Stochastic Integer Programming Models in the Management of the Blood Supply Chain: A Case Study

Transcription

1 World Joural of Operatioal Reearch 2017; 1(2: doi: /j.wjor Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study Luiaa Sibuea, Habibi Saleh, Moh Dail Hedry Gamal Departmet of Mathematic, Uiverity of Riau, Pekabaru, Idoeia addre: Lui.7bu34@gmail.com (L. Sibuea, dr.habibialeh@gmail.com (H. Saleh, mdhgamal@uri.ac.id (M. D. H. Gamal Correpodig author To cite thi article: Luiaa Sibuea, Habibi Saleh, Moh Dail Hedry Gamal. Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study. World Joural of Operatioal Reearch. Vol. 1, No. 2, 2017, pp doi: /j.wjor Received: July 11, 2017; Accepted: July 19, 2017; Publihed: Augut 14, 2017 Abtract: Thi paper preet a problem i the maagemet of the blood upply chai at the blood bak with perihability characteritic, epecially for the red blood cell ad platelet. Focu of thi dicuio i to miimize the total cot, hortage ad watage level of the blood uit. Stochatic iteger programmig approach i ued to olve thi problem by aumig the blood group ad takig ito accout the age of the blood. At the ed of thi tudy we give a imulatio to ee the reult of applyig the method i thi iue. Keyword: Supply Chai Maagemet, Blood Bak, Perihable Item, Stochatic Iteger Programmig 1. Itroductio Blood, the mot importat compoet of the body' traportatio ytem, i a liquid that carrie utriet ad oxyge to all orga of the body. Blood i categorized a a carce reource becaue it uage age i limited. From Health Media [15] i kow that the red blood cell fuctio to move oxyge to our cell, platelet fuctio to relate to the proce of coagulatio ad white blood cell fuctio to keep the body immue ytem ad defed the body agait bacteria or virue. Plama i a liquid compoet ubtace uch a albumi, blood clottig, hormoe, variou protei ad alt. If there i a hortage blood the the eed for utriet ad oxyge i ot atified ad thi will lead to death. Blood trafuio i the proce of movig blood from a pero to a pero i eed of blood. The blood ivetory i Idoeia i curretly maaged by the Idoeia Red Cro (PMI. PMI i a atioal ociety orgaizatio i the ocial field of humaity which ha bee recogized atioally with the Preidetial Decree o. 25 of 1959 from the Idoeia Red Cro [16]. The mai tak of PMI itelf i regulated i Govermet Regulatio o. 18/1980 Chapter IV, ectio 6, paragraph (1: "The maagemet ad executio of blood trafuio buie i aiged to PMI or a ititutio aiged by the Miiter of Health". Blood proceig i performed by the Blood Trafuio Uit (UTD, oe of thi i PMI UTD City of Pekabaru. Blood iput that goe ito the blood ivetory at PMI UTD City of Pekabaru come from local UTD (idepedet productio ad hipped from other UTD. Wherea the blood output i upplied to the Hopital Blood Bak (BDRS, No- BDRS ad other UTD. BDRS fuctio a a facility to make patiet eaier to get blood without paig PMI. While o- BDRS have a differet fuctio with BDRS, No-BDRS i a hopital that doe ot have blood bak. BDRS determie the level of the optimal upply of all blood compoet baed o dicrete imulatio i the BDRS agait tochatic demad ad the occurrece of damage (watage of blood that may occur i the reearch of Kataliaki ad Brailford [11]. Pierkalla [13] tate that the duty of BDRS i a a maager of the blood collectio, cromatchig procee, blood upply maagemet ad ditributio of blood compoet by eurig the blood upply chai ad the demad for blood i fulfilled. Nagurey et al. [12] aalyze the complex upply chai of huma blood coitig of collectio ite, tetig ad proceig facilitie, torage facilitie, ditributio ceter, a well a demad poit, age of the blood torage ad calculatio of the remaiig torage life whe the blood uit are retured to BDRS. I thi tudy they alo how that oe pero i every three ecod acro the coutry may wat to receive the blood trafuio. The demad for the blood cotiue to rie tochatically

2 42 Luiaa Sibuea et al.: Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study ad ha limited umber. O the other had, Belie ad Force [1] metio that the majority oly 5% from the citize populatio havig eligible coditio to do blood door. Thi alo raie the lack of the blood upply to fulfill the eed of the patiet. Accordig to the World Health Orgaizatio (WHO the miimum blood upply i each coutry i 2% of the total reidet everyday. Ariig problem i the blood upply chai icreae complexly becaue of the perihable characteritic of the blood that ha a limited uage for ue. Epecially platelet are highly perihable ice they ca oly be tored up to 5 day with torage temperature of C ad red blood cell up to 35 day with torage temperature of 2 6 C o ivetory. Therefore, everyday there are alway two poibilitie, i.e., hortage tock of blood (tock out or exce tock of blood (over tock i the ivetory. Exce blood tock ca alo caue blood become damaged (watage if it ha outdate the age limit of it ue. Belie ad Force [1] report that the everal tudie have bee doe to fid olutio to the blood upply chai problem. Such a dicrete evet imulatio, Mote Carlo imulatio, tatitical aalyi (liear regreio, urvival aalyi, logitic regreio ad ANOVA ad i operatio reearch (iteger programmig, liear programmig ad tochatic dyamic programmig. There are four mai actor i the blood upply chai, i.e., door, blood bak, hopital ad patiet. The upply chai of the blood i highly depedet o the umber of door who will doate the blood. Prataco [14] explaied that effort i blood door eed to be upported by the calculatio of the average availability of blood door ad predict the quatity of the blood upply or the blood demad. Haijema et al. [7] ue a dyamic tochatic programmig model ad a imulatio model at the blood bak i the Netherlad for productio ad the maagemet platelet of the blood upply with the aumptio of the productio cot factor of platelet. Hemmelmayr et al. [8] develop the iteger programmig model to decide which hopital a vedor (through the vehicle from the blood ceter hould viit each day give that the route are fixed for each regio. Ghadforouh ad Se [5] they formulate a oliear iteger programmig model to determie the miimum cot of platelet productio chedule for the regioal blood ceter. The iitial formulatio carrie a o-covex objective fuctio that i difficult to olve ad would ot guaratee covergece to optimality. A both the objective fuctio ad cotrait of the revied formulatio iclude quadratic term, a two-tep traformatio liear 0-1 iteger alterative i the propoed to guaratee the optimality. Gupiar ad Ceteo [6] ue tochatic iteger programmig model to reduce watage ad hortage of the blood i the hopital with the aumptio of the blood age, blood demad for two type of patiet: Type-1 patiet i the oly patiet who eed ew blood i blood trafuio while patiet Type-2 i a patiet who eed old blood or ew blood withi blood trafuio a well a uig a cromatchig trafuio ratio for oe blood bak ad oe hopital. I additio, the determiitic model i icluded i the model ad the blood ued i the blood that i quickly damaged uch a red blood cell ad platelet i geeral without coiderig the blood group. There are 3 model formulatio i their reearch, amog other are the tochatic iteger programmig model, tochatic iteger programmig model with two patiet type ad determiitic iteger programmig model with cromatch-to-trafuio (C/T ratio ad cromatch releae period. Baed o the backgroud of the iue decribed ad the importace maagemet of the blood upply chai it eed to do reearch o the maagemet of the blood upply chai for the better upply ad tock maagemet. Therefore, i thi tudy the author attemp to develop tochatic iteger programmig model i the maagemet of the blood upply chai I Gupiar ad Ceteo [6]. I thi propoed method the aumtio of blood group for a blood bak i added ad the other related aumptio are take ito accout to reduce the occurrece (watage ad hortage of blood. I thi article, author oly dicu the firt formulatio model of tochatic iteger programmig of Gupiar ad Ceteo [6] by takig ito accout the blood group with the data obtaied from PMI UTD City of Pekabaru for the year For the calculatio we ue LINGO 16 ad Microoft Excel 2010 program applicatio with four demad ceario for the year I ectio 2, the author preet the model developmet for olvig the problem of blood upply chai i PMI UTD City of Pekabaru. The, ectio 3 preet the computatioal reult of the applicatio of the propoed model developmet ad the ectio 4 preet cocluio. 2. Propoed Method I thi ectio the author develop the model of Gupiar ad Ceteo [6] by coiderig the blood group. Thi cotrait aure that the demad for blood i the hopital more efficiet ad ca prevet the occurrece of rik ad error i the delivery ad blood trafuio ito patiet with certai type of blood group. It alo make the ditributio of blood demad from PMI UTD City of Pekabaru to the hopital eaier. With thi blood group aumtio it i more eure the afety of blood trafuio ad miimize error i blood trafuio i patiet. ued i thi blood upply chai model i a blood group ubtace uch a: A, B, O ad AB without cromatchig becaue the blood beig delivered by PMI UTD City of Pekabaru i the blood that ha bee determied directly by hopital to trafu i to patiet ad the proce of cromatchig ha bee doe by hopital blood bak (BDRS. BDRS take the blood to PMI UTD City of Pekabaru i accordace with type ad blood group requet required by the patiet ad the UTD PMI Pekabaru city ever receive a retur of the blood from BDRS. Below, it i give the aumptio of the tochatic iteger programmig model ad otatio of the developmet of tochatic iteger programmig model i the maagemet of blood upply

3 World Joural of Operatioal Reearch 2017; 1(2: chai that ha bee developed i accordace with otatio i Gupiar ad Ceteo [6] ad thoe ued i blood upply chai model. 1. The followig aumptio are made i accordace with the model of Gupiar ad Ceteo [6] ad ued by author i developmet of tochatic iteger programmig model i the maagemet of blood upply chai: a. The capacity of the blood ceter i limited. b. Lead time for the blood upply are zero. I other word, the hopital order for the blood product are fulfilled i o time. c. The age of the blood uit received from the blood ceter i kow ad the varie over time. d. The lifetime of the platelet i limited to five day icludig two day of tetig. e. The lifetime of the red blood cell i limited to thirty five day icludig two day of tetig. f. Geeral blood iuig policy for the hopital i FIFO where the oldet uit o ivetory are iued firt whe the blood uit are requeted by phyicia for the patiet eed. g. If the demad i ot atified due to the uavailability of the blood uit, a hortage cot i icurred. h. If the blood uit expire, a watage cot i icurred aociated with dicardig the blood uit. 2. The followig otatio are ued for developmet of tochatic iteger programmig model i the maagemet of the blood upply chai for a blood bak ad a hopital coitig of the additio of aumptio blood group, the idice ued i the model, the parameter ued i the model a well a deciio variable ued i model developed by author i the blood maagemet. a. The additio of blood group aumptio i otatio follow: =, where deote the umber of the blood group tartig with = 1 to = 4 ad i give by Table 1. Blood Group for Model. A B O AB b. Idice ued i the developmet of tochatic iteger programmig model i the maagemet of blood upply chai for a blood bak ad a hopital are a follow: Table 2. Idice for Model. Demad ceario, =1,2,, Age of blood, =1,2,,! " Time period," =1,2,,$ c. Parameter ued i thi model are a follow: Table 3. Parameter for Model. Number of ceario! Lifetime of blood product $ Legth of plaig % Uit hortage cot of blood at the hopital & Uit purchaig cot of blood at the hopital h Uit holdig cot of blood at the hopital ( Big M (Big Number Probability of ceario,, -. = 1 / Uit watage cot of blood at the hopital Proportio of day old blood group i blood hipmet from blood bak i time period ", , =1 Blood demad group 9 at the hopital i time " (for ceario 234 Capacity of the blood bak group :;< ditributio to the 234 hopital i time period " = 1234 > 234? 1234 A 234 B 1234 C 1234 d. Deciio variable are alo ued i thi model are a follow: Table 4. Deciio Variable for Model. Auxiliary variable (afety tock aociated with age cla group i time " for ceario Number of blood hortage group at the ed of time " for ceario at the hopital Number of blood watage group at the ed of time " for ceario at the hopital Ivetory level of day old blood group at the ed of time " for ceario at the hopital Number of blood blood group ordered by the hopital from the blood bak at the begiig of time " Number of day old blood blood group received by the hopital at the begiig time " 1 if day old blood blood group ued to atify the demad i time period " for ceario, 0 otherwie The followig decribe the purpoe fuctio ad all the fuctioal cotrait that exit with additio of blood group cotrait. The purpoe fuctio i thi cae i a follow: miz = c x p hv 4 T 4 S I T tf = 1 t= 1 = 1 = 1 i= 3 t= 1 4 S T 4 S T wutf b rtf (1 = 1 = 1 t= 1 = 1 = 1 t= 1 The formulatio of the cotrait with the additio of blood group are give a follow: a. The demad of each blood group ( to be achieved at hopital i le tha the blood capacity at the blood bak at the ame time with time ", with " = 1,2,3,,$ ad = {1,2,3,4}. The model i give by 4 xtf CAP, tf t (2 = 1 b. Hopital ever receive ay uit of blood group ( i oe or two day old from the blood bak becaue two day i the time which i required i tetig after blood i collected. The model i a follow:

4 44 Luiaa Sibuea et al.: Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study y = 0, i = 1,2,, t, (3 c. Satifig the allocatio of each blood group ( hould be achieved i each age cla. The model i give by y = x θ, i = 3,4,, t, (4 tf d. Satifig the policy rule ued by blood bak for each blood group ( i defied by FIFO. The model i a follow: z z( 1, i = 3,4,,, t, (5 i tf e. Satifig the demad for each blood group ( mut be reached at the hopital. If the hortage occur afety tock i ue to atifig the demad. While there i till ome reidual tock ad at leat oly oe cla ha a ivetory amout o-zero. The model i a follow: I ( ( ( 1( 1 tf = i t f i= 3 d v y z m tf r,, t, (6 f. There i afety tock of each blood group ( that hould be achieved a a ubtitute if there i a hortage of blood from the other tock ivetorie ad doe ot exceed exitig ivetorie. The model i give by where ( ( 1 ( ( 1( 1 i tf i t f z z v y m (7 i = 3,4,, I,, t, g. Blood uit for each blood group ( aged two day i ot ued to atify the demad for blood i the Hopital. I other word, oly blood that ha a age of more tha two day received by the hopital. The model i give by z2tf = 0,, t, (8 h. The umber of hortage for each blood group ( i ot kow. The model i give by d I ( v( i 1( t 1 f y rtf,, t tf, i= 3 (9 i. Lat update of upply period for each blood group ( i each age cla ca be kow i umber. The model i give by ( 1 ( ( 1( 1 = i t f v z v y ( ( 1 i tf z z m (10 where i = 3,4,, I,, t, j. Guaratee that the old blood for each two-day-old blood group ( i ever available o ivetory. The model i a follow: v tf 0,, t, = (11 ( 2 k. There i o ivetory for each blood group ( at the begiig of time period of aalyi. The model i give by v i( 0 f = 0,, i, (12 l. The watage rate for each blood group ( i the hopital at the ed of the period ca be aalyzed. The model i a follow: utf = v( I tf,, t, (13 m. The other cotrait aociated with thi model are a follow: tf xtf tf Z, t, (14 r, u Ζ,, t, (15 y Ζ, i, t, (16 m, v Ζ,, i, t, (17 z Ζ,, i, t, (18 Cotrait (14-(17 are o-egative dicrete variable ued i the model ad cotrait (18 i a biary variable. Becaue the iteractio betwee biary variable ad dicrete variable i cotrait (6, (7 ad (10 are oliear, the the techique of liearizatio of Gupiar ad Ceteo [6] i ued to liearize thi three cotrait. Below i a tecique of liearizatio ued with the developmet tochatic iteger programmig model i the maagemet of blood upply chai: z v( i 1( t 1 f = γ, i = 3,4,, I,, t, γ z M, i = 3,4,, I,, t, (19 γ v( i 1( t 1 f, i = 3,4,, I,, t, (20 ( 1 v( 1( 1 γ M z i t f (21 i = 3,4,, I,, t, z y = α, i = 3,4,, I,, t,

5 World Joural of Operatioal Reearch 2017; 1(2: α z M, i = 3,4,, I,, t, (22 α y, i = 3,4,, I,, t, (23 ( z 1 y, M α (24 i = 3,4,, I,, t, z m = λ, i = 3,4,, I,, t, λ z M, i = 3,4,, I,, t, (25 λ m, i = 3,4,, I,, t, (26 ( λ M z 1 m, (27 i tf i = 3,4,, I,, t, z( 1 m = δ, i = 3,4,, I,, t, µ ( i 1 tf v( i 1( t 1 f, i = 3,4,, I,, t, (35 ( µ ( i 1 tf M z( i 1 tf v( i 1( t 1 f 1, (36 i = 3,4,, I,, t,,,, ( i 1 tf Z γ α ψ µ (37 λ, δ Z (38 After the firt liearizatio techique i applied, variable i cotrait (6, (7 ad (10 are replacedby the followig correpodig liearizatio variable: ( γ α I tf tf i= 3 d = m r,, t, (39 ( 1 i tf γ α µ ψ i = 3,4,, I,, t, m (40 δ z( i 1 tf M, i = 3,4,, I,, t, (28 i t f v = v( 1( 1 y γ α λ δ (41 ψ δ m, i = 3,4,, I,, t, (29 ( z 1 m, M δ (30 ( i 1 tf i = 3,4,, I,, t, z( i 1 tf y = ψ, i = 3,4,, I,, t, z( i 1 tf M, i = 3,4,, I,, t, (31 ψ y, i = 3,4,, I,, t, (32 ψ ( i tf M z( 1 1 y, (33 i = 3,4,, I,, t, i tf i t f i tf z( 1 v( 1( 1 = µ ( 1, i = 3,4,, I,, t, µ ( i 1 tf z( i 1 tf M, i = 3,4,, I,, t, (34 i = 3,4,, I,, t, 3. Computatioal Reult The reult of the computatio how that thi optimizatio model ca be ued i PMI epecially i PMI UTD City of Pekabaru. For the implemetatio, the author provide the blood platelet calculatio for the data i February 2017 by aumig four demad ceario. The four demad ceario are a follow: i Jauary, February, March ad April. The probability for each ceario are et to 0.2, 0.1, 0.3 ad 0.4 with the ceario take by the caue factor of the demad maily by hematology, ocology, traumatology ad geeral urgery i year I curret practice, the agregated of the demad modeled i almot Poio. Thi aumptio are alo jutified by the reult i Haijema et. al [7] ad Gupiar ad Ceteo [6] reearch i wich thi demad ditictio are take ito the accout. Here i the imulatio computatio example, provided by Table 5 preetig the platelet blood productio for 14 day i February 2017: Table 5. Platelet Blood Productio February No. Date Beide that, Table 6 give the requet from a hopital which became oe place of ditributio of platelet for 14 day i

6 46 Luiaa Sibuea et al.: Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study February. The demad o the firt ad the ecod day i zero. Thi i accordig to cotrai (3 that a hopital ever receive ay uit of blood group i oe or two day old. Table 6. Platelet Blood Demad February Thi calculatio alo give the damage data of watage of the blood i February 2017 at PMI UTD City of Pekabaru. Table 7 preet the watage of the blood. Table 7. Platelet Blood Watage February No. Date No. Date I additio, calculatio for the data of the ew blood tock i February 2017 i geerated by a ubtractio from Table 5 ad Table 7. For example: date 1 i Table 8 blood type A i 19 uit, thi value i got from the date 1 i Table 5 blood Table 8. New Blood Platelet Stock February type A ad date 1 i Table 3 Blood type A, for example: 20 1 = 19 uit. If thi coutig i cotiued it would be obtaied Table 8 preetig the ew blood tock Furthermore, the calculatio for the maagemet of the blood upply chai for a blood bak ad a hopital alo give the old blood for 14 day. Table 9 preet the old blood platelet tock for 14 day i February Table 9. Old Blood Platelet Stock February Table 9 i geerated by addig each cell i Table 8 ad Table 10 at the ed of each time period. For example: date 3 i Table 9 blood type A i 19 uit, thi value i got from Table 8 i date 1 blood type A accordig to cotrait (4. Thi i due to the age of the blood type A i date 1 i Table 8 ha bee 3 day old i Table 9 ad thi blood ca be ued for trafuio. Now, cotiued to date 4 i Table 9 with 35 uit, thi value i got from date 2 i Table 8 ad date 3 i Table 10 for blood type A, for example: = 3. Thi i due to the age of blood type A i date 2 Table 8 ha 3 day

7 World Joural of Operatioal Reearch 2017; 1(2: old i Table 9 ad thi blood ca be ued to trafuio. If thi coutig i cotiued it would be obtaied that the Table 9 ad Table 10. I the Table 9, how that the value for the firt ad the Table 10. Blood Platelet Ivetory Level February ecod day there i o old blood tock for each blood group. Thi i accordig to cotrait (11. Table 10 preet the blood ivetory level at PMI UTD City of Pekabaru The, Table 10 i geerated by reductio each cell i Table 9 ad Table 6 at the ed of each time period. I thi calculatio obtaied i the firt ad the ecod day there i o ivetory at the begiig of the correpodig aalyi period accordig to cotrait (12. For example: date 3 i Table 10 i 11 uit, thi value i got from date 3 i Table 9 ad date 3 i Table 6 thi i for blood type A, for example: Table 11. Blood Shortage Stock February = 11. The, cotiue thi coutig util it would be obtaied that the Table 10 for each blood group. I the calculatio of the model there i a hortage of the blood at PMI UTD City of Pekabaru. That i 18 uit for blood type O o the third day. It i caue the demad of the blood more tha the ivetory level. Thi alo give i the Table 11 preet the blood hortage Table 12 preet the four demad ceario aalyi objective fuctio of the optimal value uder the differet demad ceario accordig to equatio (1 i Idoeia moey (IDR. Figure 1 how the graph of the cot aalyi of the objective fuctio the optimal value uder the differet demad ceario. Table 12. Cot Aalyi Uig the Demad Sceario No Sceario The miimum cot of the blood (IDR A B O AB Figure 1. Cot Aalyi Uig the Demad Sceario of Blood. The miimum cot for each blood group occur o the eveth day for blood type A, third day for type B, twelfth day for type O ad the fifth day for the type AB. Baed from the calculatio, how that the average of the blood demad at PMI UTD City of Pekabaru i 3 day old atifig of the requet of the blood trafuio. Thi miimum cot for each blood group occur for uig four demad ceario. 4. Cocluio Baed o the backgroud of the iue decribed before, the author coclud that the reult from blood Supply Chai Maagemet (MRS with the ue of the tochatic iteger programmig ca reduce blood watage ad blood hortage tock at a blood bak ad a hopital. MRS could be the rigth model to icreae the blood ditributio proce a MRS ca make the blood bak (PMI eay to maage the blood upply

8 48 Luiaa Sibuea et al.: Stochatic Iteger Programmig Model i the Maagemet of the Blood Supply Chai: A Cae Study chai well, o that the blood ditributio proce become more efficiet ad effective. PMI UTD City of Pekabaru i the place choe by the author to coduct the reearch, with data obtaied from PMI UTD City of Pekabaru for the Year 2017 with 4 ceario ad LINGO 16 program to obtai the calculatio reult. I the calculatio implemet thi model ca further reduce the watage or hortage of the blood i the hopital ad i the blood bak. Referece [1] J. Belie ad H. Force, Supply chai maagemet of blood product: A literature review, Europea Joural of Operatioal Reearch, 217 (2012, [2] E. A. Beder, A Itroductio to Mathematical Modelig, Jho Wiley & So, New York, [3] J. R. Birge ad F. Louveaux, Itroductio to Stochatic Programmig Secod Editio, Beijig da Spiger-Verlag, New York, [4] S. Chopra ad P. Meidl, Supply Chai Maagemet Third Editio, Pearo Educatio, Upper Saddle River, New Jerey, [5] P. Ghadforouh ad T. K. Se, A DSS to maage platelet productio upply chai for regioal blood ceter, Deciio Support Sytem, 50 (2010, [6] S. Gupiar ad G. Ceteo, Stochatic iteger programmig model for reducig watage ad hortage of blood product at hopital, Computer ad Operatio Reearch, (2014, imulatio, Iteratioal Joural of Productio Ecoomic, 121 (2009, [8] V. Hemmelmayr, K. F. Doerer, R. F. Hartl ad W. P. Savelbergh, Vedor maaged ivetory for eviromet with tochatic product uage, Europea Joural of Operatioal Reearch, 202 (2010, [9] F. S. Hillier ad G. J. Lieberma, Itroductio to Operatio Reearch Nith Editio, Mc Graw-Hill Compaie, New York, [10] P. Kall ad S. W. Wallace, Stochatic Programmig Firt Editio, Joh Wiley & So, New York, [11] K. Kataliaki ad S. C. Brailford, Uig imulatio to improve the blood upply chai, Joural of the Operatioal Reearch Society, 58 (2007, [12] A. Nagurey, A. H. Maoumi ad M. Yu, Supply chai etwork operatio maagemet of a blood bakig ytem with cot ad rik miimizatio, Computer Maagemegt Sciece, 9 (2012, [13] W. P. Pierkalla, Supply chai maagemet of blood bak, Operatio Reearch ad Health Care: A Hadbook of Method ad Applicatio, Kluwer Academic Publiher, (2004, [14] G. P. Prataco, Blood ivetory maagemet: A overview of theory ad practice, Maagemet Sciece, 30 (1984, [15] Health Media (i Idoeia: Media Keehata, acceed i 21 Oktober 2015, at 13:00 PM. [16] Idoeia Cro Red, acceed i 22 Oktober 2015, at 20:00 PM. [7] R. Haijema, N. M. Dijk, J. Wal ad C. S. Sibiga Blood platelet productio with break: Optimizatio by SDP ad