A mathematical model for designing and sizing sow farms

Transcription

1 Intl. Trans. in Op. Res. 11 (2004) INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH A mathematical model for designing and sizing sow farms L. M. Pla` a,b, D. Babot a,c and J. Pomar a,d a Centre R&D. UdL-IRTA, Area de Produccio Animal, Spain, b Departament de Matema`tica, Universitat de Lleida, Spain, c Departament de Produccio Animal, Universitat de Lleida, Spain, d Departament d Enginyeria Agroforestal, Universitat de Lleida, Spain lmpla@matematica.udl.es [Pla`]; dbabot@prodan.udl.es [Babot]; pomar@eagrof.udl.es [Pomar] Received 10 September 2002; received in revised form 11 November 2003; accepted 4 March 2004 Abstract A semi-markov model and its application for designing and sizing swine facilities are presented. Swine production is becoming more and more specialised. Hence the sizing of a farm producing piglets is the main strategic decision of farmers who invest in sow production, since a farm comprises different facilities with different possible sizes. The classical approach is simple but the resulting design assumes some security margins without considering variations in sow performance or in the management policy. The model presented here has revealed differences with respect to the classical approach. As a result, the Markov chain approach is useful to determine more accurately the capacity, improve farm design and reduce housing cost. Keywords: sow farm design; Markov chain model; strategic decisions; simulation Introduction Recently many software programs for pig farm management have been developed and introduced for on-farm use, but they have not been widely used (Kamp, 1999; Gelb, 1999). Most of these new computing tools were intended to improve the quality of the information on which decisions are based and, consequently, increase the quality of decisions, though this process has not been developed uniformly. For instance, there are farm management areas where computing tools are less developed, especially when strategic decisions are involved. This is a surprising fact, taking into account that strategic decisions are important for farm viability, which requires getting the big decisions right and correct while making major tactical adjustments as Panell et al. (2000) noted. Such a case is farm designing and sizing, i.e. determining the capacity of swine production facilities, a typical problem when different facilities with different capacities for swine production may be built. r 2004 International Federation of Operational Research Societies. Published by Blackwell Publishing Ltd.

2 486 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) Classical methods for designing and sizing sow farms are based on the time interval representing an average reproductive cycle, which is compounded by different time intervals for each reproductive stage. Usually, operations in sow farms are ordered by reproductive stages, so these average time intervals are central for planning housing facilities and they also serve to identify the reproductive management policy adopted by the farmer (Wittemore, 1998). The final design assumes some security margins, more or less empirical, in order to avoid size problems by default or by excess; but this method does not consider possible variations in the management policy nor the stochastic nature of the swine production, and neither does it explore alternative optimal designs. Other approaches have been tested. For instance, Singh (1986) proposed a simulation model to select the optimum capacity of swine production facilities considering batch management. Later on, Lippus et al. (1996) adapted a previous Markov chain model, developed by Jalvingh et al. (1992), to support strategic planning for housing facilities for sows. They considered immediate replacement of sows, constant herd size and weekly management but without batches. It is worth pointing out that no methodological comparison was carried out in any of the cases. Thus, in this work we present an application for improving the design and the size of a sow farm based on a semi-markov chain model representing the occupancy of the facilities. Using a sample farm we show the differences between the classical approach and the new one. The sow herd model Description of the piglet production system The piglet production system is a specialisation of the swine production industry characterised by a herd of sows in a continuous process of reproduction. The commercial product is the piglets, which after weaning are sold to rearing fattening farms. This specialisation gives additional efficiency gains as Rowland et al. (1998) pointed out and it is widely extended within the Spanish swine industry. Normally, confinement facilities consist of a service facility, a gestation facility and a farrowing facility with multiple farrowing rooms. The farrowing facility is divided in multiple rooms for better disease and parasite control. All facilities may be housed in one or in several buildings. The service facility houses breeding sows, gilts (young sows) and boars. Group weaning of all litters from a farrowing room, i.e. a batch, is practised to synchronise breeding and farrowing. Uniformly during the reproductive cycle sows are culled for different reasons, and the size of batches are completed after weaning in the service facility with replacement gilts. Culled sows can remain on the farm until they are sent to the slaughterhouse. Culled or dead sows can be replaced immediately, but usually after some farm-specific delay. Replacement gilts and sows are generally kept in the service facility to be inseminated. They are moved from service to gestation facility when pregnancy is confirmed, if not, they remain to be reinseminated. Gilts and sows in the gestation facility belonging to the same batch are moved into the farrowing room one at a time approximately 1 week before parturition. To synchronise the breeding and farrowing of a group of sows, all litters from a farrowing room are weaned simultaneously and are sent to the nursery or sold. After weaning, the sows are sent back to the service facility. The farrowing room is

3 cleaned, sterilised and closed for a drying period. After the drying period, the room is ready to receive the next batch of sows. Model formulation L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) The semi-markov sow herd model used in this work has been adapted from a more general Markov decision model presented by Plà et al. (1998). In that model, sows move from one state to another through transitions. The process is represented as a Markovian decision model defined by the following elements: states S in which sows can be (usually, these states are related to reproduction states), actions A that farmer can take at each stage, transition probabilities for sows evolving from one state to another and the reward function representing the benefit ( positive or negative) associated to each transition. States and actions are finite sets. In a infinite planning horizon, O 5 {S A} 1 represents the set of all possible system paths o, i.e. all possible sequences of states and actions, o 5 (i 1, a 1, i 2, a 2,y,i n, a n,y)ao. The sequence of actions is the result of a policy or strategy D. For each stationary policy, D, P D ¼ðp D ij Þ is the transition matrix representing transitions of sows from one state ias to another jas when policy D is adopted by the farmer. The time period between states is coincident with the time period between actions and it is called a stage. Future states S are defined as being only conditioned by the present state and not by the manner to which the present state is reached (Markovian property). The reward function, r, represents the farmer s preferences in a decision theory context and it can be used in the design of a performance criterion, like the total return expected after n transitions from the initial state i, B D n ðiþ, as follows: B D n ðiþ ¼rD i þ X p D ij BD n 1 ð jþ; j2s where the reward function r D i is the expected net return from a sow in the i state when policy D is adopted. Therefore, the dynamics of a herd under any given policy can be represented as a finite irreducible and aperiodic Markov chain. The expected herd distribution at equilibrium, P t ¼ðp D 1 ; pd 2 ;...pd jsjþ, is calculated solving the following linear system of equations: p D j ¼ P p D k pd kj ; j 2 S P k2s p D j ¼ 1; j2s ð1þ where p D kj represents the transition probability for a sow to pass from state j to state k. D is the management strategy of the farmer and {p j D, jas } represents the distribution at equilibrium when policy D is adopted by the farmer. For our purpose, several changes in the above model formulation need to be introduced. Because the farmer s policy is unique and stable for a long period of time in accordance with the stationarity assumption for policies, the transition matrix is unique once the deterministic policy is set (then, from now on the symbols of policy and actions will be dropped in the notation). Furthermore, transitions to represent movements among facilities have to be introduced and therefore the state set is redefined; thus, transitions can represent the movement of sows between

4 488 facilities or sows changing reproductive state. Moreover, stages are of different time intervals depending on the current state of the sow and therefore a new group of parameters have to be added; the average time between transitions t i, i.e. the expected time in days that sows spend in state i and ias. As a result, the model becomes a semi-markov chain where steady-state distribution can be calculated. The process is in two steps; the first using (1), and the second accounting for the average time between transitions: p j ¼ p j t j; j 2 S: ð2þ Final distribution over states, fp j ; j 2 Sg, is obtained after normalising the vector obtained in (2). These values are used to estimate the capacity of different facilities. To do so, the state set has to be partitioned into three, related to service, gestation and farrowing facilities (i.e. S 5 I [ G [ F). Assigning each original state to its facility gives P p j j2f p j j2s TO F ¼ P L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) ; F 2fI; G; Fg; ð3þ where TO F is the distribution of occupation over facility F, and fp j ; j 2 Sg has been calculated through (2). Numerical comparison between approaches In order to assess the suitability and to measure the possible advantages of the semi-markov model proposed in farm designing and sizing, a comparison was established between this new approach and the classical one through a sample farm. A group of parameters satisfying the needs of both methodologies was set. These parameters represent the management policy related to facilities and production level that the farmer wants to implement or achieve. The data on biological production parameters used in this paper were derived from computerised records of Bd-Porc databank (2000), the main Spanish pig information system. Case parameters Nowadays, we can consider about sows as a reasonable size of a sow farm producing piglets in Spanish conditions. For example, the number of sows was fixed at 660. Facilities involved were service, gestation and lactation. All gilts were purchased from outside and moved into the service facility awaiting the first insemination 2 weeks after; home-grown sows are not represented in the model, since they are rare in commercial Spanish conditions. The sojourn time in service facility was fixed at a minimum of 21 days post-fertile insemination. Overall occupancy would range from 35 to 39 days, after which sows were moved to gestation dependencies. One week before farrowing, sows were moved to farrowing dependencies; as a result they had been in the gestation facility for 86 days (assuming gestations of 114 days). Lactation was fixed at 21 days, so that the total time interval in this facility was 35 days, including a drying period of 1 week. Finally, after weaning, sows re-entered service dependencies and batches were completed when

5 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) needed. Culled sows were not immediately sent to the slaughterhouse, but a time period depending on the last state visited before culling was considered. In Table 1, we can see the principal values of the parameters considered. The nutritional effects on reproduction are taken into account through the reproductive parameters. The values of the biological parameters are presented in Tables 2 4. The model assumed 11 to be the maximum number of reproductive cycles. Table 2 gives the average of time intervals for each reproductive state; these parameters are assumed to be independent of the reproduction cycle. Table 3 shows intervals from weaning to first insemination per reproductive cycle. Marginal probabilities are shown in Table 4. Marginal probabilities for conception rates can be affected by the number of unsuccessful inseminations and by the reproduction cycle. Conception rates are expressed in relation to all matings per cycle. Abortion marginal probabilities are also specific for each reproduction cycle, whereas culling marginal probabilities are specific for each reproduction cycle and state. Only conception rates for first, second and third inseminations are given in Table 4 because it was very unusual to find farms with more than two inseminations per cycle. A culling rate of 100% in the last cycle and state has been established in order to show the end of the sow lifespan. Table 1 Expected values per sow for planning housing requirements Parameter Days Reproductive rate (interval between parities) 149 Occupation of lactation facility 35 Occupation of service facility 35 Occupation of gestation facility 86 Table 2 Average of time intervals Gestation length Lactation length 21.0 Gestation with abortion length 85.0 Interval from last mating to market 21.0 Interval from last farrowing or weaning to market 7.0 Interval between matings 21.0 Age of purchased gilts Average (days) Table 3 Average of time interval from weaning to first mating by reproductive cycle Cycle Average

6 490 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) Table 4 Marginal probabilities per cycle Cycle Marginal probabilities a CR(1) (%) CR(2) (%) CR(3) (%) AR (%) CRO (%) CRG (%) CRL (%) a CR: conception rates at first, second and third matings; AR: abortion rates; CRO: culling rates for open sows; CRG: culling rates for gestating sows; CRL: culling rates for lactation. Results from the herd model Using the parameters presented in the previous section, the model was solved using (2) and (3). Two models are solved, one representing an immediate replacement of culled animals (100% of occupancy) and another one assuming the possibility of management by batches allowing empty boxes and a delay in the replacement. Thus, the herd distribution over facilities was obtained in each case just redefining conveniently t i. In Table 5, we show the results obtained for each model. Batch management is characterised because when a sow leaves its batch, the batch is not filled immediately; hence, there is a gap that is present until a new reproductive cycle restarts. In Table 5 we can see the effect of this management: the size of lactation facility increases. In absolute terms, the biggest difference is in the lactation facility, whereas for service and gestation facilities differences are small. This might be considered as normal because they are the facilities less affected by gaps generated in batch management. Other results obtained by the model are technical indicators as the time interval between farrowings, 2.4, and the real averaged time in each facility, i.e. service, 36 days; farrowing, 86 days and lactation, 35 days. Results of the classical approach The classical method used by engineers in sow farm sizing (number of boxes, number of rooms, number of buildings) for different facilities (service, gestation, farrowing lactation) assumes a fixed number of heads in the herd, a fixed time interval per facility and a homogeneous rate of occupancy (Brent, 1986; ITP, 1997; Wittemore, 1998). Usually, sow farms management is based on batches; hence, the number of batches (NB) depends on the delay between batches (d) and

7 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) Table 5 Number of boxes in each facility depending on the maximum number of reproductive cycles allowed. Two cases are presented: full occupancy and batch management allowing occupancy with gaps Cycles Full occupancy Batch management Service Gestation Farrowing Service Gestation Farrowing reproductive rate, which depends on the duration of the gestation (G), lactation (L) and the interval between weaning and conception (WCI). The delay between batches has to be adjusted according to herd size (N) and reproductive rate in order to set a reasonable number of sows per batch, i.e. to obtain an operative and practical sow management. In order to size each facility (F), the number of places (Np(F)) for a continuous farrowing system, and the number of rooms (NR(F)) in a planned farrowing system has to be calculated from: NB ¼ðG þ L þ WCIÞ=d ð4þ NpðFÞ ¼ðN TO ðfþ Þ=ðG þ L þ WCIÞ ð5þ NRðFÞ ðn p ðfþnbþ=n ð6þ Once calculations are made, results obtained for service capacity are increased to consider the needs for the replacement sows. Very often, a security margin is considered by applying a safety factor in order to assure farm operation and therefore, the prescribed final size becomes larger. Finally, results per facility are rounded up in order to obtain an integer that will be used for building purposes. The classical method was applied using the same parameters presented in Tables 1 3. Calculations were made without considering batches using (5) and considering batches in a weekly management of sows, using (4) (6). The results are shown in Table 6. Without batches, the number of places in lactation and services are the same because the occupation time of this facilities is planned in 35 days. In practice, batch management is usually implemented in sow farms and in this case, for economical purposes, the main objective is to adjust the total number of farrowing places built. We can appreciate how in the rounding up of the figures we are increasing, or decreasing, the total capacity of the farm. Usually, we round up the figures for security purposes; e.g. in case 1 in the lactation facility we only need 130 boxes, but we will have 150. In other cases, considering the

8 492 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) Table 6 Number of places, number of batches and rooms planned for housing the sow in the different facilities (reproductive rate of 149 days) Without batches Number of farrowing places 155 Number of service places 155 Number of gestation places 381 With batches Number of batches 22 Number of sows in each batch 30 Number of farrowing rooms 5 Number of farrowing places 150 Number of service places 155 Number of gestation places 381 economical restrictions, it can be preferable to decrease the expected herd size for a better use of farrowing places. Discussion The semi-markov model presented here for designing and sizing sow farms has methodological advantages over other approaches. For instance a simulation model, as that proposed by Singh (1986), would be useful to represent closely the piglet production system, to account for any exception, but such a detail is a disadvantage for practical purposes. On the other hand, the model presented by Lippus et al. (1996) was a proposal rather similar to that presented here, but Lippus et al. when considering weekly transitions introduced an artificial augmentation of the state set that made the model complex. Such a complexity was unnecessary because they were concerned in steady-state distribution of sows over facilities that can be calculated more easily with the model presented here. Furthermore, the transition matrix became bigger and hard to work with it; the semi-markov model, instead, considers transitions as they are actually observed, and they can easily be estimated through data (Plà et al., 1998). In respect to the classical method, the results presented show differences between the two approaches. The biggest difference arises when considering the batch management and lactation facility, and this could be explained because the classical approach is based on the number of farrowings per sow per year (NFY) that ignores the effect of primipara sows ( producing less piglets) and batch gaps produced along the reproduction cycle. On the other hand, herd dynamics is not considered by the classical method. In general, the result of the classical method is not very adequate either when we have a lot of gilts (e.g. with high culling rates) or with very young populations. These situations introduce instability to the herd and problems in daily operations. From the semi-markov model, it is possible to calculate different technical indexes as the NFY; hence, Fig. 1 demonstrates how the NFY varies depending on maximum lifespan of sows and maintaining the same set of remaining parameters.

9 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) NFY Max # of cycles Fig. 1. Number of farrowings per sow per year depending on a maximum number of reproductive cycles. Table 7 Comparison between two approaches Without batch management With batch management Service Gestation Farrowing Service Gestation Farrowing Classical Model Differences 18.42% 4.96% 17.42% 14.83% 5.54% 18.47% Moreover, comparing Tables 4 and 5 as shown in Table 7 reveals how a security margin equally applied among facilities did not solve the problem because the greatest differences are around plus 14 18% in the service facility, and around minus 17 19%, in the farrowing facility depending on whether batch management is considered or not. Therefore, the classical approach may cause problems to farmers when working near the farm s capacity. Finally, calculations using the herd model are simple and easy to implement in a spreadsheet; a practical model can account for around 140 states, a reasonable size to be handled on a PC. Therefore, it is worth using a slightly more elaborate model to gain accuracy in planning sow facilities. Conclusions The semi-markov model described here represents a more appropriate approach for designing and sizing sow farms. It is more accurate than the classical method, essentially because it captures the dynamics of the piglet production process. Moreover, different advantages are drawn with

10 494 L. M. Pla`, D. Babot and J. Pomar / Intl. Trans. in Op. Res. 11 (2004) respect to previously published models for the same purpose. Classical methods are based on the average time interval for each reproductive stage which is considered as a part of the reproductive management policy adopted by the farmer. The semi-markov model considers variations in sow performance, and can be adapted to possible variations in the management policy, in order to explore alternative optimal designs. We have shown how the service facility size is in general underestimated by classical methods and the farrowing facility overestimated, and how security margins cannot be applied equally to all facilities. The model can be implemented as an easy-touse tool to gain insight into the occupancy rate of the facilities. As a result, farm design is improved and housing needs can be reduced. This is a first study to explore the support that the model presented here can provide in the designing and sizing of housing facilities for sows. References Bd-Porc, Brent, G., Housing the Pig. Farming Press. Gelb, E.M., Adoption of IT by farmers Does reality reflect the potential benefit? Proceedings of the Second European Conference EFITA, September 27 30, Bonn, Germany, pp ITP Manual del porcicultor. Editorial Acribia, Zaragoza, Espan a. Jalvingh, A.W., Dijkhuizen, A.A., van Arendonk, J.A.M., Dynamic probabilistic modelling of reproduction and management in sow herds. General aspects and model description. Agricultural Systems, 39, Kamp, J.A.L.M., Knowledge based systems: from research to practical application. Pitfalls and critical success factors. Computers and Electronics in Agriculture, 22, Lippus, A.C., Jalvingh, A.W., Metz, J.H.M., Huirne, R.B.M., A dynamic probabilistic model for planning housing facilities for sows. Transactions of the ASAE, 39, Panell, D.J., Malcom, B., Kingwell, R.S., Are we risking too much? Perspectives on risk in farm modelling. Agricultural Economics, 23, Pla`, L.M., Conde, J., Pomar, J., Sow model for decision aid at farm level. In Giron, F.J. (ed) Applied Decision Analysis. Kluwer Academic Publishers, Boston, pp Rowland, W.W., Langemeier, M.R., Schurle, B.W., Featherstone, A.M., A nonparametric efficiency analysis for a sample of Kansas swine operations. Journal of Agricultural and Applied Economics, 30, Singh, D., Simulation-aided capacity selection of confinement facilities for swine production. Transactions of the ASAE, 29, Wittemore, C., The Science and Practice of Pig Production. Blackwell Science Ltd, Oxford.