a real annual interest rate of 4.5%. The price of a replacement gilt (RG) is taken to be Dfl. 550 (550 Dutch guilders) at an age of 250 days, the age when the gilt first becomes pregnant in the model.

= slaughter value of the sow in state Xi at the end of stage i = price of a replacement gilt = financial loss associated with involuntary disposal of 'a sow

during a stage

SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING 43 1

After the optimal lifespan of a sow has been determined in this way, the model calculates the total extra profit to be expected from keeping her until that optimal lifespan, compared with immediate replacement, taking into account the risk of premature disposal of retained sows. This total extra profit, called Retention Pay-Off (RPO), can be calculated for each individual sow, and is calculated as follows:

where:

RPO(Xi) = Retention Pay-Off given the initial state Xi of the sow at the end of

The other symbols have been previously defined. stage i

Predicting Litter Size In predicting litter size of a sow, a parity effect and a class effect are taken into account. Litter size, calculated as the number of pigs born alive, is determined as follows:

- LS, = LS, + zj

where:

j = parity number LS, LS, Z,

The average litter size in panty j (LSj) is given in Table 1. The class effect is calculated from information from up to 3 previous parities of the sow. To reduce the number of states in the model, litter size of each parity j is divided into 3 possible production classes Y,: Y. = 1 if litter size n in parity j varies between 0 and 8 pigs born alive; Y, = 2 if’n varies between 9 and 11; Yj = 3 if n vanes between 12 and 20. For each production class Yj the corresponding weighted average litter size W, is calculated from the normal distribution described by the average litter size of the herd and the corresponding standard deviation (see Table 2). The class effects are based on weighted average litter sizes of the production classes W. and the repeatability of litter size. The repeatability of litter size in individual sows is commonly accepted to be low, but significantly above zero (Van der Steen, 1984). In the model the class effects are calculated as follows:

= litter size in parity j (number of pigs born alive) = average litter size of sows in parity j = class effect in parity j

-

z,=o z, = 0.20 * (W, - G)

- Z3 = 0.18* (W, - W) +O.12* (Wl - w> Zj = 0.17 * (Wj-, - w> + 0.11 * (Wj-2 - w> + 0.05 * (Wj-3 - w)

432 R. B. M. HUIRNE.TH. H. B. HENDRICKS, A. A. DIJKHUIZEN AND G. W. GIESEN

where: j Z. = class effect in parity J dj = weighted average litter size of the production class in panty j (number of

pigs born alive) w = average litter size of the herd

= parity number (4 < j.< 15)

For each sow the probability distribution of litter size is calculated according to a normal distribution (ISML, 1982) from the mean litter size of that particular sow and the corresponding standard deviation of litter size (see Table 1).

Table 1 Parity-Specific Values Assumed to Represent Typical Dutch Swine Breeding Herds

Pigs born alive* Parity Probability of mean Standard Probability ofpiglet Live weight ofsow

No. involuntary disposal (~3,) deviation mortalityt ( k g P

10 1 1 12 13 14 15

I

0.18 9.4 2.78 0.18 10.1 2.74 0.17 10.7 2.72

0.27 10.7 2.69

0.38 10.2 2.69 0.41 10.1 2.69 0.62 10.0 2.69

0.15 0.14 0.15 0.15 0.16 0.16 0.16 0.17 0.17 0.17 0.18 0.18 0.18 0.19 0.19

140 160 175 188 196 200 200 200 200 200 200 200 200 200 200

* Van der Steen (1984) t Mortality between the time of artus and weaning $ At the average time of rernovayof about 1 month after weaning

For gilts, the probability distribution of litter size in the first parity is calculated from the ex ected litter size for the first farrowing (LS, =. 9.4 pigs born alive, see Table lfand the standard deviation for the first farrowing (2.78 pigs born alive, see Table 1). In Table 2 the probability distribution of pigs born alive for gilts is given.

As an example, Figure 1 a sow is considered at the end of parity 5 at stage 25 with the following class effects: in parity 5 the sow had a litter size between 9 and 11 pigs (Ys = 2), in parity 4 at stage 24 she had a litter size between 12 and 20 pigs (Y4 = 3) and in parity 3 at stage 23 she had a litter size between 0 and 8 pigs born alive (Y, = 1). In the case that the sow is replaced voluntary or involuntary at the end of parity 5 at stage 25, a replacement gilt will produce her first results at stage 26. Depending on the probability distribution of litter size, the gilt may be in production class Y, = 1 (between 0 and 8 pigs born alive), in production class Y, = 2 (between 9 and 11 pigs) or in production class Y, = 3 (between 12 and 20 pigs). These probabilities can be found in Table 1 and 2. If the SOW is not disposed of involuntarily and the farmer decides to keep her, she farrows at stage 26. The sow may be in production class of Y6 = 1, Yb = 2 0.r Y6 = 3. The probability distribution of litter size depends on the 3 previous production classes Ys, Y4 and Y3 of that sow.

433 SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING

Figure 1 The Relation Between the Decision to Keep the Sow (I) , the Decision to Replace the Sow (Voluntary Disposal) (2), and the Involuntary Disposal of the Sow ( -. - -. ) at Parity 5 at Stage25 in the Situation where the Litter Size of the Sow is Divided into 3 Production Classes.

N 4 N m N 4 h '-I o r m .rl

x

h 0

v1 Ad

C (u

C

a 0 V

4 *n

N N > N

N I

'3 m m r m

.7 w W W

I I I

I I I

r( r( r(

434 R. B. M. HUIRNE, TH. H. B. HENDRIKS, A. A. DIJKHUIZEN AND G. W. GIESEN

Table 2 Relation Between Production Class, Litter Size, Weighted Average Litter Size of Each Production Class and the Expected Frequency Distribution for Replacement Gilts (1 S j S 15)

Production Class Litter Size WeightedAverage Expected frequency distribution for f yi., (4 Litter Size (Wi) replacement gilts (Pn(0)) (%)

I

0 1 2 3 4 6.95 5 6 7 8

0.07 0.16 0.43 1.04 2.21 4.13 6.81 9.87

12.59

9 14.13 2 10 10.03 13.95

11 12.12

3

12 13 ~- 14 15 16 17 18

13.37

9.26 6.23 3.68 1.92 0.88 0.35 0.13

19 0.04 20 0.01

Results Basic Situation Given the probabilities of involuntary replacement presented in Table 1, the optimal replacement decisions result in an optimal average herd life of 3.43 panties. Voluntary culling for insufficient production accounts for 14% of the total. In the optimal situation the average income per sow per year, defined as net return to labour and management, is Dfl. 234.

In Table 3, the Retention Pay-Off (RPO) values are presented. These are calculated for average producing sows only, at the average time of conception in the herd. As shown in Table 3, the RPO is highest for young sows. At the end of parity 10 the RPO falls below zero, and hence the maximum economic life of sows of average productivity is 10 parities. The calculated RPO values also represent the maximum amount that could be spent in trying to return a sow to her previous production level in case of reproduction failure or health problems. The results obtained a ree to a great extent with those published by Bisperink (1979) and Dijkhuizenfl986).

Table 3 Retention Pay-Off (RPO) for Average Producing Sows in the Parities 1 to 10 Parity RPO (Dfr.)

1 224 2 222 3 187 ~.

132 88 55 32 19

9 8 10 <O

SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING 435

Table 4 Price and Production Variables Used in the Sensitivity Analysis Variable Low" Basic High 'L

Feeder pig price (Dfl . ) -2(1%" I22 +20% Feed price (Dfl . ) -20'% +20% Slaughter value ofculled sows (Dfl.) -2O'X 502 +20'%, Priceof areplacement gilt (Dfl . ) -20% 550 +20% Pre-weaning piglet mortality - 2 o x 15.3 +20% Litter size (number of pigs born alive) - I t + I Interval weaning-conception (days) -7 27 +7

.L

* Thechanges are relative to the basicsituation t Sow feed price: Dfl. 56 per IOOkg

$ The basicsituation is summarised in Table I Piglet feed price: Dfl. 76per I(I0kg

Sensitivity Analysis TO gain insight into the behaviour of the model and the sensitivity of the results, the effects of changes in major price and production levels and in involuntary disposal rates were analysed.

Alternative price and production levels for which optimal replacement policies were determined are summarised in Table 4. The influences of changes in these factors are summarised in Table 5. The average annual income per sow is sensitive to changes in feeder pig prices, feed costs and litter size. The average annual income is not greatly influenced by the slaughter values of culled sows, the price of a replacement gilt, pre-weaning mortality, or the interval between weaning and conception. In contrast to the average annual income, the optimal replacement policy is most sensitive to the difference between the price of replacement gilts and the slaughter value of culled sows. A reduction of this difference results in a higher rate of voluntary replacement. The optimal replacement policy is not very sensitive to changes in feeder pig prices, feed prices, pre-weaning mortality, and interval between weaning and conception. These variables affect the sow which can be replaced as well as the replacement gilt.

Table 5 Percentage Changes in Income Per Sow Per Year, in Voluntary Replacement Rate and in Average Herd Life When Variables Varv AccordineTable 4 (Relative to the Basic Situation)*

Variable

Feeder pig price Feed price Slaughter value Price replacement gilt Pre-weaning mortality Litter size Interval weaning-conception

~~

Changes in: Income per Sow Voluntary Replace- Average Herd

per Year men! Rare Life

Low High Low High Low High

-190 +19n -7 +7 -1 0 +88 -88 0 0 0 0 -21 +22 -53 +79 +10 -12 +23 -21 +95 -54 -17 + I 1 +22 -22 -9 +9 + I -1 -13 +72 +I7 -12 -2 + I + 25 23 $4 -4 0 0

* Basicsituation:- income persowperyear = Dfl. 234 - voluntary replacement rate = 14% - average herd life = 3.43 parities

436 R. B. M. HUIRNE.TH. H. B. HENDRIKS, A. A. DIJKHUIZEN AND G. W. J . GIESEN

Table 6 The Average Herd Life (Parities) and the Average Annual Income Per Sow (Dfl.) for a Situation With and Without Voluntary Disposal at Different Rates of Involuntary Disposal

No voliinrary disposal With vohinrary disposal

Involiinrary disposul rate* Herd lye Annual income Herd life Anniinl income

120% 3.63 (-0.43)t 2 15 (-1 6) 3.24(-0.19) 2 I7 (-I 7) I OO'X 4.06 23 1 3.43 234 8W" 4.58 (+O.52) 244 (+ 13) 3.58(+0.15) 251 (+17)

' Relative tothe basicsituation (defined as 100%) t Between brackets thedeviation from 100%

The consequences of a proportional decrease and increase of 20% in the marginal probabilities of involuntary disposal for the replacement policy and the annual income per sow are given in Table 6. If all sows are kept until they have to be replaced involuntarily the average herd life is 4.06 parities. A 20% decrease in the involuntary disposal rate results in an increase in average herd life of 0.52 parities and an increase in average annual income of Dfl. 13. A 20% increase in involuntary disposal leads to a Dfl. 16 reduction in average annual income per sow.

In the case with voluntary disposal, the average herd life is 0.63 parities lower, but the annual income per sow is Dfl. 3 higher than in the case without voluntary replacement. If the involuntary disposal rate decreases by 20% and voluntary disposal is permitted, the annual income per sow increases relatively more (Dfl:17 versus Dfl. 13) and the average herd life relatively less (0.15 versus 0.52 parities) than in the case where there was no voluntary disposal. Reducing involuntary disposal, therefore, is more profitable when part of this reduction is used for increasing voluntary replacement, instead of for increasing the average herd life of the sows only.

Discussion Dynamic programming is a flexible tool for determining the optimal replacement policy for sows. It has the advantage, among others, that variation in and repeatability of traits can be taken into account. Therefore, the risk that high-producingsows may have a low future piglet production, as well as the risk of replacement of a sow by alow-productive replacement gilt, can be accounted for.

On the other hand, dynamic programming models can become very large, resulting in high computing costs. This was the reason why, in this analysis, litter size in previous parities was represented by only 3 production classes. Moreover, the length of the planning horizon has been restricted to 50 stages, being the minimum required length for stabilised results.

Optimal replacement decisions result in an optimal average herd life of 3.43 parities, given the probabilities of involuntary disposal assumed in this analysis. Reducing involuntary disposal is more profitable when part of this reduction is used for increasing voluntary replacement, instead of only for increasing the average herd life (Table 6).

The average annual income per sow highly depends on feeder pig prices and feed costs, as might be expected. However, changes in these two factors do not greatly affect the optimal replacement policy, since the expected income of

SOW REPLACEMENT DECISIONS BY STOCHASTIC DYNAMIC PROGRAMMING 431

both the sows present in the herd and replacement gilts is affected (Table 5). On the other hand, changes in the price of replacement gilts or in slaughter value of culledsows have a much bigger effect on optimal replacement policy. The smaller the difference between these two prices, the higher the voluntary replacement rate.

Further research is needed on some parity-specific variables such as litter size and live weight of the sow. No consistent values could be found for these variables in the literature. Another key factor in the model is the repeatability of litter size within sows. There is surprisingly little published information on this point, and further investigation is needed to determine whether this figure is as low as commonly accepted.

The calculated RPO values for individual sows can be a useful guide in optimising replacement decisions, but has only been calculated under simplified model conditions here. For instance, insufficient piglet production is assumed to be the only reason for voluntary disposal based on economic considerations. Reasons for involuntary disposal, among others, include removal for failure to conceive, leg problems, diseases and accidents, and death of the sow. Some of these reasons for involuntary disposal can also be considered as economic decisions, especially removal for failure to conceive. The stochastic dynamic programming model, therefore, will be extended by including variation in other reproductive traits, like heat detection and conception rate. Moreover, it is the intention to use the model i n an expert system, which will be developed to support farmers in controlling the economic profitability of individual swine breeding herds.

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