Application of Dynamic Programming in Salmonella control of meat products

Transcription

1 . INTRODUCTION. Problem tatement Salmonella i a common caue of food borne dieae in human. Therefore Salmonella criteria have been laid down for minced meat, meat rearation and meat roduct by Commiion Regulation. Official control on the roduction of meat are neceary to inect on food buine oerator own control and to verify that food buine oerator comly with hygiene rule and reect criteria and target laid down in Community legilation. The community legilation require the aroval of etablihment and the cometent authority hould make an on-ite viit (EC No 85/4). Regulation (EC) No 854/4 require that modern meat inection hould be baed on rik aement and that cro contamination in the laughter hall hould be revented. However, it ha been hown that almonella i readily tranmitted from one carca to the next by the variou maniulation that are required to be erformed during the traditional meat inection rocedure (Dwinger et al., 7). The traditional ot-mortem meat inection rocedure are directed toward leion which are of more qualitative variation than ublic health imortance, and therefore may be better handled by the quality control ytem of the laughterhoue (Ellerbroek, 7). Further more, SCVM (7) tate that the quality of meat inection can be monitoring for it outcome, rather than being excluively governed by rigorouly recribed rocedure (SCVM, ). Bondt et al. (7) introduced a new governmental control ytem, uerviion of control, which i concerned with the changing role of the control activitie of government in the tranition toward increaed elf-regulation by rivate comanie. They indicated that a reduction in the rocedure of ublic meat inection may ave cot for the laughterhoue. However, it i doubtful whether the choice of food buine oerator in electing control trategie will be altered by the reduction of ublic meat inection o that the revalence level of macro organim in meat roduct i altered.. Obective and reearch quetion Thi reearch i aimed to analyze the imact of ubic meat inection on the choice of the meat laughter lant in electing control trategie; a well a to find out the factor that may alo change their choice in electing control trategie. The following quetion will be anwered in thi reearch: What i the effect of a reduction in ublic meat inection on the choice of meat laughter lant in electing the otimal control trategie? Are there any influential factor that may alo change the deciion making of meat laughter lant? If thoe influential factor are exited, then what i their effect on meat laughter lant deciion making? How doe the intenity of meat inection together with the influential factor alter the food buine oerator deciion making?

2 What i the effect on the exected revalence level of Salmonella in meat roduct when the different control trategie are elected by the food buine oerator? What i the effect of the intenity of meat inection and the influential factor on the exected Salmonella revalence level in meat roduct?

3 CHAPTER SALMONELLA CONTROL Thi chater introduce the kground of almonella control, include: the revalence tranition in herd and carcae, the control meaure and the revalence level.. Salmonella tranmiion dynamic among herd and between carcae An animal that i infected with Salmonella can tart hedding teria and therefore become infectiou within four hour ( Fedorka-Cray et al., 994). The ero-converion eriod of reaching the detectable antibody level after infection i about two week. The animal can remain in a carrier tate where the teria are in intetine or lymh node, without hedding it, and may become erological negative again. At the laughter tage, teria from the intetine or lymh node can contaminate the carca and thereby contaminate the meat roduct (Lettelier et al., 999). The read of Salmonella in the oourn time on the finihing farm i deigned by the tochatic ucetible infectiou recovered (SIR) model a decribed by Van der Gaag, et al.(4b). A illutrate in table., an animal can be aigned to one of three tate: the ucetible tate, the infectiou tate and the recovered tate. The two ucetible tate are S and S, the three infectiou tate are I, I and I ; and there i one carrier tatec. For intance, S denote Salmonella free animal that reult in a almonella free carca and will be teted negative with a erological tet; and S i the notation for that will reult in a Salmonella free carca but that will be teted erological oitive. A illutrated in figure., a Salmonella free herd in tate S can be infected and move to tate I. After the eroconverion eriod, it will be in tate I. When the animal to hedding, it then move to tate C and become ucetible again S. An animal in tate S may be infected again

4 I or become erological negative S. It i aumed that there i no infection induced morality ( e.g. Fedorka- cracy et al., 994). However, after the animal i laughtered, everal tranition are imoible to haen, uch a eroconverion. The erology of the animal doe not change anymore during or after laughter, although in the blood or meat dri of the carca the erology tatu i detectable. Therefore, the tranition between I and I i unable to occur. The oible tranition for the carca during laughter are outlined Fig. the interretation of everal tate in thi figure differ lightly from thoe in Fig.. For intance, the infected animal in Fig. hould be interreted a a contaminated carca, which may be caued by teria from the intetine or lymh node during eviceration or by cro-contamination at the laughter time. An infected herd ( I, I and I ) may become a carca free of Salmonella ( S or S ), if the eviceration i carried out very carefully and no teria contaminate the carca. Beide, almonella free ig ( S or S ) may alo become infected, becaue of cro contamination by teria of other infected carcae or by teria on the laughter equiment. The carrier tate (C ) doe not exit after laughter, due to a carrier carca i either a contaminated ( I and I ) or an uncontaminated carca ( S ).. Salmonella revalence control meaure and revalence level On the farm-level, control meaure include the urchae of Salmonella free animal, rodent control, careful cleaning and diinfection of facilitie, trict earation of grou of hog and all-in-all-out routing and acidification of feed and or water. Although, ome of thee meaure can be quite effective mot notably acidification of feed and water are cotly and of them will eliminate the oibility that ome animal are infected. On the laughter-lant level, control meaure include daily cleaning of equiment, careful eviceration and logitic laughtering ractice whereby grou of low revalence hog are 4

5 laughtered early in the day to minimize cro contamination (King, et al., 7). One or everal control meaure are combined a a control ackage, where different combination of control meaure have different cot. When the more cotly control ackage are elected, more intenive Salmonella control i achieved (Van der Gaag et al., (4b)). The revalence (i.e. ercentage of infected herd or carcae in a oulation) of almonella in herd or carcae can be meaured by erological or teriological teting. For intance, baed on erology, at oulation level the mean revalence of tiue or carcae after laughter i % in the Netherland (Swanenburg et al., ). In thi reearch, the revalence level of herd or carcae are meaured by the ercentage of infected herd or carcae in a amle of oulation. On the farm-level, the amle of finihing herd are teted by erological revalence ditribution; and on the laughter lant-level, the amle of carcae are teted by teriological revalence ditribution. Table. rereent the cot and the teriological revalence ditribution reulted from uing three laughter lant-level control ackage. Table. illutrate the cot and the erological revalence ditribution that reulted from uing four farm-level control ackage. Slaughter lant-level revalence robability ditribution for the three control ackage are hown in table.. Thee teriological revalence ditribution for one carca on a finihing laughter lant were calculated uing an eidemiological imulation model decribed by Van der Gaag et al.(4b). The algorithm i demontrated in chater 4.Some modification in calculation in the cot of control ackage are made to reflect realitie more. The table how that teriological revalence ditribution of three Salmonella control ackage are centered to a zero ercent level, but ackage ha the mot centered ditribution on thi level. The cumulative ditribution above ten ercent level of control ackage i., thi mean a laughter lant who ued control 5

6 ackage x in Salmonella control would have a. robability of exceeding the Salmonella revalence threhold of ten ercent. When comaring thi robability with the robability of exceeding ercent by uing control ackage, which i.896, control ackage roduce a higher chance to a a quality tet. Farm-level erological revalence robability ditribution for the four control ackage are hown in table.. Thee teriological revalence ditribution for one herd on a finihing farm were originally etimated uing an eidemiological imulation model decribed by Van der Gaag et al.(4b). The table how that the revalence ditribution for ackage i quite diered and i centered on 5 ercent; the ditribution for ackage i more concentrated and ha more of it robability ma at level below ercent. The ditribution for ackage 4 i till more concentrated at lower end of the revalence range, whit the robability of a tet reult of zero exceeding 7 ercent. If a roducer ued ackage would have a.8 robability of exceeding the almonella revalence threhold of ercent and having a roduction hitory level of zero in the next eriod and a.8 robability of having a roduction hitory level of one in the next eriod. 6

7 CHAPTER MODEL DESCRIPTION Thi chater introduce the concetual tructure and the theoretical kground ued for thi reearch. Dynamic Programming, teady tate ditribution theory and the cumulative exerience ytem theory are introduced baed on literature review.. Concetual tructure Aume there i a laughter lant deliver carcae in the Netherland and it i required by the official authority to articiate in a rogram of Salmonella control. During thi rogram, the laughter lant ha to elect one control ackage to reduce the revalence of Salmonella in the carcae each month. At the end of each month, the teriological revalence of carcae may be teted by the official authority. Intenity of Salmonella control i increaed with the cot of the elected control ackage. If carcae are teted negative, the laughter lant i rewarded by on extra oint on it reutation of Salmonella control. And if carcae are teted oitive, the laughter lant looe all it reutation oint and mut ay for a enalty. At the tart, all control rocedure are inected by the official authority on a daily bai. The inecting and teting cot are comletely aid by the laughter lant. To reduce the intenitie of inection and teting, the laughter lant can ave the total cot. The intenity of inection and teting both deend on the number of reutation oint. The more reutation oint, the le inection and tet are carried out. Furthermore, the adotion of effective control ackage can reduce the chance of having oitive tet reult and thu increae the number of reutation oint, with the conequence that the intenity of teting i reduced. Without an invetment on Salmonella control, the laughter lant can neither reduce the inecting cot, nor ave the teting cot. Thu, the election of control ackage to balance the cot of Salmonella control and the cot of inection and teting i eential for the laughter lant. The laughter lant roblem can be olved under two different condition: under the firt condition, there i the no governmental intervention and under the econd condition, the governmental intervention i active and influential to the laughter lant deciion making. Under the firt condition, the otimal control ackage are elected aim to minimize the laughter lant total cot; under the econd condition, the otimal control ackage are elected aim to minimize the laughter lant total cot baed on the change in governmental olicie. Table. ummarize the calculation of the model in three te: Dynamic Programming (DP) i alied to determine the otimal control ackage in the firt te; in the econd te, the exected total cot of the laughter lant and the exected Salmonella revalence level of carcae will be calculated with the teady-tate ituation; and in the lat te, a erie of incentive arameter will be introduced to imulate the imact of governmental intervention on the laughter lant deciion making. 7

8 . The otimal control trategie determination The laughter lant roblem can be analyzed in a dicrete time, dicrete ate and tochatic Markov deciion model with the time meaured in month. Miranda and Fackler (5) denoted that dicrete Markov deciion model may be olved by the dynamic rogramming (DP) method develoed by Richard Bellman (957). Alication of DP method require ix-te deciion making; include determining the tate variable, the deciion variable, the tochatic variable, the tranition, the cot function and the Bellman equation. State variable The model define the tate variable, S a the number of reuted record obtained by the laughter lant. Every month, the laughter lant may hold one of three oible tate: the tate with two reuted teting record, the ate with one reuted teting record and the ate with no reuted teting record,. For the imlicity, in thi tudy, the maximum number of the reuted teting record i two. Deciion variable The deciion variable are the Salmonella control ackage x, ( x x ) k x k =. There are x three control ackage are available in thi tudy; each control ackage x k i aociated with a cot c ( x k ), a robability of teting tet and a robability of aing the tet x ). Cot a ( k are increaed from control ackage x to control ackage x and the intenitie of control are increaed with cot. The adotion of the more intenive control ackage roduce more chance of aing the quality tet. Stochatic variable The tochatic variable r are analyzed in two hae: hae-one concern about whether carcae will be teted by the official authority, and hae-two refer to whether carcae could a the tet. Phae one i conditional on hae two. The outcome of random variable in hae one 8

9 deend on the tate i occuied by the laughter lant and the control ackage xk elected for Salmonella control. The outcome of random variable in the firt hae and the effectivene of the elected control ackage x determine the outcome of the random variable in the econd hae together. k Tranition In thi tudy, the control action of the laughter lant are aumed a irreverible. The different control ackage can be oted for different month. There i not correlated effect on outcome between the different month. In figure., Each month, the laughter lant occuie a tate i, i S. By chooing a control ackage x k, he may move to a new tate, S. Tranition among tate are monthly erformed and denoted byt i. Given the tating tate i, the elected control ackage xk and the random variable r ; one write the ending tate a:. The tranition robabilitie of moving from tate i to tate are denoted by P and determined by the tate variable, the deciion variable i x k and the outcome of the random variable r. Notation can be ued to exre thi deendency. In figure., the firt art rereent the tranition tarted in tate, the econd art rereent the tranition tarted in tate and the lat art rereent the tranition are tarted in tate. The red dot in the figure tand for carcae are teted by the official authority, and the green dot how that carcae are exemted from the tet. By chooing one of three control ackage, the laughter lant receive a robability of being teted tet and a robability of aing the tet a. If the tet are erformed and carcae are at in the tet, the laughter lant will move to tate +, when the next tranition i tarted; if the carcae are failed in the tet, the laughter i 9

10 lant will move to tate Alication of Dynamic Programming in Salmonella control of meat roduct and ay for an amount of enalty c ( ) ; in cae of carcae are exemted from the tet, the laughter lant will tay in tate i when the next tranition i tarted.. Cot function and Bellman equation The cot otimality olicie can be formulated in the form of Bellman equation V ). the model ( i define V ) in thi tudy i : ( ) c( x ) + P (, x r) V ( ) V i = max k δ i i k,, i S, S (.) S Where, i and rereent the tarting tate and the ending tate in the tranition reectively, δ, V ( ) c( x k ) tand for the cot function, δ i the dicount factor, aociated in the ending tate and P (, x, r) V ( ) S i i k ( i rereent the cot δ i the exected cot received in tranition.due to the model ha an infinite time horizon characteritic, therefore, the value function V () i not deendent on the time and the Bellman equation i a vector fixed-oint equation whoe ingle unknown i the common value function V.. Steady-tate robabilitie In an irreducible Markov chain, the teady-tate robabilitie exit after a huge number of tranition. The term teady-tate robability mean that the robability of finding the roce in tate after a large number of tranition i indeendent of the initial robability ditribution defined over the tate and tend to the valueπ. No i matter where the tranition tarted, the robabilitie of reaching all the ending tate are equivalent. With thi roerty, the teady-tate-robabilitie matrixπ can be contructed a: π π = π π π π π π π π There are three row and three column in matrixπ. The row rereent the tarting tate i and the column rereent the ending tate. After the long-run tranition, If the ditribution over tate ha reached the teady-tate ditribution rereented by the π at time n, then the ditribution over tate at time n+ i the ame. Thi

11 characteritic can be ued to derive directly the vector containing the tationary robabilitieπ, intead of making all the neceary time te. Contructing the hort-term tranition matrix with the otimal tranition robabilitie i : P = The teady-tate robabilitie π can be obtained by the olving the equation = P together with the roerty π =. Thee include: π = π = π = * π + = π + π + π * π + * π + * π + * π + * π + * π * π * π In matrix P, all the tranition robabilitie are calculated for uing control ackage x, due to the minimum cot are accrued from uing x in each tate i. = =.4 The imact of governmental intervention on the otimal control trategie The aim of the laughter lant i to reduce the intenity of the inection and the intenitie of the tet to realize cot minimization in the Salmonella control. Enlarging the robabilitie of the tet exemtion and the robabilitie of inection releae, the laughter lant i imelled to invet on the cotly control ackage. Due to diminution in either the inection or the quality tet are determined by the intenitie of Salmonella control. Incentive arameter defined in thi tudy afford the chance to the government to adut the teting and inecting intenitie on the laughter lant. The imact of governmental intervention and the change in the laughter lant deciion making will be demontrated in thi chater..4. Incentive arameter deign Bae on the incentive arameter develoed by King. et al., (5), the et of arameter defined for thi tudy i illutrated in table.:

12 Where, arameterα, α and α are defined for aduting the teting robabilitie; arameter β, β and β are defined for aduting the inecting robabilitie; arametert ero i ued to alter the trictne of the erological tet on the farm level, arameter t i for altering the teriological tet on the laughter lant level, and arameter ( ) V,. c,γ, θ and n are defined for regulating the ending tate cot ( ) The value of thee arameter are altered between the uer bound and the lower bound with the relevant te ize illutrated in table.. The network of the et of incentive arameter are demontrated in figure..

13 Figure. i read from the to down. During the tranition, arameter c ( ), γ, θ and n may directly change the cot V ( ) aociated in the ending tate. And arameter t ero, t, α, α, α, β, β and β may indirectly alter the value of tranition robabilitie i. Through changing the value of and the value ( ) i V, the government can influence the laughter lant deciion making and therefore change the hort-term total cot V ( i ), the long-run exected revalence level erev, the long-run exected cot ev ( ) and the exected inecting cot ec ( in). The focu of thi tudy i to analyze the change in the laughter lant exected inecting cot, the change in the hort-term total cot, the change in the laughter lant deciion making, and the change in the long-run exected revalence level

14 and exected total cot, through aigning the different value to the et of arameter. Chater 4 Emirical tudy 4. Tranition robabilitie The emirical calculation of tranition robabilitie and table 4.. i are illutrated in figure 4. Thi deciion tree i conited of three art: the left art rereent the deciion variable k x ; the middle art rereent the teting robabilitie ( ) tet x k ; and the right art rereent the robabilitie of aing the tet a ( x k ). The ending oint of the tree tand for the ending ate in the tranition. Value of thee tranition robabilitie i can be calculated in table 4.: 4

15 The tranition robabilitie i are ditinct from the long-run tranition (teady-ate) robabilitieπ. The long-run tranition robabilitie for calculating the exected Salmonella revalence level will be decribed later. Inecting robabilitie & teting robabilitie The model define the intenitie of official inection in are deendent on the tate variable i. The number of the reuted record inection are increaed. Thi relation i written by equation.. in e tet e = i (4.) ( i + a ( xk )) = (4.) i declined, the intenitie of In equation., the robability of teting i deendent on both of the tate variable i and the effectivene of control ackage a. Beide the imact of the tate variable i erformed on the teting robabilitie tet, the effectivene of control a may alo wing the teting intenitie. At the moment of tranition are tarted in the low tate i, uch a i =, the adotion of the intenive control ackage may comenate the teting robabilitie that enlarged by the weak tate i. Probabilitie of aing the tet (effectivene of control) The robabilitie of aing the quality tet a rereent the effectivene of the elected control ackage x k. Calculation of the aing robability a require five 5

16 vector and five matrixe that contructed from the reviou reearch conducted by van der Gaag et al.,(4b) and by king et al., (5). Table. how the required matrixe and the relevant notation a follow: Matrix A, matrix B, matrix B and matrix B are conited of the reult of the eidemiological imulation model that decribed by van der Gaag et al., (4b). Vectorb, c, c, c and matrix C and D are calculated by the exiting reult. The algorithm of the erology-teriology tranition and the effectivene a of the laughter lant control ackage xk are illutrated a follow: 6

17 Figure 4. conited of two art of calculation. The firt art i to calculate the erology-teriology tranition between herd and carcae. The econd art i to calculate the effectivene a of control ackage x k with the reult obtained in the firt art. Where, arameter t denoted in matrix D rereent the teriological threhold et to the laughter lant by the government. Exlanation for each te calculation are illutrated a follow. --- Calculation of the erology-teriology tranition Thi art of calculation i aimed to acquire the teriological revalence ditribution matrixc, through calculating the erology-teriology tranition ditribution. The relevant vector a, b, c, d, e and the required matrixe A, B, B, B in thi art will be decribed in turn. Vector a ero = t Vector a tero ( a a a a ) y y y y4 a t, ero 4 at ero R i contructed by the reult of the exiting model that develoed by king et al., (5). It i related to the farm-level erological revalence control. In that model, there are four erological control ackage y m, y m = ( y, y, y, y 4 ) are ditinguihed, each control ackage aociated with a cot. Intenitie of control are increaed with cot. From y to y 4, the intenitie of control 7

18 are gradually increaed. Each control ackage y m ha a robability of being ued on farm. For intance, the element ay rereent the robability of control ackage y i ued by farmer, and element ay rereent control ackage y i ued by farmer. Matrix A y y y y 4 a y a y a y a y4 4 = a y a y a y a y4 A M M M M M a y a y a y a y4 Matrix A alie the reult of the Van der Gaag urvey conducted in 5. it denote the erological revalence ditribution of herd for each control ackage y m.the row of the matrix rereent the eleven erological revalence level, rev ( L ). The column of the matrix rereent the four ditinct control ackage y m. The element of the matrix, for intance, a y, rereent the robability of realizing the revalence i ten in herd by uing control ackage y Vector b ( b b L ) b = a * A' = b The inner roduct of vector a and matrix A' generate vectorb. There are eleven element contained in vector b, rereent the exected erological revalence ditribution in the long-run tranition. For intance, b rereent the robability of realizing the revalence i ten in herd in the long-run tranition, and b rereent robabilitie of realizing the revalence i twenty in herd in the long-run tranition. Matrix B, B and B 8

19 B = M M Alication of Dynamic Programming in Salmonella control of meat roduct er er L L L er M ero Matrix B, B and B are the erology-teriology tranition matrixe. B, B and B rereent the revalence tranition ditribution for uing teriological control ackage x, x, and x on the laughter-lant level reectively. The tranition matrixe are unit matrixe. There are eleven row and eleven column in all the tranition matrixe: row rereent the erological revalence of the laughtered herd and column denote the teriological revalence of carcae. For matrix B, the element er rereent the teriological revalence ditribution of carcae on the ten er cent level when the erological revalence of the herd laughtered i twenty er cent by uing teriological control ackage x on the laughter lant-level. Vector c, d, e and matrixc c = M y y d = M y y e = M y y C = [ c d e] The inner roduct of vectorb and matrix B roduce vector c ; the inner roduct of vectorb and matrix B roduce vector d and the inner roduct of vector b and matrix B roduce vector e. There are eleven element in all the vector c, d and e rereent the teriological revalence ditribution after erology-teriology tranition by uing teriological control ackage on the laughter lant-level. Vector c denote the ditribution for control ackage x ; vector d denote the ditribution for control ackage x and vector e denote the ditribution for control ackage x. Contructing matrix C with vector c, d and e, matrix C i the teriological revalence ditribution matrix after revalence erology-teriology tranition. -- Calculating the robabilitie of aing the tet 9

20 Thi art of calculation i aimed to get the effectivene (aing robabilitie) of each control ackage x k. Symbol t in matrix D denote the threhold level of the allowable revalence in carcae. The official authority can aign the different value to t to alter the trictne of the quality tet. C = M x x x c c M c d d d M e e e M In matrix C, there are eleven row and three column. The row rereent the teriological revalence robability ditribution and the column rereent the teriological control ackage x k. For intance, column c tand for the revalence ditribution of uing control ackage x and column d i the revalence ditribution of uing control ackage x.the element c in column c rereent the robabilitie of carcae revalence reache in the econd level, ten, by uing control ackage x. And the element d in column d rereent the robabilitie of carcae revalence reache in fourth level, thirty, by uing control ackage x. The row of matrix D tand for the different threhold level t and are calculated by umming u one or everal row in matrixc. For intance, the econd row of matrix D i calculated by umming u the firt two row of matrix C. And the third row of matrix D i calculated by umming u the firt three row of matrix C. The cumulative robabilitie in matrix D exre the effectivene a of the elected control ackage x k. The firt column of matrix D tand for the effectivene of control ackage x, and the econd and the third column of matrix D trand for the effectivene of control ackage x and x. When the allowable threhold t i one, the laughter lant receive the lowet cumulative robabilitie a of aing the tet (ymbolized by the firt row of matrix D ). And when the threhold t i four, the larget robabilitie a (ymbolized by the fourth row of matrix D ) are

21 received by the laughter lant. 4. Following tate and aociated cot During the ate tranition, each following (ending) ate aociated with one or everal cot, include the teting cot c ( tet), the exected inecting cot ec ( in) and the enaltie c ( ). The amount of the cot are altered with the ending ate a illutrated in table.. The model define the teting c( tet) and the exected inecting cot ( in) ec are aociated in the ending tate + ; the ingle ayment of the exected inection i ec( in) i aociated in the ending tate i and the teting cot c ( tet) ec( in) together with the enaltie ( ), the inecting cot c are aociated in the ending tate. Therefore the highet cot are aeared in the ending tate and the lower cot are occurred in the ending tate i and i +. Beide, the exected inection cot are decreaed a the tate variable i imroved. In the high ending tate = i+, the intenitie of inection are droed down to in = e i, wherea, in the low ending tate = + lowet level ( ) in = e. inection are raied to the highet level ( ), the intenitie of

22 4. The total exected cot in the long-run tranition The long-run exected cot are denoted by ev ( i ) and calculated by multilying the teady-tate robabilitie ev i = = ( i ) V ( i )* π i= = π with the total cot ( ) = (4.4) V : It i notable that the total cot V ( i ) aeared in equation 4.4 are the minimum cot that generated by uing the otimal control ackage x in the current tranition. 4.4 The exected revalence level in the long-run tranition i The exected revalence level in the long-run tranition are denoted by erev and calculated by multilying the vectorl with the inner roduct of the revalence vector brev and matrixc : erev = brev * C * l (4.5) Where, brev rereent the eleven teriological revalence level: ( L ) brev =. Matrix C denote the teriological revalence ditribution calculated in figure 4., and vector l, l = ( l l ) l rereent the robabilitie each control ackage x k will be ued by the laughter lant. Algorithm of l can be demontrated with figure 4. and figure 4.4 a follow:

23 The green line of figure 4. rereent the otimal control ackage x, the red line rereent the otimal control ackage x and the yellow line rereent the otimal control ackage x. In cae of figure 4., the otimal control ackage i x in tate, x in tate and x in tate. The teady-tate robabilitie for tate, and are π, π and π reectively. Therefore, The robability of uing control ackage x i l = ; π The robability of uing control ackage x i l = ; π The robability of uing control ackage x i l = π. However, in the cae calculated in thi tudy, the otimal control ackage i x for all the tate i, a illutrated in figure 4.4 below. Therefore: The robability of uing control ackage x become l = π + π + π = ; The robability of uing control ackage x become l = ; The robability of uing control ackage x become l =. In equation 4.5, Vector f rereent the eleven teriological revalence level: f = ( L ) From zero to a hundred, the increaed te of the revalence level i ten. The inner roduct of matrix C and vector f generate the hort-term exected revalence

24 vector v, v R. v = f = Alication of Dynamic Programming in Salmonella control of meat roduct ( v v ) * C v Each element of v rereent the exected revalence level for uing control ackage x k in a hort-term tranition. After multilying vector l with v, the long-run exected revalence level erev i calculated. 4.5 Incentive arameter alication 4.5. Teting and inecting robabilitie arameter Aending the arameter β, β and β to equation 4. and the arameterα, α and α to equation 4., the new exreion of inecting robabilitie ( in) and teting robabilitie ( tet) are defined a: β* in) = max( β * e, β ) 4.6 ( α*( + ( k )) ( ) α * a x tet e, α ) = max( 4.7 ( in) β * e = β β * if if β * e β * e β* β* β β 4.8 ( tet) α * e = α α*( + a ( xk )) if if α * e α * e α* α* α α 4.9 In equation 4.6, there are four variable determine the inecting robabilitie ( in) : the maximum inecting robability β, the reduction robability β, the minimum inecting robability β and the tarting tate of tranition i. If kee the tate variable i contant, then the inecting robabilitie ( in) are determined with β* β and β, when β * e β exit, and determined with β, β* β* when β * e β exit. In cae of β * e β, the inecting robabilitie 4

25 ( in) are decreaed with β and increaed with β. The maller value of β together with the larger value of β generate the lower inecting robabilitie ( in) β*. In cae of β * e β, the inecting robabilitie ( in) are increaed with β. Leening the value of β reduce the robabilitie of inection ( in). The ame theorem i alo alied to the equation 4.8 and 4.9 for arameterα, α andα Tet threhold arameter Serological threhold arametert ero The erological threhold arameter tero ha four oible value, (,,, 4) t ero =. Each value aociated with a vector a t ero which i obtained from the exiting model. The firt element in each vector rereent the robabilitie of erological control ackage y i ued on farm; the econd element rereent the robabilitie of erological control ackage y i ued on farm; the third and the fourth element rereent the robabilitie of y and y 4 are ued reectively. When the allowable threhold t ero i et by one, only the mot intenive control ackage y 4 can be 5

26 elected and the ret have no oibilitie of being ued on farm. When the allowable threhold tero i et by the highet value, four, the leat intenive control ackage y aociated with the larget robability of being ued on farm, wherea, the mot intenive control ackage y 4 will not be ued on farm. Therefore, the robability of adoting the mot intenive control ackage i reduced a the value of threhold arameter tero i increaed. It i notable that the value of vector a t ero determine the teriological revalence ditribution of carcae. Remember the algorithm of teriological revalence ditribution matrix C demontrated in figure. The lower threhold value roduce the fewer chance of aing the tet, and thu increae the laughter lant incentive of adoting the cotly control ackage. Bacteriological threhold arametert Parameter t i related to the matrix D which i illutrated in figure.. When the threhold equal t four, the robabilitie of aing the teriological 6

27 tet a for uing each teriological control ackage x k are rereented by the fourth row of matrix D: t 4 a = = + t = ( x ) ct = c + c + c c4 t 4 a = = + t = ( x ) dt = d + d + d d4 t 4 a = = + t = ( x ) et = e + e + e e4, robability of aing the tet by uing x, robability of aing the tet by uing x, robability of aing the tet by uing x When the threhold t equal two, the robabilitie of aing the teriological tet a for uing each teriological control ackage x k are rereented by the econd row of matrix D: t a = = + t = ( x ) ct = c c t a = = + t = ( x ) dt = d d t a = = + t = ( x ) et = e e, robability of aing the tet by uing x, robability of aing the tet by uing x, robability of aing the tet by uing x Therefore, the value of threhold t determine the trictne of the teriological tet and influence the robabilitie of aing the teriological tet a. the lower threhold value bring more incentive to laughter lant to aly the more intenive control ackage Cot haring arameter and enaltie After adding the cot haring arameter γ and θ into the model, the teting cot c ( tet) and the inecting cot ( in) ( tet) γ c( tet) c = * teting cot ( in) θ c( in) c = * inecting cot c become: Increaing the enaltie c ( ) and the value of arameterγ and θ, the laughter lant incentive in adoting the more intenive control ackage are increaed. 7

28 4.5.4 State arameter n The model decribed above, there are three tate are available for the tate tranition. The arameter n equal two and the minimum teting and inecting robabilitie are e and e ( + a ) reectively. If the government determine that there are ix tate can be available for tranition, then the arameter n become five and the minimum teting and inecting robabilitie are reduced to 6 e and e ( 6+ a ) reectively. Therefore, increaing the value of the tate arameter n, the laughter lant get the more incentive of adoting the intenive control ackage to obtain the leat robabilitie of being inected and teted Altering tendencie of arameter The imact of governmental intervention i analyzed for each teriological threhold level. From the left to the right, the value of the arameterα, β and c( ) are gradually increaed, wherea, the value of arameter α, β andt ero are gradually declined and the arameterα, β γ,θ and n are ket on contant. The larger value of arameter α and β and the maller value of arameterα, β and c( ) are alway et under the maller value of erological threhold. By thi way, the government ha increaed the laughter lant incentive of adoting the cotly control ackage on each teriological threhold level from the left to the right. 8

29 5. RESULT 5. No governmental intervention The reult illutrated in thi ection rereent the ituation when there i no governmental intervention. It i reented to comare the reult of other cenario. 5. With governmental intervention Threhold for laughter lant t Threhold for farmer t ero Incentive arameter Maximum teting arameter α Reduction teting arameter α Minimum teting arameter α Maximum inecting arameter ß Reduction inecting arameter ß Minimum inecting arameter ß Penaltie c() Sharing of teting cot γ Sharing of inecting cotθ Maximum tate n Performance meaure Otimal ackage control State x * State State Short-term total cot State

30 V ( i ) Alication of Dynamic Programming in Salmonella control of meat roduct State State Steady-tate robabilitie π State State State Exected inecting cot Long-run exected total cot Exected revalence level for uing each control ackage in the hort-term tranition x x x Probabilitie of each control ackage being ued by the laughter lant l x x x Steady tate exected revalence level in the long-run erev tranition Table.8 From the reult reent in ection 5. and the reult reent in ection 5., we can ee the change of the laughter lant in electing the otimal control trategie. In ection 5., the laughter lant chooe the cheaet control ackage for each tate and generate the relative high exected revalence level. After adding the governmental intervention, in the ection 5., the laughter lant kee uing control ackage one when teriological threhold value t i relaxed and the erological threhold value t ero i trict. However, when the teriological threhold i trict and the erological threhold i relaxed, the laughter lant changed the otimal control trategie to the more cotly control ackage x and x, and thu reduce the exected revalence level in carcae. On each teriological

31 threhold level, form the left to the right, the robability of meat inection i gradually reduced, but thi reduction i conitent with the adotion of the cotly control ackage. 5. Imact of the governmental intervention on the laughter lant otimal control deciion The reult how: ) the adotion of the more cotly Salmonella control ackage at the laughter lant level increae a the teriological threhold i reduced; ) when erological threhold i et trict to the farm, the laughter lant looe the incentive of adoting the cotly control ackage, wherea, ) when the erological threhold i et relaxed to the farm, the laughter lant i imelled to invet on the cotly control ackage. In table.7, the leat incentive of adoting the cotly control ackage to the laughter lant aear in the firt column and the mot incentive to the laughter lant aear in the lat column. By etting a higher value for the erological threhold and a lower value to the teriological threhold, the government increae the incentive of adoting cotly control action at the laughter lant. When the teriological threhold i et to four, the otimal control ackage for the laughter lant i alway x for all the tate, no matter how the ret of the arameter will be changed. When the teriological threhold i et to three and the erological threhold i et to one, two or three, the otimal control ackage i ket on x for all the tate. However, a long a the erological threhold i changed to four, the otimal control ackage will become to x. When the teriological threhold i et by two and one, thi change i even more obviou. Under the teriological threhold i two and the erological threhold i four, the otimal control ackage for tate S and S i x, and for tate S x. When the teriological threhold i one and the erological threhold four, the otimal control ackage i x for all tate. Setting a evere erological threhold for intance, t ero = to the farm make the government to increae it olicy to aim for the adotion of the intenive control action at the farm-level (ee table 4.4), reduce the exected teriological revalence of carcae in the laughter houe and lower the incentive to adot the cotly control ackage to the laughter lant. Setting a evere teriological threhold for intance, t =to the laughter ant

32 make the government to increae the robability to tet oitive (ee the algorithm of matrix D of figure.), and thu imel the laughter lant to invet in a more intenive control ackage to reduce the robability of getting the enalty. Beide, when arameter minimum teting robability arameter α i zero, increaing the value of arameter α and reducing the value of arameter α, the laughter lant may receive an increaed robability of being teted and getting the enalty, thu, he i treed to imrove the outcome of control by adoting the more cotly control ackage to avoid getting the enalty. When minimum inecting robability arameter β i zero, increaing the value of arameter β and reducing the value of arameter β, the robability of inection i enlarged, eecially in the lower tate. Here, the laughter lant face on increaed exected inecting cot if he fail the tet and conequently dro down to the lowet tate. Therefore, adoting the more cotly control ackage to enlarge the robability of aing the tet become neceary for the laughter lant to avoid the enlarged exected inection cot in the future. 5.4 Imact of the governmental intervention on the exected revalence level in the long-run tranition The exected revalence level in the hort-term tranition are increaed with the value of erological threhold. The more the erological threhold to the farm i relaxed, the higher the teriological revalence the laughter lant i exected. When a relaxed erological threhold i et to the farm and the mot evere teriological threhold i et to the laughter lant, the laughter lant i treed to adot the cotly control ackage. Becaue of the a relaxed erological threhold i et to the farm, farmer looe the inenitive of adoting the cotly erological control ackage. Thi reult in a high erological revalence in the delivered herd. The teriological threhold i retrict make the adotion of the cotly teriological control ackage more neceary to the laughter lant to avoid the failure in teriological tet. If the intenity of meat inection i reduced it will not influence the efficiency in Salmonella revalence control if the government ae incentive the laughter lant on teting intenitie, enaltie and the allowable teriological threhold.

33 6. CHANGES BASED ON THE EXISTING MODEL Comared with the exiting dynamic rincial model that develoed by King et al., (7), the model develoed in thi tudy ha made ome imrovement: Firt, the incentive ytem tudy from the laughter lant to the farm level wa extended to an incentive ytem from the governmental to the laughter lant level. Second, the otimal control ackage wa analyzed when the intenitie of meat inection i reduced, which wa not included in the former model. Third, more incentive arameter were added, uch a the inecting arameter β β and β, and teriological threhold arameter t Fourth, the algorithm of the exected teriological revalence in carcae on the laughter lant-level wa develoed and added. Fifth, the deign of the tranition matrix wa changed. Sixth, the leat influential arameter, rice remium wa removed from the model. A illutrated in figure 6., the former tranition matrix aume only two oition: carcae are teted oitive (rereented by the green dot) and carcae are teted negative (rereented by the red dot). By thi definition, the laughter lant i imoible to tay in the original tate in the tranition. In the develoed tranition matrix, the blue dot denote the ituation of carcae are exemted form the tet, and therefore, the changing the rule of tate tranition a well a the calculation of tranition robability. Seventh, the obective function wa changed from rofit maximization to cot minimization.

34 7. CONCLUSIONS Thi reearch focued on undertanding the effect of governmental intervention on the laughter hoe manger in chooing the otimal control trategie. The focu wa ecifically on the intenity of ublic inection, the intenity of teting, the enaltie and the allowable threhold of revalence level. the concetual framework of the market wa baed on interview with three exert in uerviion of control and one quality manager in a laughter houe integration in the Netherland. The analyi baed on in thi model demontrated the value of conidering erformance hitory when roducer make reeated deliverie and the imact of the reduction in ublic inection on the conequence of almonella control in meat roduction. Thi tudy how how the government inherence the roduction active of the laughter lant through aing incentive to the relevant influential factor. The mot influential factor that may change the choice of the food buine oerator i the allowable threhold et by the government. When the farm-level threhold i relaxed and the laughter lant-level threhold i trict, the laughter lant manager are induced to aly more tringent laughter lant Salmonella control meaure and the exected Salmonella revalence in meat will be reduced. Reduction in ublic inection doe not have an effect on the quality of the roduct. Converely, it may increae the incentive of laughter lant in adoting more cotly control ackage to award the reuted roduction hitory and reduce the inecting cot. However, the teting arameter have le effect on the deciion made by the laughter lant. The control cot are quite enitive and robut to the final reult. It imlicate that the change in control cot are extremely influential to the laughter lant deciion making. Shortening the rice difference between teriological control ackage x and x with. Euro, the otimal control ackage of the laughter lant become x for all the tate when the teriological threhold i two and the erological threhold i four. 4

35 8. DISCUSSION The model i olved with Matlab routine; the inecting robability written by in Matlab language did not comletely reent the deign of tate tranition in thi reearch. In tate, the laughter lant i imoible to move to tate, and therefore, the minimum inecting robability hould be e, wherea, with Matla routine, the minimum inecting robability i automatically calculated a e. The model analyzed the controlled Markov chain with an infinite time horizon; it aume the tate tranition of the laughter lant are tationary over time. However thi i not comletely conitent with the reality. In tead, one can alo analyze the laughter lant roblem a the controlled Markov chain with a finite time horizon. The reult obtained in thi reearch i conitent with the anwer got in the interview with three exert in uerviion of control. The quality of roduct influence a laughter lant reutation and brand name. It determine the utainable rofitability of the comany in the future. Therefore, with or without ublic inection will not change the attitude of laughter lant in reducing Salmonella revalence and imroving the roduct quality. Viual meat inection romote the information integration between the farm and the laughter lant. Information haring ytem may imrove the efficiency of Salmonella control but may alo bring uncertaintie in moral hazard. Therefore, to analyze the rik of Salmonella control in a integrated chain could be the toic in the future reearch. 5

36 9. REFERENCES Robert P. King, Ge B. C. Backu, and Monique A. van der Gaag Incentive ytem for food quality control with reeated deliverie: Salmonella control in ork Production Mario. J.Miranda Paul. L. Fackler, Alied Comutational Economic and Finance. Monique A. van der Gaag, Fred Vo, Helmu W. Saatkam, Michiel van Boven, Paul van Beek, Ruud B.M. Huirne A tate-tranition imulation model for the read of Salmonella in the ork uly chain Euroean Journal of Oerational Reearch 56 (4) Final reort of Salmoella in Pork: re-harvet and Harvet Control Otion baed on Eidemiologic, Diagnotic and Economic Reearch Editor: Danilo M.A.Lo Fo Wong; Tine Hald Proect reonible: Preben Willeberg A.A. Dikhuizen R.S. Morri Animal health economic rincile and alication. Bondt, N., Wangenberg, C.P.A. van, Ooterkam, E.B., and Backu, G.B.C. control of control roceeding of the firt annual congre. 6