Effectiveness of quantity support in combinatorial auctions

Transcription

1 Mat-2.8 Sovelletun matematkan erkostyö Effectveness of quantty support n combnatoral auctons Valtter Ervast 565N

2 INTRODUCTION AUCTION SETTINGS MATHEMATICAL FOUNDATIONS Wnner determnaton problem Decson support problems AUCTION DESIGN SIMULATION DESIGN... 3 VARIABLE INPUT PARAMETERS PRODUCTION COST FUNCTION SETTINGS: ECONOMIES OF SCOPE PRODUCTION CAPACITIES INITIAL BID GENERATION Method Method Method Settng the ntal bd prce QUANTITY SUPPORT: AN EXPRESS VERSION NUMBER OF BIDDERS FIXED INPUT PARAMETERS NUMBER OF ITEMS VARIANCE BETWEEN THE BIDDERS PRODUCTION COST FUNCTIONS EXPECTED PROPORTION OF FIXED COSTS OTHER FIXED INPUT PARAMETERS AND SETTINGS EXPERIMENT DESIGN FACTORIAL ANALYSIS VIEWPOINT THE EFFICIENT ALLOCATION OUTPUT VARIABLES RESULTS EFFECTS ON Q/EFF, Q/SEFF, AND QC/EFF EFFECTS ON P2/EFF, P2/SEFF AND P2C/EFF EFFECTS ON P2-Q AND P2C/QC EFFECTS ON INI-Q EFFECTS ON QSUCC EFFECTS ON EFF/SEFF AUCTION LENGTH NUMBER OF WINNING BIDDERS INTERPRETATION OF RESULTS CONCLUSION REFERENCES

3 Introducton Ths study etends the results obtaned n my Master s thess [], where the effect of prce support and quantty support on the results of a combnatoral aucton were studed by means of smulaton. An aucton s defned (Pekec & Rothkopf 23) to be combnatoral f: ) several, dstngushable tems are sold at the same tme, and 2) ndvsble, all or nothng bds are accepted for mult-tem combnatons. When the bdders valuatons for the tems are epected to be superaddtve, a combnatoral aucton desgn s epected to reach a hgher level of effcency than a noncombnatoral one (de Vres and Vohra 23). It s a consequence of property 2) that wnner determnaton s comple n combnatoral auctons. In the auctons of ths study as well as many others, the wnners are determned by solvng an nteger lnear programmng (ILP) problem, whch has been proven (de Vres and Vohra 23) to be NP-complete. Another consequence of 2) s that n combnatoral auctons, the tem combnaton s another attrbute by whch bds are evaluated, n addton to the prce and other possble attrbutes. In forward combnatoral auctons, two or more bds cannot be smultaneously accepted f they overlap: n reverse ones, they cannot be smultaneously accepted f they do not add up to total demand. Prce support and quantty support are decson support tools for the bdders partcpatng n a combnatoral aucton. Ther task s to help the bdders place the rght bds n order to enter the wnnng group. Decson support s needed n combnatoral auctons where competton s possble not only wth prce, but wth tem combnaton as well. It s especally needed n sealed bd auctons, where the bdders cannot see each other s bds, and therefore have no way of knowng whch combnaton should be bd on. Nevertheless, not many studes have addressed decson support n combnatoral auctons. Those that have done so nclude Adomavcus and Gupta (25) and Tech et al. (26) and Leskelä et al., whch ntroduced prce and quantty support. Three aucton types were defned n []. They were dfferent to each other n the decson support system that was used, and smlar n every other respect. In each smulaton, the three auctons were smulated usng the same nput data, makng parwse comparsons possble. The same aucton and smulaton desgns are used n ths study as well. The results of [] are etended by studyng the effects of some factors that were fed n []. Ths tme, there are: Two new methods to generate the ntal bds. Dfferent economes of scope. An epress verson of quantty support. The hghest number of bdders s now 3 nstead of 5. The bdders can have equal producton capactes. 3

4 Contrarly, some factors that were varable n [] are now fed. The fnal cost to the auctoneer s consdered the most mportant aucton result: n a reverse aucton, the cost s beng mnmzed. Some other results, such as aucton length and the number of wnnng bdders, are also studed. The smulatons are run wth Matlab as was done n []. A more effcent optmzaton engne, the Lndo API, s now beng used to solve the wnner determnaton and prce/quantty support problems. Usng the Lndo API also allows to calculate the so called effcent allocaton and cost n each smulaton. In ths study, Chapter 2 defnes the optmzaton problems related n wnner determnaton and decson support, as well as the aucton and smulaton desgns used n ths study. The factors for whch dfferent levels are to be tested (also called nput parameters), are descrbed n Chapter 3: Chapter 4 descrbes the nput parameters that have fed values. Chapter 5 presents the eperment desgn and defnes the output varables to represent the results. 4

5 2 Aucton settngs 2. Mathematcal foundatons The auctons consdered n ths study are reverse auctons. Ths means that the auctoneer s the buyer, and the bdders are sellers. In an aucton, there are: N tems, each {,2, K, N} of whch D unts are demanded by the auctoneer, n bdders, m bds. Each bdder s requred to submt at least one bd, so m n. The bds entered are of the form B ;, where q s the quantty of tem ncluded n bd, and p [ ] T = q L qn p s the prce at whch the bdder s wllng to supply ths tem combnaton. Together, all bds n the aucton comprse an ( N + ) m bd matr: q M bdmatr = qn p q q M N 2 p 2 2 L O L L q q m M p Nm m (2-) Durng the course of an aucton, the bdders can submt new bds and remove old ones. To keep track of whch bd belongs to whch bdder, a separate n m key matr s defned: key keymatr = M key n key M key 2 n2 L O L key M key m nm, (2-2) where key = f bd belongs to bdder, and key = otherwse. As s seen n the net secton, the key matr can also be used to restrct the mamum number of bds that can be accepted from the same bdder. When used to restrct the number of bds, t does not matter what values the nonzero elements of the key matr assume as long as they are equal wthn each row. In fact, f they all had value the effect of the correspondng constrants would be eactly the same. But ths way the key matr becomes easer, for the human eye, to nterpret. 5

6 2.. Wnner determnaton problem Wth nformaton about the bd matr, the key matr and the :s, the wnners of a reverse, combnatoral aucton can be determned by solvng the wnner determnaton problem (WDP) D mn m = p s. t. m = m = m = M q q key N bnary D M D N. (2-3) The WDP mnmzes the auctoneer s total cost whle satsfyng the overall demand as defned by the :s. The constrants based on the key matr ensure no more than one bd D becomes actve from each bdder. Reflectng bd ndvsblty, each decson varable assumes value f bd s accepted, and f t s reected. Bds cannot be partally accepted. When solved, the WDP produces the bnary vector *, ndcatng whch bds are to be accepted and whch are not. In a notaton adopted from [], the bds for whch the correspondng = are called actve. Conversely, the bds for whch = are called nactve. In a smlar way, the sets of actve and nactve bdders are defned. The total cost C to the auctoneer equals the value of the obectve functon at optmum: m C = = * p The WDP can be lnearzed by leavng the bnary constrants of the s out of (2-3). The dual of the lnearzed WDP can then be solved to obtan N dual or shadow prces d,,. These are later used to solve the quantty support problem. L d N 2..2 Decson support problems Once the wnners of a gven aucton round have been determned, prce and quantty support can be used by any nactve bdder, n order to fnd a bd that wll become actve f submtted. The total cost to the auctoneer s requred to decrease, so we set C new =, 98 C. The sze of ths decrease s a compromse: as t s effectvely nflcted on a sngle bd, a larger decrease would more lkely be unproftable for the bdder. 6

7 Conversely, a smaller decrease n total cost would probably cause the aucton to last longer. Prce support fnds a new prce P for a bd whose tem quanttes Q, L,Q N have already been set: f placed wth ths new prce, the bd would become actve. In the correspondng prce support problem (PSP), P s mamzed whle the constrants ensure that the bd becomes actve, the auctoneer s total cost doesn t eceed C 2 new, and that the overall demand s met. ma P s. t. m' = m' = m' = m' = q q p N + Q + Q key + P N C D D new M M (2-4) N wth P C. new bnary * As a soluton, the PSP produces the vector {,, } bd [ Q Q N P L ; * ] T, t wll become actve. P L. If the bdder now submts the ; m ' In quantty support, the tem quanttes Q, L,Q N are not fed. Instead, they are decson varables lke P and the s. They are requred to be non-negatve but not to eceed the bdder s producton capactes l 3, L,l N. Other constrants are the same as n the PSP. The obectve functon to be mamzed s now the bdder s proft, obtaned by subtractng from P the cost to produce Q, L,Q N. The auctoneer does not know the bdder s producton cost functon, so t s appromated by the dual prces. 2 Ths constrant ensures the new bd wll become actve f submtted, snce there s no other bd combnaton whose combned prces would amount to less than C. 3 See Secton 3.2 for more nformaton about the producton capactes. 7

8 ma P N = d Q s. t. m' = m' = m' = p q + Q key + P C D Q P new l bnary =, K, N J (2-5) * * * After a soluton { P Q,, Q ;, L, } * * * T [ Q L Q N ; P ] ; N m ' K to the QSP has been obtaned, the bd wll become actve f t s submtted. 2.2 Aucton desgn Three types of auctons are defned, each wth a dfferent type of decson support: Aucton wth prce support (referred to as a P aucton) Aucton wth ntellgent prce support (P2 aucton) Aucton wth quantty support (Q aucton) Prce support s used as the decson support method n both P and P2 auctons. The dfference s that n the P aucton t s only used for the bdder s estng bd, whle n the P2 aucton t s also used for the correspondng subcombnatons, obtaned by settng some of the s to zero. Ths s done n an attempt to elmnate possble surplus tems q from the bd, ncreasng ts chances to become actve. Intellgent prce support should not be thought of as a separate decson support tool. Instead, t attempts to smulate the behavour of the nactve bdders as they try usng prce support on dfferent tem combnatons. For eample, suppose a bdder had submtted the bd T [ ] (2-6) and t s currently nactve. In a P aucton, he would use prce support for ths bd only, to obtan an acceptable prce P < 2. But n a P2 aucton, he would use prce support for the bds 8

9 P P P P 4 3 P P 6 3 P 7 5 2,5. (2-7) 5 P8 The collecton (2-7) ncludes the orgnal bd, the orgnal bd wth the tem quanttes halved, and the one- and two-tem subcombnatons, of whch there are s. The collecton of bd and prce propostons s called the shortlst. In addton to the P2 auctons, a shortlst s also created n the Q auctons: ths s descrbed n Secton 3.4. In ths contet, t can be sad that the shortlst s also created n the P auctons, but t only contans one entry. Snce the orgnal bd (2-7) now contans three nonzero tem quanttes, the shortlst has only = 8 If all the tem quanttes were nonzero n the orgnal bd, the shortlst would contan = 32 entres. The auctons can now be descrbed algorthmcally as follows. The graphc outlne of an aucton s presented n Fgure 2.. Step. Intalzaton. Before the aucton actually begns, the set of bdders s defned: ths ncludes defnng the bdders producton capactes and cost functons. After ths the bdders place and prce ther orgnal bds. The ntal bd matr now consttutes n columns.. Wnner determnaton. The auctoneer solves the wnner determnaton problem, leavng some bds actve and the rest nactve. A set of actve bdders s created, as well as a set of nactve bdders. Solvng the WDP n ts lnear form wll also produce a set of dual prces d,,d N for the N tems. Proceed to Step 2. K 2. Usng prce or quantty support. One nactve bdder, wth dentty, s chosen at random to use prce or quantty support. Ths s where the auctons dffer from each other: In a P aucton, the PSP s solved for bdder s estng bd. In a P2 aucton, the PSP s solved for bdder s estng bd and the correspondng subcombnatons as defned above. In a Q aucton, the QSP s solved several tmes as defned n Secton

10 As a result, a shortlst of bd propostons s then presented to bdder. The bdder then compares bd prces p l n the shortlst to the costs of producng the tem quanttes nvolved 4 : f = 2K c q (2-8) l ( q) F N, + c ql N Nl and fnds the most proftable bd, that s, the bd for whch pl fl (q) reaches ts mamum value. If p f ( q) >, ths bd s added at the end of the bd matr, and l another column s added to the key matr, wth bdder dentty number at the correct row. (In a P aucton, no new bd s added n the bd matr: the orgnal bd prce s altered nstead.) If bdder has prevously placed a bd whose tem quanttes are dentcal to the new one but the prce s hgher, the older bd s deleted from the bd matr. Also the correspondng column s deleted from the key matr n ths case. The frst round of the aucton has ended. Return to Step. If no bds on the shortlst are proftable, the bdder doesn t choose any of them and s removed from the set of nactve bdders. If the set has now become empty, proceed to Step 3. If not, the turn to use quantty support s then passed on to another bdder n the nactve set. Repeat Step 2. Step 3. End aucton. Comng here means all nactve bdders have used prce/quantty support and reected ts propostons for new bd, whle the bd matr hasn t changed. The aucton has ended, and currently actve bdders are the fnal wnners. The wnners, and the tem combnatons ncluded n ther bds, are called the wnnng allocaton. Fgure 2.. The outlne of an aucton. 4 For nformaton about the producton cost functons, see Secton 3..

11 To summarze, an aucton round can end n one of two ways:. A bdder accepts a proposton of prce or quantty support and places t as a new bd (or n a P aucton, updates the prce). 2. The set of nactve bdders becomes empty. The former case means the aucton moves nto the net round, where the WDP s solved agan to obtan a new set of wnners and a new, decreased total cost to the auctoneer. The latter case means the aucton has ended. In ths aucton desgn, the round length s varable, as s the number of rounds. Fnally, t should be noted that the man nterest now s to study the dfferences n results between the P2 and Q auctons. However, the P auctons are ncluded n each smulaton. 2.3 Smulaton desgn A smulaton s defned as an ndependent set of three auctons that s ntalzed by a set of nput parameter and produces a set of output varable values. Each smulaton contans three phases: Intalze the smulaton. Equals Step descrbed above. o Defne the set of bdders, producton cost functons and capactes o Calculate the effcent allocaton o Submt the ntal bds Run the auctons Record the results Because the ntal bd matr s the same n all three auctons, also the ntal wnnng bdders (frst results of Step ) are the same. The auctons begn to dvert when a randomly selected bdder uses prce/quantty support n Step 2. The same nput parameter values are used n all three auctons, makng parwse comparsons possble wthn a smulaton. An outlne of a smulaton s presented n Fgure 2.2.

12 Fgure 2.2. The outlne of a smulaton. Several parameters must be assgned values before a smulaton can take place: these are called nput parameters. The ntenton s now to run many smulatons wth dfferent nput parameter values, to see what effect ther varaton has on the smulaton results. Those nput parameters that are assgned multple values are defned n Chapter 3. The rest of the nput parameters are assgned fed values, whch reman unchanged throughout the smulatons. These are defned n Chapter 4. The output varables, used to represent the aucton results, are defned n Chapter 5. For more detals on the mathematcal foundatons, the auctons and the smulatons, see Chapters 3 and 4 of []. 2

13 3 Varable nput parameters Ths chapter defnes the nput parameters that are assgned multple values. 3. Producton cost functon settngs: economes of scope Each bdder has hs producton cost functon defned as f =, c q, (3-) ( q) FI + c q N N where c = varable cost to bdder to produce one unt of tem. = fed cost to bdder to produce nonzero quanttes of the combnaton F I, I = { }, 2,L. Therefore, each bdder has a well defned producton cost for each tem combnaton Q, L,Q N. As stated n Secton 2.2, the bdders evaluate the propostons of prce and quantty support by lookng at the dfference of the proposed bd prce P and the producton cost f (q). If P f ( q) >, the proposton s proftable. Economes of scope are defned to est f the cost of producng a combnaton of tems s lower than the sum of the costs of producng each tem separately. Formally defned, f s the fed cost nvolved n producng the tem combnaton I, economes of scope F I, est whenever + F > F s true for all I J. F I, J, I J, Two levels of economes of scope are defned and tested n ths study. In practce, ths means assgnng dfferent values to the s. When I s a combnaton of the two tems and 2, the epected value of ( E( F )) ( E( F )) F I, s F I, E ( FI, ) =,5, wth Scope, and E F ) =,4 wth Scope 2. ( I,, Wth larger combnatons, each enterng tem adds,5 E ( F I, ) when Scope s used. Wth Scope 2,,4 ( ) of the s are presented n Table 3.. F I, E F I, to the epected fed cost s added nstead. The numercal values 3

14 Table 3.. Fed cost parameters and ther dstrbutons under dfferent economes of scope. Economes of scope: normal Economes of scope: large F,I Mnmum Epected Mamum F,I Mnmum Epected Mamum tem tem tems tems tems tems tems tems tems tems Unlke the F I, s, the varable cost parameters c are fed. They are drawn from the unform dstrbuton [ 53,33 66,67], wth E ( ) = 6. c 3.2 Producton capactes As was done n [], the producton capactes of each bdder and tem are denoted wth l s: l = the largest quantty of tem that can be produced by bdder. The total demand capactes, the l D for each tem s defned to be 6 unts. Wth nequal producton :s are drawn from the dscrete dstrbuton: l = 3 wth 5% probablty. l = 225 wth 25% probablty. l = 5 wth 25% probablty. Ths dstrbuton s ntended to model the dvson of the bdders nto small, medum, and large players n the ndustry. The s drawn do not affect each other n any way, so a bdder may well be a small player n one tem and large n another. The mamum value = 3, equalng 5% of total demand, also smulates the auctoneer not wshng l to be too dependent on one suppler. l Wth equal producton capactes, we set for all l s: l = 3 When the producton capactes are equal, all bdders can bd on every combnaton, and the compettveness of the aucton s epected to ncrease. 4

15 3.3 Intal bd generaton After the producton cost functons and capactes have been defned, the net step s to place the ntal bds. It conssts of two phases: settng the tem quanttes, and settng the prce. In [], three methods to generate the ntal tem quanttes were defned. These were: Method : the tem quanttes were drawn from the unform dstrbuton [, ]. Method 2: the tem quanttes were drawn from normal dstrbuton wth parameters: o Epected value l adv ) ( o Standard devaton,25 l. Method 3: the tem quanttes were dscrete: ether q = or q =, dependng on the varable producton costs. l In Method, the producton cost ddn t affect the tem quanttes of the bd at all. In Method 2, the varable costs were consdered so that f a bdder s varable cost to produce tem was lower than average, he was more lkely to bd for a large quantty of ths tem. In Method 3, the bdder would always bd for a large quantty f he had an advantage n varable costs. The ntal bds were found to sgnfcantly affect the outcome of an aucton, wth Method 3 generally amountng to the best results. In ths study, t wll be compared to two other methods to generate ntal bds. The net three sectons descrbe the bd generaton methods tested n ths study. l 3.3. Method 3 Wth N tems n the aucton, each bdder has N varable cost parameters values have been drawn from the unform dstrbuton 3 adv are then defned by c, whose [ a b ]. Cost advantage ndces adv 3 = F a b a. (3-2) The tem quanttes q are then set at zero f the bdder has hgher than average c, and at mamum capacty f he has lower than average : q = f adv, 5. q = l f adv <, 5. c 5

16 3.3.2 Method 4 Method 4 consders fed cost parameters rather than varable ones. Other than that, t s smlar to method 3. Wth N = 5 tems n the aucton, each bdder has 2 5 = 3 fed cost parameters F, one for each tem combnaton I. They are drawn from the unform dstrbuton 4 [ A I BI ]. As a result, each bdder wll have 3 cost advantage ndces adv I, defned by I adv 4 I = F b I I a a I I. (3-3) 4 For each bdder, the mnmum of adv s found subect to I. Then we set q = l for all I mn 4. I Method 5 The dea of method 5 s to fnd the combnaton each bdder s most effcent producng the mamum amount of, when both fed and varable costs are consdered. For each tem combnaton I, there s the bdder s true producton cost f TRUE, I = F I + for each bdder: I c f = E( F ) + E( c ) l EXP, I I I l EXP, I, and the epected producton cost that all bdders share:. Based on these, we now defne 3 cost advantage ndces FTRUE I adv 5, I =. (3-4) F As n the prevous method, the mnmum of q = for all I mn 5. l 5 adv I s found subect to I, and we set Settng the ntal bd prce 6

17 Once the tem quanttes q,, prce p L q N of a bd have been set, the bdder has to decde the at whch he offers to supply ths tem combnaton. As stated earler, the bdder s cost to produce the combnaton equals f + ( q) = F, 2K N + c q +... c N q N. The ntal prce p s then set 2% hgher than the producton cost: Ths creates an ntal proft margn of 6,67%. p =,2 f ( q). 3.4 Quantty support: an epress verson In Secton 2.2, the quantty support problem (QSP) was defned as ma P N = d Q s. t. m' = m' = m' = p q + Q key + P C D Q P new L bnary =, K, N J (2-5). To appromate the bdder s producton cost functon, the dual prces d are used n the obectve functon. As ths s only an appromaton, t can happen that the optmal soluton to the QSP s not the most proftable one to the bdder. In the worst case, the QSP can produce a soluton that s unproftable to the bdder, whle a proftable one would have been found usng a dfferent obectve functon. To account for the naccuracy of the appromaton, alternatve solutons of the QSP can be generated by addng etra constrants of the types = and Q = one or more at a tme. The collecton of these solutons s called the shortlst. It s now studed how the effectveness of quantty support s affected by the length of the short lst. We defne the full verson of the shortlst to contan: The orgnal soluton obtaned wth no addtonal constrants. Solutons obtaned wth all the = constrants, one at a tme. Solutons obtaned wth the Q = constrants for all,2, K, n -tem combnatons. 7

18 The epress verson of the shortlst does not apply the = constrants, and therefore contans: The orgnal soluton obtaned wth no addtonal constrants. Solutons obtaned wth the Q = constrants for all,2, K, n -tem combnatons. These two versons of the shortlst are now compared, partcularly to see f the transton from the full verson to the epress verson sgnfcantly worsens the aucton results. In the smulatons, the orgnal soluton, obtaned wth no addtonal constrants, was placed at the end of the shortlst. Ths s done to make t easer to montor the performance of quantty support. 3.5 Number of bdders The number of bdders was a varable nput parameter n []: ts assgned values were and 5. Changes n the number of bdders were found to affect many output varables: for eample, the dfference n the fnal cost of a P2 aucton and a Q aucton decreased when there were more bdders. It would be nterestng to see f ths trend contnues when the number of bdders s further ncreased. Ths s straghtforward and t doesn t affect the wnner determnaton or the quantty support problems, although t does make the calculatons more tme-consumng. In ths study, the smulatons are run wth 5 and 3 bdders. The varable nput parameters are summarzed n Table 3.2. Table 3.2. Varable nput parameters and ther values. VARIABLE INPUTS Short lst Full Epress Producton capactes Equal Dfferent Intal bd generaton B3 B4 B5 Economes of scope Normal Large Number of bdders 5 3 8

19 4 Fed nput parameters In the prevous chapter, fve nput parameters were assgned multple values n order to study the effect of ther varaton on the aucton results. To keep the sze of the eperment reasonable, the rest of the nput parameters are assgned fed values. The frst three sectons of ths chapter specfy those nput parameters that were varable n [], but are fed now. The fourth secton brefly descrbes the rest of the nput parameters. For a thorough descrpton of the nput parameters and aucton rules, see Chapter 4 of []. 4. Number of tems The smulatons n [] were run wth 3 and 5 tems n the aucton. Whle t would be nterestng to further ncrease the number of tems, ths could only be done wth some smpler form of producton cost functon. Ths s because a separate fed cost parameter s defned for each bdder, and each tem combnaton I. The number of F I, combnatons ncreases eponentally wth the number of bdders: wth 5 tems, there are 2 5 = 3 such combnatons, wth 7, there are 2 7 = 27, and wth 9, there are 2 9 = 5. In ts present form, the effcent allocaton (see Secton 5.2.) would be very dffcult to calculate wth more tems. Therefore n ths study, the number of tems s set at Varance between the bdders producton cost functons a,. The varance s the length of ths nterval, as a percentage of the lowest possble cost: b a %. It represents the hghest percentage by whch one bdder s cost to produce a some tem combnaton can be hgher than another s. The varable and fed cost parameters are drawn from the unform dstrbuton [ b] In [], varance was tested at 25% and %. It was found that wth % varance, the fnal cost of the aucton tended to be hgher, and that quantty support s success rate tended to be lower. It seems unnecessary to test hgher levels of varance. In ths study, the varance s set at 25%. 4.3 Epected proporton of fed costs Ths nput parameter s defned as the proporton of fed costs out of epected total producton costs, when,5 D unts of tem are produced by one bdder. In [], the values of % and 5% were tested for ths nput parameter. Ths tme, there are no 9

20 resources to test ts effect, as other parameters have been chosen for closer study. The epected proporton of fed costs s set at 5%. 4.4 Other fed nput parameters and settngs The remanng nput parameters are fed at the same values as they were n []. Most of these have already been mentoned n the prevous sectons, but a summary s presented here for clarty. Total demand D for each tem s set at 6. The mamum number of unts of each tem that can be allocated to one bdder s set at,5 D = 3. Ths s taken nto account n the producton capactes l, whch never eceed 3. At most one bd s to become actve from any one bdder. The prce decrement on each round s set at 2%. In other words, each aucton round decreases the total cost by 2%. The proft margn ncluded n the ntal bds s set at 6,67%. On each aucton round, one nactve bdder at a tme s selected to use prce or quantty support. Ths selecton s made at random. When evaluatng the bd propostons of prce and quantty support, the bdders are prepared to accept prces that nclude, at least, a prce equal to the producton cost. Therefore, the mnmum proft margn s %. The bdders evaluate the bd propostons n terms of absolute proft rather than relatve. The number of aucton rounds s unlmted. The aucton ends when no bdder has accepted the propostons of prce or quantty support. All fed nput parameters are descrbed n Table 4.. Table 4.. Fed nput parameters. FIXED INPUTS & AUCTION SETTINGS Number of tems 5 Cost varaton 25 % Proporton fed cost / total 5 % Total demand of each tem 6 Ma. allocaton gven to one bdder 3 Ma. accepted bds from one bdder Prce decrement 2 % Intal proft margn 6,67 % Mnmum proft margn % Selecton of PS and QS users Random Proft calculaton Absolute Short lst generaton, nt. prce support See 2.2 Number of rounds Unlmted 2

21 5 Eperment desgn Wth the nput parameters defned n Chapters 3 and 4, t s possble to ntate and run an ndvdual smulaton. The purpose of ths study s to run many smulatons wth dfferent nput parameter values, to see what effect ther varaton has on the results. In ths chapter, Secton 5.. defnes how the smulatons are arranged. The output varables that represent the smulaton results are defned n Secton Factoral analyss vewpont In the contet of factoral analyss, the varable nput parameters defned n Chapter 3 are referred to as factors. The effect of a factor s defned to be the change n response 5 produced by a change n the level of the factor (Montgomery 99). In Chapter 3, a total of fve factors were defned to be tested. Letters A to E are now ntroduced to represent the factors as descrbed by Table 5.. Factor C (Intal bds) was prevously assgned three levels, whle the other four were assgned two levels. All factor level combnatons are tested, equalng a = 48 factoral desgn. Each set of factor levels s called a block. To gan statstcal sgnfcance for the results, each block contans 5 smulatons as they dd n []. The total number of smulatons then becomes 48 5 = 24. Table 5.. The factors and ther levels. FACTOR LEVELS A: Quantty support Full Epress B: Producton capactes Equal Dfferent C: Intal bd generaton B3 B4 B5 D: Economes of scope Normal Large E: Number of bdders 5 3 Of the factors presented n Table 5., the type of quantty support and the number of bdders defne rules that can be drectly used to ntalze and run an aucton. The remanng three factors defne dstrbutons from whch the actual nput data are drawn for each smulaton. Therefore, any two smulatons wll have somewhat dfferent nput data even f ther factor levels are the same. The output data s dvded nto groups sorted by dfferent factor levels. Means and medans are then calculated for each group, and the medan test s used to determne f the dfferences between groups are statstcally sgnfcant. The medan test s nonparametrc, meanng t does not assume the data to be normally dstrbuted, or the varances n dfferent subgroups to be equal. The data n [] was found n general not to 5 Here, the response s referred to as output. 2

22 meet such assumptons: the same s epected here. For more nformaton about the medan test, see Conover (99). Before defnng the output varables, a bref outlook nto the effcent allocatons s n order. 5.2 The effcent allocaton When the producton capactes and cost functons have been defned for all the bdders, t s possble to determne the set of bdders who can produce the total tem quanttes D, D2, K, D N at the lowest cost. Ths can be done by solvng a specfc lnear optmzaton problem: a detaled descrpton can be found n Secton 3.4. of []. Ths set of bdders, and the tem quanttes each of them s supposed to produce, s called the effcent allocaton. The sum of the producton costs s called the effcent cost. Gven the partcular set of bdders, the effcent cost s the lowest possble total cost that can theoretcally be reached n an aucton. In [], the effcent allocaton was not calculated because of nsuffcent computng capacty. Instead, t was appromated by a sem-effcent allocaton where the tems were allocated to the bdders one by one, gnorng the fed cost parameters of mult-tem combnatons. There was no nformaton of how close the sem-effcent cost was to the real effcent cost, and ts man purpose was to serve as a benchmark wth whch the aucton results could be compared across the whole range of factor levels. In ths study, the true effcent cost s calculated n every smulaton. 5.3 Output varables The followng abbrevatons are used when denotng the smulaton results: In = Intal aucton cost P = Fnal cost of a P aucton P2 = Fnal cost of a P2 aucton P2C = The sum of the producton costs for the wnnng allocaton of a P2 aucton Q = Fnal cost of a Q aucton QC = The sum of the producton costs for the wnnng allocaton of a Q aucton SEff = Producton cost of the sem-effcent (heurstc) allocaton Eff = Producton cost of the true effcent allocaton It should be noted that the fnal cost varables P, P2 and Q are not drectly comparable across all factor levels. For eample, the dfferent levels of factor D (Scope) assgn dfferent values to the fed cost parameters, affectng the epected producton costs of all tem combnatons: wth Scope 2, the :s are generally lower than wth Scope. F I, Therefore, also P2 and Q are epected to be lower. 22

23 In a smlar way, the effcent cost s also epected to vary wth dfferent factor levels: for eample, Eff s epected to assume lower values wth Scope 2 than wth Scope. As a consequence, P2/Eff and Q/Eff are epected to be comparable across all factor levels, wth beng the lowest value they can possbly assume. In other words, Eff serves as a benchmark that makes the fnal costs comparable at all factor levels. SEff s calculated so t can be compared wth Eff, and also to make the results of ths study easer to compare to those of [], where only SEff was calculated. The producton costs P2C and QC are nterestng because they can be drectly compared to SEff and Eff whch also represent producton costs. It s true that QC = Q ( proft ncluded n Q). If a value of s observed for QC/Eff n a smulaton, t means the effcent allocaton has been found n the Q aucton. Ths would not be not be notced ust by lookng at Q/Eff, because even when the effcent allocaton s reached, there s usually stll some proft margn ncluded n the total cost Q. In such a case the varable Q/Eff would assume a value hgher than. In other words, the varable QC/Eff contans nformaton about how close the wnnng allocaton s to the effcent allocaton, and the varable Q/Eff contans nformaton about how close the fnal cost s to the effcent cost. Both varables are nterestng n ther own rght. In the net chapter, mean and medan results are presented for the followng output varables: Q/Eff, Q/SEff and QC/Eff represent the fnal cost of the Q auctons. P2/Eff, P2/SEff and P2C/Eff represent the fnal cost of the P2 auctons. P2 Q P2C QC and represent the parwse dfference between the fnal costs P2 P2C of the P2 and the Q auctons of the same smulaton. In Q represents the cost decrease acheved durng the course of a Q aucton. In Qsucc represents the success rate of quantty support. Eff/SEff represents the accuracy of SEff n appromatng Eff. Aucton length and the number of wnnng bdders are also observed. 23

24 6 Results Ths chapter presents the mean and medan results sorted by the dfferent levels of the fve factors. The results for all 48 factor level combnatons are avalable n Append. 6. Effects on Q/Eff, Q/SEff, and QC/Eff The mean and medan results for the varables Q/Eff, Q/SEff and QC/Eff are shown n Table 6.. The results are sorted by dfferent factor levels: also the standard devatons are shown. Table 6.. Factorwse mean and medan results of Q/Eff, Q/SEff, and QC/Eff. Q/Eff Mean Medan St.Dev. All,62,48,48 Q/SEff Mean Medan St.Dev. All,892,88,64 A: Shortlst Epress,66,49,53 Full,58,46,42 B: ProdCap Equal,25,22,4 Inequal,99,92,4 C: Bd 3,65,5,5 4,6,46,48 5,6,47,45 D: Scope,62,48,48 2,62,49,48 E: Bdders 5,7,55,54 3,54,4,39 A: Shortlst Epress,894,883,68 Full,889,88,59 B: ProdCap Equal,844,842,32 Inequal,939,934,52 C: Bd 3,95,898,67 4,885,875,63 5,884,874,59 D: Scope,897,886,64 2,886,876,64 E: Bdders 5,9,898,69 3,873,867,52 QC/Eff Mean Medan St.Dev. All,26,6,28 A: Shortlst Epress,27,7,29 Full,25,5,27 B: ProdCap Equal,8,5, Inequal,45,43,28 C: Bd 3,27,7,28 4,25,6,26 5,27,5,29 D: Scope,26,6,28 2,26,6,28 E: Bdders 5,28,7,3 3,25,5,25 Judgng by the observatons obtaned for Q/Eff, t can be seen that the fnal cost of a Q aucton was around 5 to 6 per cent hgher than the theoretcal mnmum Eff. Ths seemed to be very stable across dfferent factor levels, wth the ecepton of factor B (Producton 24

25 capactes). When all the bdders had equal producton capactes, the fnal cost Q was lttle more than 2% hgher than Eff. Wth nequal producton capactes, ths dfference was around 9% to %. Also factor E (Bdders) had a mnor, but statstcally sgnfcant effect, wth Q/Eff decreasng when there were more bdders. The other factors were labeled by the medan test as statstcally nsgnfcant. The varables Q/SEff and QC/Eff behaved n a smlar way to Q/Eff. The factor effects were qualtatvely smlar, but larger for Q/SEff and smaller for QC/Eff. The results of QC/Eff show that the wnnng allocaton of a Q aucton was usually qute close to the effcent one, wth the producton cost, on average, only 2,6% hgher than the effcent cost. l 6.2 Effects on P2/Eff, P2/SEff and P2C/Eff The mean and medan results for the varables P2/Eff, P2/SEff and P2C/Eff are shown n Table 6.2. Table 6.2. Factorwse mean and medan results of P2/Eff, P2/SEff, and P2C/Eff. P2/Eff Mean Medan St.Dev. All,98,78,2 P2/SEff Mean Medan St.Dev. All,6,987,5 B: ProdCap Equal,28,,69 Inequal,267,26,3 C: Bd 3,68,43,98 4,64,42,9 5,26,257,7 D: Scope,96,78, 2,2,77,3 E: Bdders 5,239,23,2 3,57,33,85 B: ProdCap Equal,929,95,66 Inequal,83,73, C: Bd 3,993,975,7 4,972,956, 5,53,46,2 D: Scope,,993,2 2,,982,7 E: Bdders 5,54,43,2 3,958,94,86 P2C/Eff Mean Medan St.Dev. All,8,67,55 B: ProdCap Equal,5,47,3 Inequal,,,59 C: Bd 3,65,5,48 4,67,58,42 5,9,96,62 D: Scope,79,66,55 2,82,69,56 E: Bdders 5,96,82,64 3,65,59,39 Overall, the fnal cost of a P2 aucton was about 8% to 2% hgher than Eff, wth factors B and E havng vsble effects. The behavor of P2/Eff was somewhat smlar to that of Q/Eff, but wth hgher standard devatons, and factors B and E havng more effect. The 25

26 bg qualtatve dfference was the effect of factor C (Bds). Whle the ntal bds had no sgnfcant effect on Q/Eff, the values of P2/Eff were clearly ncreased when the ntal bds had been generated wth Method 5. The medan test labeled factors B, C and E as statstcally sgnfcant. Table 6.2. demonstrates that the results obtaned for P2/SEff and P2C/Eff qualtatvely resembled those of P2/Eff. Factor effects and the standard devatons were somewhat smaller for P2C/Eff. 6.3 Effects on P2-Q and P2C/QC P2 Q The varable (shortened to P2-Q) ests to represent the decrease n fnal cost P2 when quantty support was used nstead of ntellgent prce support. Addtonally, the P2C QC varable (shortened to P2C-QC) was observed to study the decrease the P2C correspondng decrease n producton cost. The results are shown n Table 6.3. Table 6.3. Factorwse mean and medan results of P2-Q and P2C-QC. P2-Q Mean Medan St.Dev. All,8 % 9,6 % 6,45 % P2C-QC Mean Medan St.Dev. All 4,83 % 4,27 % 4, % A: Shortlst Epress,59 % 9,6 % 6,48 % Full,2 % 9,6 % 6,42 % B: ProdCap Equal 8,84 % 7,76 % 5,29 % Inequal 2,77 % 3,9 % 6,9 % C: Bd 3 8,4 % 7,76 % 5,59 % 4 8,52 % 7,76 % 5, % 5 5,5 % 4,92 % 5,98 % D: Scope,69 % 9,6 % 6,46 % 2,93 % 9,6 % 6,44 % E: Bdders 5 3,6 % 3,9 % 6,75 % 3 8,56 % 7,76 % 5,25 % A: Shortlst Epress 4,74 % 4,8 % 4,7 % Full 4,92 % 4,35 % 3,95 % B: ProdCap Equal 4,4 % 3,78 % 2,9 % Inequal 5,62 % 5, % 4,74 % C: Bd 3 3,44 % 3,4 % 3,5 % 4 3,8 % 3,48 % 3,8 % 5 7,24 % 6,64 % 4,22 % D: Scope 4,68 % 4,8 % 3,95 % 2 4,98 % 4,38 % 4,6 % E: Bdders 5 5,95 % 5,2 % 4,43 % 3 3,7 % 3,53 % 3,7 % It can be seen that overall, the decrease P2-Q was about %, but less (about 8%) wth more bdders or equal producton capactes, and more (about 5%) when the ntal bds were generated wth Method 5. Judgng by the results obtaned for P2/Eff n Secton 6.2, the effect of factor C on P2-Q can be accounted to P2 beng hgher wth C (Bds) = 5. When, nstead, the producton costs ncluded n the wnnng allocatons of the P2 and Q auctons were compared, the dfference was not as large but stll postve. Qualtatvely, the effects of the factors on P2C-QC were smlar to those on P2-Q. If effcency s defned as the ablty to produce the tems at the lowest cost, t can now be stated that the Q auctons allocated the tems more effcently. 26

27 6.4 Effects on In-Q In Q The varable (shortened to In-Q) represents the prce decrease that occurred In durng a Q aucton. The mean and medan results are shown n Table 6.4. Table 6.4. Factorwse mean and medan results of In-Q. Mean Medan St.Dev. All 8,56 % 8,29 % 3,8 % A: Shortlst Epress 8,28 % 8,29 % 3,87 % Full 8,83 % 8,29 % 3,73 % B: ProdCap Equal 8,8 % 8,29 % 2,55 % Inequal 9,3 % 8,29 % 4,7 % C: Bd 3 7,77 % 6,66 % 3,74 % 4 7,45 % 6,78 % 3,2 % 5 2,45 % 9,93 % 3,8 % D: Scope 8,5 % 8,29 % 3,79 % 2 8,62 % 8,29 % 3,83 % E: Bdders 5 9,6 % 8,5 % 4,42 % 3 7,5 % 7,78 % 2,69 % The values of In-Q were farly stable, wth the means and medans varyng between 6% and 2%. Larger decreases n total cost were observed when there were more bdders (factor E) and when the ntal bds were generated wth Method 5 (factor C). Ths latter case s suspected to have occurred because In was generally hgher wth C = Bd 5 than wth other bd generaton methods. 6.5 Effects on QSucc The varable Qsucc measures the success rate of quantty support. It s defned by: ( QS succeeded ) N( QS used ) N Qsucc =. As defned n Secton 3.4, the last entry of the shortlst corresponds to the soluton obtaned for the QSP wthout any addtonal constrants. Quantty support s therefore defned to have succeeded when the bdder usng t dentfes the last entry of the shortlst as the most proftable one. When ths happens, the bdder s producton cost functon f = c ( q) F, 2K N + c q N q N has successfully been appromated by the obectve functon 27

28 N P = d Q of the quantty support problem. The mean and medan results of Qsucc are shown n Table 6.5. Table 6.5. Factorwse mean and medan results of Qsucc. Mean Medan St.Dev. All,758,783,242 A: Shortlst Epress,78,89,227 Full,736,737,254 B: ProdCap Equal,938,984,74 Inequal,578,588,48 C: Bd 3,748,769,246 4,75,772,25 5,775,829,229 D: Scope,757,797,245 2,758,778,239 E: Bdders 5,74,755,253 3,776,827,229 Large varaton was typcally observed for Qsucc, as demonstrated by the hgh standard devatons. The overall mean was,783. Very hgh success rates occurred when all the bdders had equal producton capactes: wth unequal producton capactes, the results were closer to those obtaned n [], where the producton capactes were also unequal. Factor A (Shortlst) had statstcal sgnfcance, as dd E (Bdders). 6.6 Effects on Eff/SEff The relaton between the true (Eff) and the appromatve (SEff) effcent cost s very nterestng, but t was not nvestgated n [], as Eff was not calculated. In ths study, both Eff and SEff were calculated n each smulaton, and the results for Eff/SEff are shown n Table 6.6. Factors A and C are not shown, as 28

29 Table 6.6. Factorwse mean and medan results of Eff/SEff. Eff/SEff Mean Medan St.Dev. All,839,837,32 B: ProdCap Equal,824,822,28 Inequal,854,853,29 D: Scope,844,843,32 2,833,83,32 E: Bdders 5,85,849,33 3,828,827,28 Table 6.6 shows that the means and medans of Eff/SEff were very stable between,82 and,86. The standard devatons were also low. Indvdual values observed for Eff/SEff were also qute stable, usually varatng between,75 and,9. All three factors were labeled statstcally sgnfcant by the medan test, but the effects were very small. Based on these results, t can be stated that under the cost functon settngs of ths study, SEff was about 2% hgher than Eff. 6.7 Aucton length Aucton length can be measured by the number of rounds. But snce each round conssted of a varable number of bdders usng prce or quantty support, the round length was varable. Another, perhaps more accurate measure of aucton length would be the number of tmes prce or quantty support was used durng the entre aucton. These are referred to as teratons. Nevertheless, the number of rounds s nterestng n ts own rght, as t corresponds to the decrease n total cost (because each round decreased the total prce by 2%). It should also be noted that the number of teratons s not comparable across all factor levels: when the number of bdders ncreases, so does the number of teratons. On the other hand, the number of rounds can be compared across all factor levels. Table 6.7 shows the means, medans and standard devatons of the numbers of teratons and rounds. Overall results are shown, as well as those obtaned wth dfferent levels of B (Producton capactes) and E (Bdders), and wth C (Bds) = 5. It can be observed that changng from equal to unequal producton capactes ncreased the number of teratons n the Q auctons, but decreased them n the P and P2 auctons. To a lesser etent, the same s true of the number of rounds. 29

30 Table 6.7. Aucton length measured n rounds and teratons. AUCTION LENGTH Overall Equal producton capactes Inequal prod. capactes Mean Medan St. Dev, Mean Medan St. Dev, Mean Medan St. Dev, Iter(P ) 4, ,3 46, ,75 34, ,3 Iter(P2 ) 42, ,84 45, , 39, ,8 Iter(Q ) 47, ,96 4,4 39 3,95 53,5 49 2,4 N(P2 ) 5,42 6 2,54 6,23 7 2,3 4,6 4 2,66 N(Q ),9 2,37,86,58,53 2,9 5 bdders 3 bdders Bd 5 Mean Medan St. Dev, Mean Medan St. Dev, Mean Medan St. Dev, Iter(P ) 22, 2 2,3 58, ,38 33, ,7 Iter(P2 ) 24,95 24,74 6, ,5 35, ,88 Iter(Q ) 34, ,6 59, ,43 48, ,9 N(P2 ) 4,8 5 2,63 6,3 7 2,29 3,92 4 2,28 N(Q ),86 2,77,52,62 2,36 2 2, Number of wnnng bdders Quantty support was found to decrease the number of wnnng bdders. As descrbed n Table 6.8, the mean number of wnnng bdders n a Q aucton was 2,56, as opposed to about 4, n the P and P2 auctons. The number of wnners also vared less n the Q auctons. The most effectve factor was B (Producton capactes): when all bdders had equal producton capactes, there were less wnners n all three auctons than when the producton capactes were unequal. But even n ths case, the Q auctons nearly always fnshed wth less wnnng bdders than the P and P2 auctons. It can also be noted that when the ntal bds were generated wth Method 5, the number of wnners ncreased strkngly n the P and P2 auctons, from around 3,6 to around 5. Meanwhle, the Q auctons were largely unaffected. Table 6.8. The number of wnnng bdders NUMBER OF WINNING BIDDERS Overall Equal producton capactes Inequal prod. capactes Mean Medan St.Dev, Mean Medan St.Dev, Mean Medan St.Dev, P 4, 4,25 3,4 3,87 4,8 5,8 P2 4,3 4,2 3,45 3,84 4,82 5,4 Q 2,56 2,62 2,6 2,24 3,7 3,44 Bd 3 Bd 4 Bd 5 Mean Medan St.Dev, Mean Medan St.Dev, Mean Medan St.Dev, P 3,65 4,94 3,6 3,89 5,7 5,26 P2 3,73 4,92 3,62 3,86 5,4 5,26 Q 2,58 3,6 2,58 3,62 2,54 2,62 3

31 6.9 Interpretaton of results The length of the shortlst (factor A) dd not greatly affect the results. For the varables Q/Eff, Q/SEff and QC/Eff, denotng the fnal cost of a Q aucton, smlar results were observed wth both levels of factor A. The success rate of quantty support was slghtly ncreased wth the epress verson of the shortlst. It can therefore be sad that the full verson of the shortlst was not necessary to mnmze the fnal cost. Instead, the smulatons where the full verson was used usually took 25%-5% more tme, due to the ncreased amount of calculatons. The producton capactes (factor B) were found to strongly affect the results. The effectve dfference between ntellgent prce support and quantty support, denoted by the varables P2-Q and P2C-QC, decreased when the bdders had equal producton capactes. But even then, the Q aucton tended to reach a lower fnal cost: n fact, the medan of QC/Eff n such cases was as low as,5, suggestng the effcent allocaton was nearly always found. Wth equal producton capactes, the effcent allocaton usually conssted of two bdders, each producng 3 unts of each tem. The success rate of quantty support was also strongly affected The ntal bds (factor C) had lttle effect on the outcome of the Q auctons. But n the P2 auctons, method 5 was clearly the one producng the worst results, ncreasng the fnal cost and the number of wnnng bdders. Meanwhle, methods 3 and 4 were vrtually ndstngushable from each other. Economes of scope (factor D) had very lttle effect on any of the results. One reason could be that the dfference between the two levels was rather small: wth Scope, the fed cost to produce two tems was,5 tmes the cost to produce one tem, and n Scope 2 the rato was,4. A hgher rato could have been tested for Scope : a lower one for Scope 2 seems unrealstc. Nevertheless, there was an ncentve to study the effect of economes of scope, as combnatoral auctons are only epected to have an advantage over non-combnatoral ones when economes of scope est. The number of bdders (factor E) had an effect on the fnal costs of both the Q and the P2 auctons: the fnal cost was observed to decrease when there were more bdders. The decrease was sharper n the P2 auctons. Smlar results were obtaned n [], suggestng that quantty support ncreases competton n an aucton, otherwse acheved by allowng more bdders to enter. But even wth 3 bdders, the Q auctons produced better results than the P2 ones, as denoted by the postve mean and medan of the varable P2-Q. 3

32 7 Concluson In my Master s Thess [], evdence was obtaned that quantty support mproved the results of a combnatoral aucton n many ways. Ths was done by smulatng combnatoral auctons wth and wthout quantty support, usng the same nput data. Fve nput parameters were consdered partcularly mportant, and were assgned multple values n order to study what effect ther varaton has on the aucton results: these parameters were referred to as factors. As the effects of many potentally nterestng factors could not be studed n [], ths study has attempted to etend ts results. The effects of three entrely new factors were studed now, and two old ones were assgned new values. To allow for a drect comparson, the aucton and smulaton desgns of ths study were dentcal to those of []. The new optmzaton engne enabled the so called effcent allocaton to be calculated, as well as auctons to be smulated wth up to 3 bdders. The results were qute smlar to those of []. The auctons wth quantty support were found to fnsh wth lower total cost to the auctoneer, and to allocate the tems more effcently. Some of the new factors were found to affect the results, but n all cases quantty support was preferable to prce support. Quantty support also had a stablzng effect on the aucton results, wth less varaton wthn each block of certan factor levels. The appromated effcent cost, ntroduced n [] when the true one could not be calculated, was found to be about 2% hgher than the true effcent cost. 32