THE STOCHASTIC SCOUTING PROBLEM. Ana Isabel Barros

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1 THE STOCHASTIC SCOUTING PROBLEM Ana Iabel Barro TNO, P.O. Box 96864, 2509 JG The Hague, The Netherland and Faculty of Military Science, Netherland Defence Academy, P.O. Box 10000, 1780 CA Den Helder, The Netherland Lanah Ever TNO, P.O. Box 96864, 2509 JG The Hague, The Netherland and Faculty of Military Science, Netherland Defence Academy, P.O. Box 10000, 1780 CA Den Helder and The Netherland Econometric Intitute, Faculty of Economic, Eramu Univerity Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherland ABSTRACT Efficiently couting talented port player i of key importance in the commercially oriented port world. Beide being able to recognize talent, it i important to plan the couting activitie given the available time, in order to maximize the potential for couting talented player. Thi problem relate to the orienteering problem. In reality, not only are the time required to travel between chool and the time required to cout at thee chool uncertain, but alo the potential of the port player i yet to be determined by the cout. The approache found in the literature do not model all thee uncertaintie imultaneouly. Therefore, we will introduce the tochatic verion of the couting planning problem by extending an exiting approach to fit the practical feature of thi problem. We illutrate the benefit of thi new approach uing a cae tudy. KEYWORDS. Sport couting problem, Orienteering problem, uncertainty, tochatic programming Main area: OA - Other application in OR, MP- Mathematical Programming

2 1. Introduction Due to the increaing commercialization of port, huge amount of money are involved in the tranfer of player. Conequently, port couting or headhunting i an important iue in human reource port management. A uch, reearch ha been conducted in order to develop ytem that can effectively upport couting deciion. In a team port context, wherein the aim i to find new team member to fill open poition and/or to enhance the quality of the team, Boon and Sierkma (2003) and Sierkma (2006) have developed a deciion upport ytem to help football coache and manager to ae the potential or actual contribution of particular player to their team. Thi approach upport not only couting and purchaing new player but alo team election deciion. Cooper et al. (2008) howed that a Data Envelopment Analyi can be ued to develop performance indice of baketball player. Their approach to baketball enable the aement of player according to a previouly pecified profile. On the other hand, Papić et al. (2009) preent a fuzzy expert ytem for couting and evaluating young port talent. Baed on the knowledge of everal human port expert, variou motoric kill tet, morphologic characteritic meaurement and functional tet, thi ytem help to identify a talented youngter and to elect the mot appropriated port for him/her. The above reearch focue mainly on how to cout talented port player individually or in a team context. Another facet of the couting problem correpond to the planning of the couting activitie in the mot effective way. Butt and Cavalier (1994) were the firt to explore thi problem and focued on the couting planning of high chool football player. They conider the high chool couting approach wherein high chool campue in a given area will be viited. Uually, the time available for couting i inufficient to viit all chool in the area. Therefore the goal i to viit a ubet of the chool, which maximize the potential for couting future player. The author model thi problem a an Orienteering Problem (OP). The OP i a generalization of the well-known Traveling Saleman Problem (TSP). It i defined on a graph where a profit i aociated to each node and a cot i aociated to each arc. Contrary to the TSP, the OP doe not require all node to be viited. The aim of the OP i to find a maximum profit tour, tarting and ending at a depot location, where the total cot of the arc elected in the tour doe not exceed ome predefined cot limit. The couting problem correpond to an OP wherein the node repreent the chool and the aociated profit value correpond to the couting potential of the chool in term of talented player. The cot of the arc repreent the travel time between the chool, combined with the time required for couting at each chool. The cout need to leave and return to the college campu, repreented by the depot location, within a deired time limit, which i modeled a the total cot limit in the OP. In the model of Butt and Cavalier (1994) all parameter are aumed to be determinitic. In reality however, the travel time, the time required to cout, a well a the couting potential of the chool are uncertain. Therefore, the olution to the determinitic OP might turn out to be infeaible or uboptimal in reality. Although reearch ha been conducted on how to incorporate uncertainty in the OP already in the modeling tage, none of the approache fit the pecial cae of our Stochatic Scouting Problem (SSP). The exiting reearch incorporate uncertainty either only in the travel and viiting time of the OP, or only in the profit value. To better model the SSP, we extend one of the model of Ever at al. (2012b) by including profit uncertainty. We adjut the model by auming that the viiting time are influenced by the actual profit realization. That i, when oberving the port player at a chool, the cout may decide to leave earlier or tay longer than planned, depending on the potential of the player. Thi approach fit the SSP bet with regard to the way uncertainty i handled, a we will dicu in the next ection. Thi paper i tructured a follow. In the next ection we will provide ome background on model that incorporate uncertainty in the OP and we will dicu their uitability to the SSP, which we will formally introduce in Section 3. In Section 4 we will decribe the olution approach we applied to the SSP and we will illutrate it application in Section 5. We will conclude in Section

3 2. Orienteering problem under uncertainty The determinitic OP ha been well tudied in the literature. Although in mot practical application ome or all of the parameter are uncertain, the amount of reearch on uncertain variant of the OP i limited. In thi ection we will dicu how the SSP relate to the exiting variant of the OP. The two main approache to incorporate uncertainty in an optimization problem are tochatic programming and robut optimization. Mot of the uncertain variant of the OP ue a tochatic programming approach, where the uncertain parameter are aumed to follow a predefined probability ditribution. Robut optimization, on the other hand, ue more general aumption on the behavior of the uncertain parameter, like interval uncertainty. We will ue the term viiting time to define the time pent at a node, which in application of the OP may repreent ervice time (like the unloading time of a truck), or like in cae of the SSP, the time required to cout port player at a chool. In mot uncertain variant of the OP, uncertainty i conidered only in the travel and viiting time and not in the profit value (Teng et al. (2004), Tang and Miller-Hook (2005), Campbell et al. (2011), Ever et al. (2012a) and Ever et al. (2012b)). In many application of the OP it i indeed reaonable to aume that profit value are determinitic. Ilhan et al. (2007) on the other hand, provide example of application of the OP where it eem more appropriate to aume that profit value are uncertain while travel and viiting time are determinitic. For the SSP however, travel and viiting time, a well a the profit value are uncertain, a we will dicu in the next ection. Regarding the way to incorporate uncertainty in an OP, we ditinguih between three approache ued in the literature. Both Teng et al. (2004) and Campbell et al. (2011) penalize the objective function when travel and viiting time realization are uch that the tour cannot be finihed within the time limit. A a penalty, Teng et al. (2004) ue the amount of time in exce of the time limit to repreent the cot aociated to overtime paid to driver, while Campbell et al. (2011) penalize the node in the planned tour that cannot be viited, repreenting a lo in cutomer goodwill. For the SSP both penaltie do not eem uitable, ince the cout might decide to return to college campu before completing the planned tour to meet the time limit. Lo in goodwill indeed play a role in a cutomer - ervice provider relation, but thi doe not reflect the aim of the cout who focue only on maximizing the couted player potential. The econd approach to incorporate uncertainty in an OP, i by balancing the probability of atifying the time limit contraint under travel and viiting time uncertainty, againt the profit that will be obtained in all feaible cae. Thi can be done by chance contraint (Tang and Miller-Hook (2005)) or by a robut optimization approach (Ever et al. (2102a)). Thee approache are appealing when feaibility of the initial plan play an important role, like in military miion or when trying to meet ervice level agreement with cutomer. A drawback of thee approache however, i that they do not take into account the effect of infeaibility on the realized objective function value. For example, when the lat node of the planned tour ha a high profit, thi high profit will not be obtained when realization of travel and viiting time from previou node turned out to be higher than accounted for. The third approach overcome the iue jut mentioned. The Two Stage Orienteering problem (TSOP) by Ever et al. (2012b), explicitly model the effect of not reaching one or more of the final node in the planned tour due to travel and viiting time uncertainty. The TSOP i a two tage tochatic model, where the firt tage deciion i the contruction of a tour and the ocalled econd tage cot i the expected unobtainable profit, baed on predefined travel and viiting time ditribution. In determining thi econd tage cot, the TSOP take into account that one hould return to the depot within the time limit. The objective of the TSOP i to maximize the total profit of the firt tage tour minu the expected econd tage cot. A imilar objective i uitable for the SSP, ince feaibility of the initial plan doe not play an important role and maximizing the expected value of the profit (potential of the player) i the cout aim. To model the SSP, we will extend the TSOP firt by including profit uncertainty, and econdly by introducing a relation between the actual profit realization and the viiting time realization. Thi

4 relation model the cout deciion on when to leave the chool, depending on the level of the port player he/he oberve. 3. The tochatic couting problem In thi ection we will give a formal problem decription of the Stochatic Scouting Problem (SSP). In Section 3.1 we will firt provide the formulation of the determinitic OP ued by Ever et al. (2012b) while in Section 3.2 we will preent the SSP model. 3.1 The Orienteering Problem Conider a graph G(N +,A) where N + = N {0} repreent the et of chool N combined with the college campu {0} from where the cout will tart hi/her tour. To each chool we aociate a profit value p i, repreenting the expected potential to cout talented port player at chool i. To each arc (i,j) A we aociate a parameter t ij repreenting the travel time from chool i to chool j and to each node we aociate a viiting time parameter v i repreenting the time required to cout at chool i. The total time limit, wherein the cout hould finih the couting tour, i denoted by T. We introduce the deciion variable x ij, which are aigned the value 1 in cae arc (i,j) i elected in the tour and 0 otherwie. We ue an auxiliary variable u i to denote the poition of node i in the tour. Baed on thee definition, a Mixed Integer Programming (MIP) formulation of the couting problem in a determinitic etting i the following: (OP) max p, (1) Subject to i x ij j N + \{ i} x 0 i = xi0 = 1, (2) ij ji + + \{ j} \{ j} x = x 1 j N, (3) ( tij + v j ) xij T, (4) ( i, j) A ui u j + 1 ( 1 xij )N i, j N, (5) 1 u i N i N, (6) x ij { 0,1} ( i, j) A. (7) Contraint (2) guarantee that the tour tart and end at the depot. Contraint (3) are the flow conervation contraint and enure that a node i viited at mot once. Contraint (4) i the capacity contraint. Finally, Contraint (5) prevent the contruction of ubtour. 3.2 The Stochatic Scouting Problem The couting problem can be modeled a an OP only when parameter t ij, v i and p i are determinitic. However, in reality all thee parameter are uncertain and therefore SSP will incorporate uncertainty in the modeling tage. Before the actual viit to the chool, the cout ha only etimate about the potential of the port talent (i.e. the profit value) at each chool. If the cout would have been certain about their potential beforehand, the couting proce would not have been neceary in the firt place. Regarding the travel time of the SSP, unpredictable traffic caue uncertainty in the travel time between chool. Similarly, the viiting time or couting time at a chool i alo uncertain. Furthermore, for the SSP the uncertainty in the profit value will influence the viiting time, i.e. depending on the cout obervation the cout can decide how long to tay at the chool. For example, when the cout encounter a player with a very high

5 potential, the cout might want to tay longer and talk with the player about the future poibilitie. On the other hand, the cout i likely to leave earlier than planned in cae he/he oberve player of low potential. Thi extra ource of uncertainty cannot be fully captured in the TSOP model of Ever et al. (2012b) and therefore, a relation between the realization of the couting potential value at a pecific chool and the couting time at the chool will now be taken into conideration. We model thi relation and the inherent uncertainty of the couting time by v i = f(p i,ε i ), where ε i repreent a random deviation in the couting time, independent from the realization of the couting potential. Function f( ) can be derived uing hitorical data on previou couting tour. We aume p i, t ij and ε i to follow a certain predefined probability ditribution. Given the definition provided in Section 3.1, we define the SSP-objective a follow: t, p, ε i ij ε ), (8) j N + \{ i} (SSP-Objective) max E p x g( x, t, f ( p, ) where g(x,t,v) i a function that expree the um of the value of the couting potential of the chool in the tour decribed by x that cannot be viited a a reult of a given vector of travel time realization t and a given vector of couting time realization v. The function E t.p,ε ( ) correpond to the expected couting potential to be obtained, baed on the probability ditribution of the travel time t, the profit value p and aociated influence on the couting time and the inherent viiting time uncertainty ε. To formulate the complete SSP model, SSP-Objective (8) hould be combined with Contraint (2) and (3) and Contraint (5) to (7). Note that the time limit Contraint (4) i not included in thi model. When travel and viiting time are uncertain, it will depend on the actual realization of the travel and viiting time whether or not Contraint (4) will be atified. Therefore, the SSP capture the expected effect of the actual travel and viiting time realization on the total profit value to be obtained by the econd term within the expected value function of (8). Note that thi formulation will produce planned tour that are long, ince a tour will not be optimal a long a node exit that have a trictly poitive probability of being reached, and ince adding one of thoe node to the tour increae the objective value given by (8). Concluding, the firt tage deciion boil down to contructing a tour before the actual travel and viiting time realization are known. However, thi firt tage deciion i baed on the expected effect that the realization in the econd tage will have on the profit value to be obtained. 4. Sample average approximation Sample Average Approximation (SAA) i a well-known olution approach in tochatic programming (Norkin et al. (1998), Mak et al. (1999), Kleywegt et al. (2001)). It ue random ample of data realization generated by Monte Carlo imulation to approximate upper and lower bound on the objective value of the aociated tochatic optimization problem. We will now dicu how SAA can be ued to provide an approximation of the olution to the SSP. To apply SAA to the SSP we will contruct a et of cenario S. Thee cenario will be ued to etimate the SSP-Objective (8). For each cenario S we draw one realization t ij for every arc (i,j) A, a well a one realization p i and ε i for every node, baed on the aociated probability ditribution. The total profit value that will be obtained for one pecific cenario, for a given tour decribed by x, can be calculated uing a linearization propoed by Ever et al. (2012b). In that linearization a deciion variable y tv i wa introduced, aigned the value 1 in cae node i belong to the planned tour, decribed by x, and it cannot be reached due to the cenario pecific travel and viiting time realization t and v. Uing thi linearization, the SAA objective function of the SSP now become:

6 (SAA-SSP-Objective) max pi xij 1 S S j N + \{ i} p i y t f ( p, ε ) i The contraint of the SAA-SSP model are Contraint (2) and (3) and (5) to (7), which are related to the firt-tage problem: contructing a tour before the realization of the uncertain parameter are known. Additionally, for each cenario the econd tage problem can be olved by the linearization propoed by Ever et al. (2012b) yielding the cenario pecific olution y i tv. In thi econd tage problem, the cenario pecific travel and viiting time realization are ued to determine for how long the planned tour can be continued, and conequently which of the node in the planned tour cannot be viited and thu will get aigned the value y i tv =1. The reulting SAA-SSP model i a large MIP, which aim to find the maximum total etimated expected profit, which i the average of the firt-tage profit reduced by the econdtage cot over all predefined cenario. Thu, the reulting tour i baed on the expected profit, reulting from the realization of travel and viiting time, a well a the realization from the profit value (and their effect on the viiting time realization), etimated by the ample generated in the SAA procedure.. (9) 5. Cae tudy In thi ection we will report the reult of a cae tudy to illutrate the purpoe of the SSP model and to compare the reult of the SSP to two alternative model: the OP where travel and couting time a well a the couting potential are modeled a determinitic value and the a TSOP approach where uncertainty only in travel time and viiting time i taken into account. 5.1 Data For thi cae tudy we ue the firt eight node from the benchmark data et Tiligiride problem 2 (Tiligiride 1984). In thi data et the node are poitioned in a rectangular area of 15 by 15 unit. The profit value aociated to the node are 10, 15, 15, 20, 20, 20, 20 and 30. Thee profit value repreent the expected couting potential of the chool in our cae tudy. Since the original data et doe not contain data uncertainty, additional etting regarding the uncertainty in the travel time will be ued a decribed in Ever et al. (2012b). To repreent the expected value of the travel time, we ue the Euclidean ditance between the node. The travel time uncertainty will follow a uniform ditribution. Such a ditribution help illutrating the effect of very uncertain ituation with repect to the kind of realization that one can expect. Note that other ditribution could alo be teted eaily by the SAA olution approach. The bound of the uniform ditribution were determined by taking a 15 percent deviation from the expected travel time. For couting potential uncertainty, we ue a uniform ditribution a well. The bound of the aociated interval however, are contructed differently. The uncertainty in the couting potential cannot only be repreented via an equal deviation from the expected couting potential for every chool. For two chool with the ame expected couting potential, hitorical data could indicate that the deviation from thi expected level i larger for one chool than the other. In contructing the ize of the uncertainty interval, we firt conider a 20 percent deviation from the expected couting potential for each chool. Additionally, to model the variation in the ize of the uncertainty interval between chool, we add a random element to each chool individually. Thi random element i drawn from a uniform ditribution defined between 0 and 80 percent of the expected couting potential. Note that in thi way the maximum deviation from the expected couting potential could turn out to be 100 percent. Finally, the function decribing the relation between the realized couting potential and the realized couting time need to be defined. We like to tre that in application of thi model,

7 thi relation hould be derived baed on empirical data. For illutrational purpoe we ue the following imple linear relation between the couting potential and the couting time, combined with an additional random term: v = f ( p ε ) = p + ε. i i, i i i where the random term ε i follow a uniform ditribution within [ 0.15 f ( pi,0);0.15 f ( pi,0) ]. Finally, we et the time limit for the tour to be performed by the cout at 40 time unit. 5.2 Experiment To olve the SSP for the data intance jut decribed, we implemented the SAA olution procedure in Java and performed the optimization with CPLEX The MIP i contructed baed on S =8 cenario. Note that the olution found by the SAA procedure give an approximation of the SSP objective. That i, the SAA-SSP Objective (9) i an etimation of the real SSP Objective (8). Therefore, in order to better etimate the SSP objective, the reulting SAA tour wa evaluated over 10,000 cenario, independent from the 8 cenario previouly defined to find the olution. Since the SAA olution i feaible for the SSP, but not necearily optimal, the etimated objective value derived in thi way provide an etimate of a lower bound on the actual SSP olution. Thi whole procedure of olving an MIP baed on 8 cenario and evaluating the aociated olution baed on 10,000 cenario wa repeated for 10 run and we will report the bet etimated lower bound of the SSP olution found in thee 10 run. We will compare the SAA olution of the SSP both to the olution of the determinitic OP, and to the olution of a TSOP approach to thi problem. For the OP, no uncertainty i taken into account and all parameter are defined by the expected value of the ditribution decribed in the previou ubection. Since in the tandard OP and the TSOP a relation between couting potential and couting time i not taken into account, we conider the ame expected couting time for all chool. Thi expected couting time correpond to the average couting time over all chool baed on our data aumption ued for the SSP. For the TSOP we take travel and viiting time uncertainty into account and diregard profit uncertainty. Similar to the SSP, in the TSOP we conider uniformly ditributed travel time, with bound et at 15 percent deviation from the expected value. For the couting time uncertainty in the TSOP, we take a uniform ditribution a well, bounded by a 15 percent deviation from the overall expected couting time. To olve the TSOP, the ame SAA olution approach i ued a decribed for the SSP. When taking the expected value of all parameter in an optimization problem, the reulting olution will ue the available capacity to the highet extend poible. In the cae of the determinitic OP, thi mean that the expected duration of the reulting tour i likely to almot fully ue the total available time. When evaluating the reulting determinitic olution under data uncertainty, the tour might therefore often turn out to be infeaible. In thoe cae, thi implie that the cout will return to the college campu before he/he ha viited all chool in the planned tour. Conequently ome of the planned couting potential will not be obtained. The SSP and the TSOP tour on the other hand, explicitly take into account the effect of not reaching one or more of the final node in the planned tour due to travel and viiting time uncertainty. Similar to the SSP tour evaluation, we will ue 10,000 cenario to evaluate the OP tour and the TSOP tour, and we will report the average unobtainable couting potential value due to the realization of the uncertain parameter. We will refer to the average unobtainable couting potential value over all cenario a the profit hortage. On the other hand, we will alo conider the ituation where low realization reult in left-over time at the end of the determinitic tour. In that cae we conider the poibility of adding chool after the final chool of the initial tour according to the agile trategy decribed by Ever et al. (2012a). We will refer to the average of thi additionally obtained couting potential value over all cenario a profit urplu. Note that uch a urplu will not be attained for the olution of the SSP and the TSOP, due to it problem formulation. Recall that adding chool to an exiting olution doe not decreae the objective

8 value of the TSOP and the SSP and therefore the reulting tour will be relatively long. To compare the reult fairly, we ue the ame 10,000 cenario to evaluate each olution. 5.3 Reult Table 1 contain the reult for the bet SSP tour and the bet TSOP tour found in the aociated 10 SAA run, a well a the reult for the tour found by the determinitic OP. The three model each reult in a different tour. The average total profit value denote the average couting potential of all chool in the tour over the 10,000 cenario. However, due to the uncertainty, not all chool in thee tour can be viited in every cenario. The reulting average couting potential of the chool that could not be viited i reported in the next column profit hortage. A mentioned in the previou ubection, for the determinitic OP tour, we include the poibility of extending the tour with additional chool if poible. The average additional couting potential obtained in that way i given in the column profit urplu. The expected profit i the reult of the average profit of all chool in the tour, reduced by the profit hortage, and increaed by the profit urplu. Tour Expected Average total Profit Profit urplu (by index) profit profit hortage OP tour 3,2,4, TSOP tour 1,2,3,5, SSP tour 1,2,4,5,6, Table 1: reult for the SSP, the TSOP and the determinitic tour, evaluated over the ame 10,000 cenario of realization of the uncertain parameter, rounded to two decimal. A hown in Table 1, the average profit value of the SSP tour and the TSOP tour i quite large compared to the average profit value of the OP tour. Thi can be explained by the SPP and TSOP problem formulation that yield tour with a relatively large number of chool. Therefore, the um of the imulated value of the couting potential of thee chool i higher than for the tour obtained for the determinitic OP. Thi effect i to ome extent corrected by the profit hortage. But till, the expected couting potential of the SSP tour far exceed the expected couting potential of the other two tour, ince it take into account all uncertaintie of the problem, the relation between thee uncertaintie, and the effect that thee uncertaintie have on to eventual couting potential to be obtained. Furthermore, the average tour duration of the OP tour i A expected, thi determinitic olution turned out to be very cloe to the available capacity of 40 unit. Conequently, in 47% of all cenario the couting potential value of one or more of the chool of the planned tour turned out to be unobtainable due to a large realized tour duration. Concluding, thi cae tudy illutrate the benefit the SSP formulation over the ue of either a determinitic or a TSOP approach by incorporating the effect of uncertainty in travel and couting time a well a in the value of the couting potential, already in the modeling tage. 6. Concluion Sport couting i an important iue in human reource port management. In thi paper we addreed the planning of the couting tour in the mot effective way. The determinitic approach to thi problem correpond to the Orienteering Problem (OP). However, in reality everal uncertaintie play a role. Not only are the time required to travel between chool and the time required to cout uncertain, but alo the potential of the port player i yet to be determined by the cout. Therefore, we introduce a more realitic extenion of the OP: the Stochatic Scouting Problem (SSP). Since previou reearch on related problem focue only on incorporating uncertainty either in the travel and viiting time or in the value that would model the couting potential of the chool, we developed a eparate approach to fit the reality of the SSP, baed on

9 one of the available model from the literature: the Two-Stage Orienteering problem (TSOP). Our model of the SSP not only conider uncertainty in all of the input parameter of the problem, but it alo take into account a relation between the potential of the port player at a chool and the remaining time that the cout will tay at the chool. Uing a cae tudy we how that thi model perform much better than a determinitic model and a TSOP approach with repect to the expected couting potential that will be achieved. Thi model i alo uitable to other application, like the tourit tour planning problem. In thi problem a tourit encounter uncertain travel time between ighteeing location, and the atifaction that the tourit will get from viiting the location, i uncertain beforehand a well. In thi example it i alo reaonable to aume that a relation exit between profit value and viiting time. When a ighteeing location i more beautiful than the tourit expected, the tourit might decide to tay at the location longer than planned. Acknowledgement We would like to expre our gratitude to prof.dr. A.P.M Wagelman and dr. H. Monuur for their valuable remark on a previou verion of thi paper. Reference Butt, S.E., Cavalier, T.M. (1994) A heuritic for the multiple tour maximum collection problem, Computer of Operation Reearch, 21(1): Campbell, A.M., Gendreau, M., Thoma B.W. (2011) The orienteering problem with tochatic travel and ervice time, Annal of Operation Reearch 186(1): Chao, I.M, Golden, B.L., Wail, E.A. (1996) The team orienteering problem, European Journal of Operational Reearch, 88(3): Cooper, W.W., Ruiz J.L., Sirvent, I. (2009) Selecting non-zero weight to evaluate effectivene of baketball player with DEA, European Journal Operational Reearch, 195(2): Ever, L., Dollevoet T., Barro A.I and Monuur, H. (2012a) Robut UAV Miion Planning. Under review at Annal of Operation Reearch. Ever, L., Glorie, K., Ster, S. van der, Barro, A.I., Monuur, H. (2012b) The Orienteering Problem Under Uncertainty, Stochatic Programming and Robut Optimization compared. Under review at Computer and Operation Reearch. Ilhan, T., Iravani, S.M.R., Dakin, M.S. (2008) The Orienteering problem with Stochatic Profit, IIE Tranaction 40: Kleywegt, A.J., Shapiro, A., Homem-De-Mello, T. (2001) The ample average approximation method for tochatic dicrete optimization, SIAM Journal on Optimization 12(2): Mak, W.K., Morton, D.P., Wood, R.K. (1999) Monte Carlo bounding technique for determining olution quality in tochatic program, Operation Reearch Letter 24(1): Norkin, V.I., Pflug, G.Ch., Ruzczynki, A. (1998) A branch and bound method for tochatic global optimization, Mathematical Programming 83(3): Papić, V., Rogulj, N., Pleština,V. (2009) Identification of port talent uing a web-oriented expert ytem with a fuzzy module, Expert Sytem with Application, 36(5): Sierkma, G. (2006) Computer upport for coaching and couting in football, in The Engineering of Sport 6, 9(4): , Springer New York. Tang, H., Miller-Hook, E. (2005) Algorithm for a tochatic elective travelling aleperon problem, The Journal of the Operational Reearch Society 56(4): Teng, S.Y., Ong, H.L, Huang, H. C. (2004) An integer L-Shaped algorithm for the time contrained traveling aleman problem with tochatic travel time and ervice time, Aia- Pacific Journal of Operational Reearch 21(2): Tiligiride, T. (1984) Heuritic method applied to orienteering, Journal of the Operational Reearch Society 35(9):