Scheduling Jobs with a Common Due Date via Cooperative Game Theory

Transcription

1 Amerca Joural of Operato Reearch, 203, 3, Publhed Ole eptember 203 ( chedulg Job wth a Commo Due Date va Cooperatve Game Theory Irel Draga Uverty of Texa at Arlgto, Mathematc, Arlgto, UA Emal: draga@uta.edu Receved October 8, 202; reved December 30, 202; accepted Jauary 7, 203 Copyrght 203 Irel Draga. Th a ope acce artcle dtrbuted uder the Creatve Commo Attrbuto Lcee, whch permt uretrcted ue, dtrbuto, ad reproducto ay medum, provded the orgal work properly cted. ABTRACT Effcet value from Game Theory are ued, order to fd out a far allocato for a chedulg game aocated wth the problem of chedulg job wth a commo due date. A four pero game llutrate the bac dea ad the computatoal dffculte. Keyword: chedule; Effcet Value; Egaltara Value; Egaltara Noeparable Cotrbuto; hapley Value; Cot Excee; Lexcographc Orderg; Cot Leat quare Preucleolu. A chedulg Game ad mple oluto A mache may proce job, J, J 2,, J, wth the completo tme p, p2,, p, all potve umber. No two job ca be multaeouly doe, ad for all job there a commo due date d potve. Ay chedule a equece of job, ad o preempto allowed. The chedule determed by the umber C, N, the completo date of the job J. Ay devato from the due date wll be pealzed, ether a early or a late completo relatve to the due date. The total tme devato a chedule gve by C d. () N The uual chedulg problem to: fd out the chedule * for whch the total devato mmal. I [], J. J. Kaet olved the problem for the cae whe the um of completo tme maller tha, or equal to d, ad gave a algorthm for computg a chedule wth a mmal devato. Of coure, th algorthm may be ued to fd the total pealty for th chedule ad alo to fd the total pealty for ay mmal devato correpodg to ay ubet of job. Th make ee the cae whe the cot of the devato, early or late, are proportoal to ther ze. I the followg, we aume that the cot are equal to the pealte. A more geeral cae other tha Kaet ha bee olved by M. U. Ahmed ad P.. udararaghava [2]. I [3], N. G. Hall ad M. E. Poer coder mlar problem. The lterature coected to more geeral cae huge, ad the cocluo obtaed the preet paper ca be appled to mot other cae. For the preet dcuo, the mplet cae offered by Kaet algorthm, wth the above aumpto, good eough to ugget mlar approache all other cae, coecto wth a ew problem to be troduced below. Aumg that the grad coalto ha bee formed ad the total pealty for early ad late devato, w N, ha bee computed by ome algorthm, a ew problem : how much hould be a far dvdual pealty for each job? To awer the queto, we ow buld the followg cooperatve game wth traferable utlte: let N, 2,, be the et of player, the player be the cutomer orderg the job J, for each, 2,, Coder ay coalto of cutomer,, N,, ad otce that f, the the mmal chedule tart the correpodg job at d p, ad there wll be o devato from the due date. Therefore, f we deote the devato for coalto, by w, we have w 0. A algorthm for computg the mmal devato, for example Kaet algorthm, wll provde the total devato w 0, whe 2. I th way, we get a cooperatve TU game Nw,, whch we wat to dvde farly w N. To make the paper elf cotaed let u ketch Kaet algorthm whch wll be ued the example how be- Copyrght 203 cre.

2 440 I. DRAGAN low. Let be ay coalto ad deote by B, (before ), a equece of job whch were already elected, wth o-creag proceg tme ad the lat job edg at d ; alo, deote by A, (after ), aother equece of job whch were already elected, wth odecreag proceg tme ad tartg at d. Now, aume that we have B A, B A, Kaet algorthm cotug to buld the partto of a follow: f B A, elect the o-elected elemet of wth a maxmal proceg tme ad take t a the lat job B, (ow we have B A ). Further, f wth the ew B, we have B A, the chooe the o-elected job wth a maxmal proceg tme ad take t a the frt job A, (that tartg at d ). Repeat the procedure, electg alteratvely player B, the A, utl exhauted; the, the tme devato computed by formula () for the correpodg chedule ad the ame way for ay ubet of player. Example : Let J, J2, J3, J4 be a et of job to be proceed o a gle mache, wth the proceg tme p 2, p2 0, p3 8, p4 5, ad the due date d 39, uch that the Kaet codto how above hold. We ca compute for the et of player N, 2, 3, 4, ad t ubet, the total pealte. Kaet algorthm wll geerate the game w w 2 w3 w 4 0, w, 2 0, w,3 w 2,3 8, w w, 4w2, 4 w, 2, 38, w, 2, 45, w w 3, 4 5, w,3,4 2,3,4 3,, 2,3,4 28. Th correpod to the total devato of all coalto, ad our problem to: fd out how we hould dvde farly wn 28 amog the player? We tart by howg two mple oluto: the Egaltara allocato ad the Egaltara o-eparable cotrbuto allocato. Deotg the frt by x *, we get x* 7,7,7,7. Deotg the ecod by y*, whch gve geeral by formula y * w N w N wn wnw N j, N, jn (2) we get the margal cotrbuto M j wnwn j, j N, equal to 5, 5, 3, 0, o that the um make 53, ad from each margal cotrbuto we hould ubtract 25/4, to obta y*,,, Lookg at the charactertc value of our game, how example, we ee that the player ad 2 eem to be equal, whle player 3 ad 4 are weaker, hece the lat two hould be aked to pay maller dvdual pealte. The frt oluto doe ot eem to how th, whle the ecod eem more far, we hall ee a method below to compare the fare of the oluto. 2. Idvdual Pealte: et oluto, hapley Value To evaluate the fare of a poble oluto z, we may ue the exce fucto; however, here t eem more approprate to ue ome mlar fucto that we hall call the cot exce fucto. For ay coalto, N,, ad ay allocato z, the cot exce fucto, z z w. (3) Thee are the egatve of uual exce fucto, obvouly, we have 2 2 uch fucto, becaue for the grad coalto, for ay allocato, by defto we have N, z 0. I word, the cot exce the dfferece betwee what the coalto wll pay z to cotrbute a cloe a poble to the total pealty for herelf, whle t alo cotrbutg to the total pealty for N. The, what we wat to do to chooe the allocato z whch mmze all cot excee o the et of allocato. Note that the um of all cot excee a cotat, becaue for ay allocato z we have, z 2 wn w. N N Moreover, we ca defe the average cot exce w, z, 2 N whch by formula (4) doe ot deped o the allocato z Th mea that f the allocato of ome coalto creaed, the the allocato of at leat oe other coalto wll be decreaed. How the cot excee are ued to compare the fare of two allocato llutrated below. Example 2: Retur to example, ad wrte the cot excee for that game Nw,, ad ay allocato z z z 2, 3, z z z 8,, z z, 2, z z, 2 3, z z, 4, z z, 3 4, 2, 0, 3 (4) (5) Copyrght 203 cre.

3 I. DRAGAN 44 4, 4, z z z 5, 2,3, z 2, 4, z z z 5, 3, 4, z 2 4, 2, 4, z, 3, 4, z, 2, 3, z z z z2 z 8, z z 5, z z z 3, 3 4 2,3, 4, z z z z z z 8, 2 3 z z 5, Our problem to mmze all cot excee, whle we are o the et of allocato, that the effcecy codto hold. I other word, we wat to mmze the maxmal cot exce, ubject to effcecy codto, or to ue aother method to olve a mult objectve lear programmg problem. Let u try to evaluate the fare of the two allocato offered utl ow. For the Egaltara oluto, we ca compute the cot excee ad put them a vector of o creag excee, whch may be called the vector of uhappe, a the compoet are take the order of o creag uhappe x * 9,9,9,8,8,7,7,7,7,6,6,6,4,3. It follow that the mot uhappy coalto are the two pero coalto whch oe of the player player 4. For the Egaltara o eparable cotrbuto, we fd the vector of uhappe y * ,,,,,,,,,,,,,, ad 2. Moreover, we have alo larger tha y ad the mot uhappy coalto are x * *, that the mot uhappy coalto x * more uhappy tha the mot uhappy coalto y *. We ca ay that y *. better tha x *, or more far. Note that the ame thg could be ad f ome par of correpodg compoet the two uhappe vector are equal, but the frt oe whch dfferet maller y * tha y * L x*, x*. I th cae, we alo wrte where L mea the lexcographc order, ad read y * better tha x *. Utl ow, we have ee two mple oluto belogg to Game Theory, they are oe pot oluto becaue each oe provdg a uque oluto. Oe of the et oluto from Game Theory the CORE, whch for a cot game Nw, lke our the et CON, w (6) zr : N, z 0,, z 0, N,. Ay elemet of the CORE codered a a good allocato, becaue uch a allocato cover the total pealty for each coalto. Lookg at the two mple oluto of example, whch a ee example 2 have all excee o egatve, we ee that both are the CORE, but, of coure, there are other wth the ame property. Moreover, we ca alo ee that the um of excee equal 96, that the worth obtaed formula (4) for 4. The mot famou oe pot oluto, whch may alo be the CORE, the hapley Value, troduced [4], ad defed by a et of axom, decrbg ome bac properte requred for a far oluto. The hapley Value wa proved there to be gve by the formula N, w H :!! vv, N.! (7) Example 3: For the game codered example, we get H N, w 8,8,7,5. Computg the cot excee ad orderg them, we obta H 8,8,8,8,7,7,7,7,7,7,6,6,5,5. H y x We get * *, L L 2 Mmze f w, z, z w, N becaue the frt compoet of the uhappe vector are th order, hece the hapley Value better tha the Egaltara o eparable cotrbuto, whch better tha the Egaltara oluto. Note that th may ot be the cae for other game. Note alo that f the game large, the the hapley Value may ot be eay to compute. A algorthm baed upo the o called Average per capta formula, gve by the author [5], may be ued, a t wll be explaed the ext ecto ad the algorthm allowg eve a parallel computato of the hapley Value. mlar tuato may occur coecto wth the other value. 3. The Cot Leat quare Preucleolu I the followg, we may coder a a oluto the Leat quare Preucleolu of the game, troduced by L. Ruz, F. Valecao ad J. Zarzuelo [6]. Th mlar to the Preucleolu, troduced coecto wth the Nucleolu, due to D. chmedler [7], except that th wa defed by mea of the followg quadratc programmg problem ubject to (8) N, z 0. (9) Copyrght 203 cre.

4 442 I. DRAGAN By ug the Kuh-Tucker codto, (8), (9), t ha bee how that our problem ha a uque oluto, that the author called the Leat quare Preucleolu, amely w N LN, w aw a jw, N, jn (0) where 2 a w w, N,. () : A the cot excee are replacg the excee, we called th value the Cot Leat quare Preucleolu, but the expreo (0) ad () are the ame eve our cae. Example 4: Computg by (0) ad () th oluto for our game, we get L N, w,,, The vector of uhappe L * (ee foot-ote). Notce that the um of cot excee alo 96. Now, checkg the comparo of the ew oluto wth the other three oluto, we obta, LNw, H Nw y* x*, L L L that the hapley Value eem to be more far tha the other three value. Of coure, the chmedler Preucleolu may alo be computed; the computatoal method, due to A. Kopelowtz [8], alo how [9]. However, th clude a log computato for olvg a equece of lear optmzato problem. The Preucleolu would be the bet, by the defto of the value. Aother prcple may be ued to chooe the approprate allocato: for each allocato avalable, compute the dfferece betwee the cot for the mot uhappy coalto ad the happet coalto ad chooe the allocato that gve the mallet dfferece. uch a allocato would ot gve a hgh dfferece of cot betwee the happy ad the uhappy coalto. I our cae, we have dh 3, dl, dy* 5, d x * Notce that by the lat prcple the four value are ordered the ame way. 4. Cocluo The techology put together the preet paper apple to other chedulg problem whch the aocated chedulg game ca tll be geerated by ome algorthm. ome of the followg remark may help: ) The above dcuo wa llutrated by example, 2, 3, 4 relatve to a four pero game. If we have 5 job, uder the ame codto lke above, Kaet algorthm tll apple whe the Kaet aumpto hold. If the objectve dfferet, for example to mmze a weghted combato of devato relatve to a commo due date, the the algorthm by Hall ad Poer hould be ued to get the chedulg game. 2) A oo a the game avalable, the problem of dvdg farly the worth of the grad coalto the problem of choog a effcet value from Game Theory, for whch the computato could be doe. A all gleto have a zero worth, the Ceter of the mputato et [9] become the Egaltara value, whch geeral ot far. The Egaltara o-eparable cotrbuto may be a alteratve allocato. The hapley Value, whch ha a lot of properte derved from the axom, the mot preferred by almot all cett, but for more tha te player t become dffcult to compute. For uch large game t may be better to ue the formula gve by the author [5], called the Average per capta formula w w H N, w, N, (2) where w the average worth of coalto of ze, ad w the average worth of coalto of ze, that do ot cota player, wth w 0, N. It obvou that the tak ca be performed by team, ad each oe compute oe rato for oe. 3) I the computato of the Cot Nucleolu by Kopelowtz method [8,9] the paage from oe LP problem to the ext ot decrbed detal mot ource. The complemetary codto how whch cot excee hould rema cotat o all optmal oluto of the curret problem, ad hould be kept cotat the ext problem. Thee equato hould replace the correpodg equalte of the curret problem ad rema atfed utl we meet a problem whch ha a uque oluto. The oluto of the quadratc programmg problem eem eaer to compute. The geeralzed ucleolu may be ued a a oluto of ay Multcrtera Lear Programmg problem, a how by the author [0], workg wth a three pero game. The bac dea appeared a former paper of the author [], a well a the more recet paper by E. March ad J. A. Ovedo [2] L *,,,,,,,,,,,,, Copyrght 203 cre.

5 I. DRAGAN 443 REFERENCE [] J. J. Kaet, Mmzg the Average Devato of Job Completo Tme about a Commo Due Date, Naval Reearch Logtc Quarterly, Vol. 28, No. 4, 98, pp do:0.002/av [2] M. U. Ahmed ad P.. udararaghava, Mmzg the Weghted um of Late ad Early Completo Pealte a gle Mache, IEEE Traacto, Vol. 22, No. 3, 990, pp do:0.080/ [3] N. G. Hall ad M. E. Poer, Earle-Tarde chedulg Problem, I: Weghted Devato of the Completo Tme about a Commo Due Date, Operato Reearch, Vol. 39, No. 5, 99, pp [4] L.. hapley, A Value for -Pero Game, Aal of Mathematc, Vol. 28, 953, pp [5] I. Draga, A Average per Capta Formula for the hapley Value, Lberta Mathematca, Vol. 2, 992, pp [6] L. Ruz, F. Valecao ad J. Zarzuelo, The Leat quare Preucleolu ad the Leat quare Nucleolu, Two Value for TU Game Baed o the Exce Vector, Iteratoal Joural of Game Theory, Vol. 25, No., 996, pp [7] D. chmedler, The Nucleolu of a Charactertc Fucto Game, IAM Joural o Appled Mathematc, Vol. 7, No. 6, 967, pp do:0.37/0707 [8] A. Kopelowtz, Computato of the Kerel of mple Game ad the Nucleolu of -Pero Game, RM 3, Hebrew Uverty of Jerualem, Jerualem, 967. [9] G. Owe, Game Theory, 3rd Edto, Academc Pre, New York, 995. [0] I. Draga, A Game Theoretc Approach for olvg Multobjectve Lear Programmg Problem, Lberta Mathematca, Vol. 30, 200, pp [] I. Draga, A Game Theoretc Approach for olvg Multobjectve Lear Programmg Problem: A Applcato to a Traffc Problem, Quader de Grupp d Rcerca CNR, Pa, 98 [2] E. March ad J. A. Ovedo, Lexcographc Optmalty the Multple Objectve Lear Programmg: The Nucleolar oluto, Europea Joural of Operatoal Reearch, Vol. 57, No. 3, 992, pp do:0.06/ (92)90347-c Copyrght 203 cre.