A Robust Hierarchical Approach To Multi-stage Task Allocation Under Uncertainty

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1 Proceedngs of the 44th IEEE Conference on Decson and Contro, and the Euroean Contro Conference 5 Seve, San, December -5, 5 TuIB8.5 A Robust Herarchca Aroach To ut-stage Tas Aocaton Under Uncertanty Dongxu L, Student ember, IEEE and Jose B. Cruz, Jr., Lfe Feow, IEEE Abstract A mut-stage tas aocaton robem s dffcut to sove when the search sace s arge. The ntrnsc uncertanty mbedded n mtary oeratons maes the robem more chaengng. Scaabty and robustness are recognzed as two man ssues. A new herarchca agorthm s roosed to attac ths robem. The agorthm has two eves. The uer eve rovdes mutua coordnaton among a decson-maers; whe the decson-mang at the ower eve s decentrazed. The agorthm s not ony a tasng anner but aso an onne feedbac controer. Comutatona demand of the agorthm s dvded nto two arts. The most comutatonay ntensve art can be memented off-ne, and the comexty of the onne art s reduced sgnfcanty. Smuatons show that ths s otentay a good method for sovng mut-stage resource aocaton robems nvovng a arge number of vehces and tass wth robust erformance. U I. ITRODUCTIO nmanned Aera Vehces (UAV) have shown great otenta vaue n reducng the human woroad n future mtary oeratons. A feet of UAVs can wor more effcenty and effectvey than a snge UAV, but requres a comex hgh-eve tactca annng too. Ths nd of robem nvoves sgnfcant chaenges, whch ncude team comoston, sensor nformaton, otma tas aocaton and otma ath annng etc. In ths aer, we focus on a mut-stage tas aocaton robem, where ony target stre s concerned. A tyca scenaro s to assgn dentfed targets to a team of avaabe UAVs n a artay nown envronment. The utty of every ossbe assgnment ar s detered accordng to the oston and the caabty of each UAV and each target. An otma aocaton scheme s searched by maxmzng the sum of the uttes of a assgnments. In the case when, more than one target may be assgned to a UAV, and that UAV roceeds to the assgned targets n sequence. We ca t a mutstage tas aocaton robem, whch has characterstcs of both resource aocaton and schedung. Ths robem s very dffcut to sove due to a arge search sace, such that scaabty s of concern. Another chaenge resuts from envronmenta uncertantes. In a batte fed, t s ey that new o-u targets are dentfed or some UAVs are anuscrt receved arch 7, 5. Ths research was sonsored by the Coaboratve Center for Contro Scence at the Oho State Unversty, under Grant F from the Ar Force Research Laboratory (AFRL/VA) and the Ar Force Offce of Scentfc Research (AFOSR). D. L (Te: , ema:.447@osu.edu) and J.B. Cruz (ema: bcruz@eee.org) are wth the Deartment of Eectrca and Comuter Engneerng, The Oho State Unversty, Coumbus, OH 43 USA. destroyed n the mdde of oeratons. The agorthm must be robust enough to comensate for these changes. Extensve research has been done recenty n ths fed []-[8]. In []-[4], tas aocaton has been formuated n the form of xed-integer Lnear Programg (ILP). The tg ssue s addressed [3]-[4]. In ths aroach, the robem s treated as a deterstc otmzaton robem wth nown arameters. Athough the soutons reserve goba otmaty, the ILP suffers from oor scaabty due to the P-hardness of such robems []. On the other hand, mtary stuatons are nherenty dynamc and uncertan because of the UAV s sensng mtaton and adversara strateges. Thus, reannng s needed whenever the nformaton s udated. Wthout consderng the comutaton demand, the extreme behavor of churnng henomenon s dscussed n [5]. A otenta dsaster of the sme reannng aroach s ustrated, where tass may never be accomshed. Thus, heurstcs and ad-hoc methods are consdered durng reannng n []. In [5], an aroach to mtng the rate of change by reannng n the frequency doman s ntroduced. In [6]-[7], uncertanty s taen nto account n terms of otmzaton arameters, and rs management technques n fnance are utzed. In [6], a nonnear nteger rogramg robem s formuated wth a constrant based on a rs measure by condtona vaue-atrs. In [7], a robust aroach based on the Soyster formuaton on the exectaton of the target scores s roosed. These aroaches are based on sovng combnatora otmzaton robems, where soutons are conservatve and scaabty s st a bg ssue. An aternatve aroach n deang wth uncertantes s to formuate a stochastc otma contro robem based on the method of ode Predctve Contro (PC) [8]-[9]. Wth onne otmzaton n PC wth necessary constrants, contro becomes feedbac of the state. However, due to the mted horzon, the actua an s no onger otma and feasbty can be a otenta robem [9]. In ths aer, dfferent from the aroaches mentoned above, a new method of sovng a mutstage UAV-tas aocaton robem s ntroduced. The aroach combnes otmzaton and feedbac contro. We assume suffcent communcaton caabtes among UAVs. Our goa s to desgn an agorthm that must be: ) comutatonay scaabe,.e., caabe of deang wth a reatvey arge number of UAVs and targets; ) adatve,.e., the agorthm s abe to resonse to envronmenta changes romty; 3) robust, where contros are feasbe under ossbe /5/$. 5 IEEE 3375

2 uncertantes. In the next secton, we formuate the robem and derve ts comexty. otvaton from a seca case of the robem wth a snge stage s dscussed, and a herarchca agorthm s ntroduced n secton III. Greedy decson-mang under uncertanty s nvestgated and the comutatona comexty of the agorthm s anayzed. In secton IV, exames are resented to demonstrate the erformance. In the ast secton, we concude the aer wth suggestons for the future wor. II. PROBLE FORULATIO AD AALYSIS Gven UAVs and targets, the obectve functon of a mut-tas aocaton robem s formuated as ( ) max J ( u, u,, u ) max m D t. () u,, u Here u denotes the ordered target set assgned to UAV,,, u T T n wheren T denotes a target; ( ) s the robabty for target of beng destroyed by UAV ; m s the th target s vaue that mes ts reatve mortance; t denotes the tme that t taes for target to be destroyed, whch t s estmated by the dstance to trave by the UAV to whch target s assgned; D s a arge number such that Dt for. ote that the order n u ndcates the tasng sequence for UAV, and assume that u T and Defne G G, G,..., G A a a a u u,, where T s the set of targets. Accordng to (), t s desred that more mortant targets be damaged wth ess tme, so that the msson rs at ater stages s reduced. The robem can be generazed as foows. Consder the resource set A a,..., a and the tas set B b,..., b. wth G B and Ga Ga for, where G a s an ordered set assocated wth a. Defne G G a and GA GA. Defne mang c : c (): G,. The obectve s,,..., J u u u c u. Ceary, J mas eements n G A to. The corresondng otmzaton robem s max Ju, u,..., u max c u. () u, u,..., u u, u,..., u otce that robem () s P-hard []. Generay seang, there s no effcent agorthm for arge and. We derve the comexty usng the sze of the search sace as a measure. Theorem : () If s n n! C C a, the comexty of robem (),,, where C n n!!( n )! n. () <, e e asymtotcay wth. Proof: () Suose ony n resources have assgnments. The n number of n -arttons of set B s C. The number of n cases for choosng n resources out of A s C. Then, the number of aocaton schemes s the summaton from to n, n C n n C. The tota number of ermutaton of the tass n n n B s!. It foows that! C C. n n n n C C C n n. () On one hand, On the other hand, by Cauchy-Schwarz Inequaty, C C C C C C n n n n n n n n n n n n n C C n n n n Thus, C C. n By Strng s aroxmaton,! e < ( ) ( ) ( ) e e, then asymtotcay wth. Fnay, et us comare the formuaton of () wth the ILP. In the ILP, the obectve can be J c x wth some necessary constrants. Here s the ndex for x,, the bnary decson varabe, and x stages; when b s assgned to a at stage and x f t s the ooste case. In a robem wth resource eements, tass and K ossbe stages, the search sace assocated wth the ILP formuaton has K onts. Ths number s much arger than the uer bound derved n theorem. By tang nto account the secfc structure of the tas aocaton robem, the formuaton of () creates an easer robem by eatng a number of nfeasbe soutons. III. HIERARCHICAL ROBUST ALGORITH Tas aocaton s a centrazed robem, whch requres goba nformaton and ntensve comutaton caabty. Its souton s gobay otma, but senstve to oca changes. A sma dsturbance may cause drastc changes n a goba otmum. In addton, oca faure may be fata for centrazed systems. However, n a decentrazed aroach, decson- mang s based on oca nformaton, such that a system has more romt resonse to the envronment wth ess comutatona demand. Systems become ess senstve to sma dsturbance and more robust aganst oca faures. Wth these advantages, otmaty s comromsed. In genera, centrazed contro s more sutabe for deterstc or acatons wth md changes; whe a decentrazed 3376

3 aroach s advantageous n acatons nvovng rad changes and sgnfcant uncertantes. Due to the uncertan nature of UAV acatons, we tae a artay decentrazed aroach, where a tradeoff between otmaty and robustness s taen nto account. In ths aer, a herarchca structure of decson-mang s created, n whch each UAV s vewed as an ndeendent Decson-aer (D) wth ts own sensng and decson-mang caabty. A. A Snge Stage Assgnment Probem In ths secton, we focus on a smfed assgnment robem wth a snge stage, where each resource eement a n A has no more than one assgnment from B. Ths robem can be wrtten as max J max cx (3) Subect to x,, x and x. Here, x s the decson varabe; c denotng the ndvdua utty when a s matched to b. In our aroach, each a s vewed as an ndeendent D. Ceary, wthout addtona coordnaton, a greedy D a chooses tas b by maxmzng ts ayoff. To overcome otenta confcts that more than one a taes the same b, we ntroduce a rorty sequence a,..., a for coordnaton. Gven a sequence, Ds n A mae choces sequentay among the currenty avaabe resources n B. After each of the a s has one assgnment, obectve (3) can be evauated. In ths way, a mang between sequences and can be estabshed. A two-eve agorthm can be desgned as foows. The uer eve s to fnd a roer sequence; whereas at the ower eve, each a maes an ndvdua decson n sequence. The foowng theorems rove that a roer sequence rovdes a vad coordnaton for a Ds. We frst consder and then other cases. Lemma : If, n souton X x of robem (3), there exsts x,, c max c. such that m Proof: Prove by contradcton. Suose there exsts no such x. For any,, there s an unque such that x n X. Defne functon f () for any. There s at east one ' wth c max c. Defne g() '. Ceary, ' m m f () g(),. Both f () and g() are one-to-one functons and have a unque nverse,.e., f ( ) ( g ( ') ). Startng from any, construct a sequence as: g f g g f f f g f g. In ths sequence, a crcuaton nvovng u to eements taes ace. Consder one comete crcuaton as f g f f g f Then, c c c f ( ), f g f ( ), g f ( ) f g, g. c f ( ), g f c ( ) f g f ( ), g f g f c ( ) f g, Thus, f x, x f ( ), f g f ( ), g f ( ), x are reaced f g, g by x f ( ), g f ( ), x f g f ( ), g f g f ( ),, x f g, the, ayoff s ncreased. Thus, X x cannot be otma. Ths s a contradcton and the theorem s roved. Remar : Lemma states that n an otma souton of (3), at east one D acheves ts maxmum ayoff. Theorem 3: If, a, there exsts a sequence a A, such that a souton to the foowng mut-eve rogramg robem n (4) s a souton to robem (3). max J( a ) max, subect to cx x x J, subect to n a a n a c n n x x n x n n max (... ) max c, subect to x Subect to x,, and x. (4) = Proof: Start wth resources, ay emma to fnd a that acheves ts best utty by choosng tas from the currenty unassgned tass. Eate resource a and the th tas and the robem s reduced to a smar one wth one ess dmenson. Foow the same rocedure unt a resources have one tas. The resutng sequence of a s s a vad sequence. Remar 3.: Theorem 3 states that wth a roer sequence, the sequenta greedy decson-mang deteres an otma souton of the robem n (3). Remar 3.: Prorty sequences vrtuay rovde vad coordnaton for ndeendent Ds. Ths sequence does not my the order that a s mae decsons n reaty. Smar concusons can be extended to other cases. Coroary 4: If or, X x s a souton to robem (3), then there ) If exsts c x n X, n and m such that max c. m () There exsts a sequence of the eements n A, such that a souton to (4) s a souton to (3). Proof: () For ( ), create dummy resources n A (tass n B ), and a of the reated uttes are 3377

4 zero. Then the same argument n emma and theorem 3 can be aed. Part () foows. B. Herarchca Partay Decentrazed Panner For a mut-stage robem, sequenta decson-mang and cooeraton ay mortant roes n the agorthm. Consder the foowng ndvdua obectve functon for D a, max max. (5) J u B c u B u In (5), subscrt denotes the D a ; B s the set of the unsgned tass at stage when a s mang decsons. Argument u n (5) mes that D a sees the best strategy to fuf a the unsgned tass. Accordng to (5), a secfc otmzaton robem reated to () can be wrtten as ( ) max J ( u B ) max m D t B. (6) u u Here, denotes the number of tass n B. Ths s a schedung robem. The herarchca agorthm s descrbed as foows. We start wth the ower eve. Gven sequence a, a,..., a, the frst D a s the frst to choose and suose b,..., b s the best sequence for robem (6). Then, D a roceeds to the frst tas b. eanwhe, there are Ds for cooeraton. At the moment, a s gong to b and ths nformaton s shared by a other Ds. The second D a faces a smar robem wth tass, and t goes to the frst tas n the souton sequence. D a 3 foows the same rocedure and so on unt a Ds have exacty one assgnment. There are st unsgned tass ( ). The tme to fnsh the current assgned tas s O Search for a better sequence of decson-maers A decson-maers have at east one assgnment? O ext decson-maer from the rorty sequence Indeendent decson-mang A tass are assgned? YES Evauate obectve J O Terate? YES Outut the assgnment scheme YES Uer eve Lower eve The next decson-maer wth the mum tme on the current tas SAVE the best aocaton resut Fg.. Fowchart of the Herarchca Panner cacuated for each D. The D wth the mum tme s the next to mae a decson. The same rocedure s foowed unt a tass are assgned. In the case when more than one D fnshes ts current tas at the same tme, the next D s chosen randomy. Wth the ower eve, obectve () can be evauated rovded a sequence of Ds. The uer eve s to sove for an otma sequence accordng to (). The fowchart s shown n Fg.. C. On-ne Decentrazed Feedbac Controer Fg. ustrates the rocedure of the agorthm for annng under the assumton that each UAV can successfuy fuf ts assgned tass, whch may not be true n reaty. Durng the onne mementaton, decsons at the ower eve are made based on the atest nformaton. In ths case, the ower eve acts as an onne feedbac controer. Wthout unexected events durng actua oeratons, the mementaton w concde wth the an. However, t s ey that some uncertan event, such as oss of UAVs or o-u of new targets, mght haen. The feedbac mechansm at the ower eve hes comensate for the uncertantes such that the agorthm s robust. Fnay, the otmzaton for each D s smar to the rocedure of Oen-Loo-Feedbac Contro []. A D soves a comete otma tasng sequence but ony executes the frst contro n the sequence. D. Greedy Decson-ang at the Lower Leve At the ower eve, greedy decson-mang can be aed, where ony the next tas s soved nstead of the whoe sequence. Then, the otmzaton robem n (6) becomes ( ) max J ( b B ) max m D t B. (7) b B Under uncertanty, a decson based on (7) may be the same as that based on (6). In ths secton, a bound on uncertanty s gven such that a souton to (7) s the frst tas n a sequence that soves (6). In ths aer, uncertanty s characterzed by ( ) the factor, whch ncudes the robabty of survva for UAV and the robabty of destructon for target. Both robabtes deend on target and UAV. For smcty, we consder UAV and use notatons D and to reresent the robabtes resectvey. Suose that the events of survva and destructon are ndeendent, ( ) D S then,. Equaton (6) can be rewrtten as Here, n n B ( ) max max ( m D t J u B / b ) b / (8) B b s the set of the unassgned tass wthout b ; t denotes the tme t taes for UAV to reach target. z Dm D t, and denote Z maxz. Then, Defne defne Z max z,, z, z,, z wth m m m arg maxz. Let the uer bound of the survva S robabty at any stages be max S. Ceary, 3378

5 ( ) D. Defne Z Z. Theorem 5: If ( ) ( ), a souton to (7), b q, s a souton to (8). ( ) Proof: For any s q, because D, ( ) s ( ) s s s msdtmax mdt t D s s zs max mdt t zs Z zs Z zs Z Z (9) Because ; functon s ncreasng wth resect to and zs Z, Z Z Z (9) Z Z ( ) Z Z Z Z Z ( ) q q q, Snce Z m D t () ( ) q ( ) q q q q ()< m D t max m D t t Z ( ) q ( ) q q q mqdt max mdt t. Thus b soves the robem (8). q By theorem 4, wth a decrease of robabty of survva, the decson-mang of UAV becomes more and more myoc. Fnay, soutons to (7) and (8) become the same. E. Agorthm Anayss The two-eve aroach s subotma from the otmzaton ont of vew. It s because a seca structure has been mosed on the aocaton cacuaton, and a best an s searched wthn a subsace of the orgna search sace. From the motvaton robem wth a snge stage, ths structure catures some characterstcs of otma aocatons. On the other hand, the sacrfce of otmaty rovdes a greater degree of freedom for mrovng robustness and adatabty, because the decentrazed decson-mang at the ower eve acts as a feedbac controer durng the msson. So far, we have assumed that the nformaton about the battefed s shared by a UAVs through suffcent communcaton ns. It s worth notng that the herarchca aroach s concetuay dentca to a two-eve Staceberg game robem [3] or b-eve rogramg [4], where the otmzaton at the uer eve deends on that at the ower eve. In ths aroach, a best rorty sequence deteres a startng ont for the msson, and the decentrazed feedbac contro s used at the ower eve to mrove robustness. Suose a greedy decson-mang s aed at the ower eve. The comexty of the ower eve s L. The comexty of the uer eve s! L. Athough ths s a hard robem, the herarchca method s st acabe because: ) the number of UAVs s normay sma comared to the number of tass; ) the otmzaton at the uer eve can be memented off-ne. IV. SIULATIO RESULTS AD AALYSIS Sma battes were created to ustrate the erformance of the agorthm. A scenaro wth 3 UAVs and 8 targets were seected wth the necessary arameters sted n Tabe I and II. TABLE I PARAETERS OF UAVS AD TARGETS UAV UAV UAV UAV 3 Poston (5, ) (5, ) (5, ) Veocty Target Poston (,) (,) (3,7) (7,) (8,8) (4,3) (5,9) (6,5) Vaue Consder that the robabty of survva s and the goba otma an generated accordng to obectve () s ustrated n Fg.. In Fg. 3, the resut of the herarchca agorthm s resented. In both fgures, the tasng sequence of each UAV s ndcated by arrows, where UAV and target number are ndcated. The erformance ndex of the goba otma an ustrated n Fg. s.3, and that for the subotma an n Fg. 3 s.7 wth D TABLE II PROBABILITY OF DESTRUCTIO T T T3 T4 T5 T6 T7 T8 UAV UAV UAV Target (UAV) (UAV) Goba Otma Assgnment Scheme Target 3 Target 6 Target 7 (UAV) Target 8 (UAV) Target UAV Target 5 Target (UAV) Target 4 (UAV) Fg. Goba Otma Assgnment Scheme ext, we created 5 smar robems based on the arameters used n ths scenaro, where the ostons of the targets and UAVs were randomy generated. The ostons of the targets were assumed to be unformy dstrbuted n a square, and the ostons of the UAVs were unformy dstrbuted on the nterva [,] on the x-axs. A 3379

6 of the three UAVs were assumed to have the same nta oston. The vaues of the obectve functons acheved by the herarchca agorthm are otted n Fg. 4, whch are reatve vaues normazed by the goba otmum for each robem. To eate the nfuence of the constant D n (), the orgn of the erformance ndex s set as the worst an wth the east obectve evauaton accordng to () n each random robem. The ots shown n Fg. 4 are movng averages wth a wndow sze of 5. The averages of erformance ndex of the subotma an for a tota of 5 robems by the agorthm are.85 and.88 for non-greedy and greedy Ds. Let the CPU tme t taes to cacuate the goba otma assgnment an be. The CPU tmes used by the herarchca agorthm are.47 and 5 5 for non-greedy and greedy cases resectvey Assgnment Scheme by the Herarchca Panner Target Target (UAV) Target 3 Target 7 Target 6 (UAV) (UAV) (UAV) Target 8 Target UAV Target 5 (UAV) Target Fg. 3 Assgnment Scheme by the Herarchca Panner Obectve Evauaton for Randomy Generated Probems Herarchca Agorthm wth on-greedy Ds Herarchca Agorthm wth Greedy Ds Goba Otmum Fg. 4 Obectve Evauaton for Randomy Generated Probems For the scenaro seected, the subotma resuts obtaned by the herarchca agorthm can reach about 9% of the goba otmum on average. The resut wth greedy decson-mang at the ower eve s a tte better. Ths s because the decson-mang of each UAV deends not ony on ts own choces but aso on the decson-mang of other UAVs at ater stages. However, ony the former deendence s consdered n (8). Furthermore, the herarchca agorthm hes reduce comutaton dramatcay. The comutatonay ntensve art can be memented offne, and the onne otmzaton can be memented to comensate for ossbe uncertantes. The faures n the extreme cases roosed n [5] can be revented. Ths can otentay be a good onne tactca oeratona too for UAV acatons. V. COCLUSIOS AD FUTURE WORK The herarchca aroach s motvated by a seca robem wth a snge stage. It s a hybrd of otmzaton wth decentrazed feedbac contro. Comutaton has been dvded nto two arts, where the comutatonay ntensve art can be done offne. Otmzaton at the uer eve rovdes centra coordnaton; whereas the decentrazed feedbac at the ower eve enabes romt resonses to envronmenta changes. Prorty sequences cature the maor characterstcs of tas aocaton robems. The method generates good assgnment ans wth sgnfcant savngs on comutatons. In ractce, feedbac at the ower eve can comensate for ossbe uncertantes such that robustness s mroved, esecay when onne reannng based on a centrazed aroach s comutatonay rohbtve. The foowng ssues need to be further nvestgated: ) a fast agorthm for sovng the uer eve otmzaton robem, ) coordnaton among Ds or reannng agorthm durng oeratons when there are dramatc changes, 3) ncudng ractca constrants on tg or ath annng. REFERECES [] P.R. Chander and Pachter, Herarchca Contro for Autonomous Teams, Proceedngs of the AIAA Gudance, avgaton, and Contro Conference, ontrea, Quebec, Canada, August,. [] J. Bengham,. Terson, A. Rchards, and J. How, ut-tas Aocaton and Path Pannng for Cooeratng UAVs, Second Annua Conference on Cooeratve Contro and Otmzaton, ov. [3]. Aghanbard, Y. Kuwata and J. How, Coordnaton and Contro of ute UAVs wth Tg Constrants and Loterng, Proceedng of the Amercan Contro conference, 3. [4] C. Schumacher, P. Chander,. Pachter, and L. Pachter, Constraned Otmzaton for UAV Tas Assgnment, AIAA Gudance, avgaton, and Contro Conference, Provdence, RI, August 4. [5]. Aghanbar, L.F. Bertucce, and J.P. How, Fter-Embedded UAV Tas Assgnment Agorthms for Dynamc Envronments, AIAA Gudance, avgaton, and Contro Conference, Provdence, RI, August 4. [6] P. Krohma, R. urhey, P. Pardaos, S. Uryasev, and G. Zrazhevsy Robust Decson ang: Addressng Uncertantes n Dstrbutons, In: S. Buteno et a. (Eds.) Cooeratve Contro: odes, Acatons and Agorthms, Kuwer Academc Pubshers, 65-85, 3. [7] L.F. Bertucce,. Aghanbar and J.P. How, Robust Pannng for Coued Cooeratve UAV ssons, 43 rd IEEE Conference on Decson and Contro, Atants, Paradse Isand, Bahamas, 4. [8] C.G. Cassandras and W. L, A Recedng Horzon Aroach for Sovng Some Cooeratve Contro Probems, Proceedngs of the 4 st IEEE Conference on Decson and Contro, Las Vegas, V,. [9] A. Rchards and J.P. How, ode Predctve Contro of Vehce aneuvers wth Guaranteed Cometon Tme and Robust Feasbty, Amercan Contro Conference, Denver, CO, 3. []. Padberg, Lnear Otmzaton and Extensons, nd Edton. Srnger, ew Yor, 999. [] C.H. Paadmtrou and K. Stegtz, Combnatora Otmzaton: Agorthms and Comexty. Prentce-Ha, Inc., Engewood Cffs, ew Jersey, 98. [] D.P. Bertseas, Dynamc Programg and Otma Contro: Voume, nd Edton. Athena Scentfc, Bemont, assachusetts,. [3]. Smaan and J. B. Cruz Jr., On the Staceberg strategy n nonzero-sum games, Journa of Otmzaton Theory and Acatons, V., , o. 5, 973. [4] B. Coson, P. arcotte and G. Savard, Beve rogramg: A survey, A Quartery Journa of Oeratons Research, 3, 87 7,