Multi-Period Resource Allocation for Estimating Project Costs in Competitive Bidding

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1 Department of Industral Engneerng and Management Techncall Report No Mult-Perod Resource Allocaton for Estmatng Project Costs n Compettve dng Yuch Takano, Nobuak Ish, and Masaak Murak September, 2014 Tokyo Insttute of Technology Ookayama, Meguro-ku, Tokyo , JAPAN

2 Mult-Perod Resource Allocaton for Estmatng Project Costs n Compettve dng Yuch Takano School of Network and Informaton, Senshu Unversty Hgashmta, Tama-ku, Kawasak-sh, Kanagawa , JAPAN Nobuak Ish Faculty of Informaton and Communcatons, Bunkyo Unversty 1100 Namegaya, Chgasak-sh, Kanagawa , JAPAN Masaak Murak Department of Industral Engneerng and Management Graduate School of Decson Scence and Technology, Tokyo Insttute of Technology W9-73 Ookayama, Meguro-ku, Tokyo , JAPAN Abstract To establsh a proft-makng strategy n compettve bddng for project contracts, t s crucal for contractors to estmate project costs accurately. Although allocatng a large amount of resources to cost estmates allows contractors to prepare more accurate estmates, there s usually a lmt to avalable resources n practce. To maxmze a contractor s expected proft, ths paper develops a mult-perod resource allocaton method for estmatng project costs n a sequental compettve bddng stuaton. Our resource allocaton model s posed as a mxed nteger lnear programmng problem by makng a pecewse lnear approxmaton of the expected proft functons. Numercal experments examne the characterstcs of the optmal resource allocaton and demonstrate the effectveness of our resource allocaton method. Keywords: Resource allocaton, Cost estmaton, Project management, Compettve bddng, Mxed-nteger lnear programmng 1 Introducton Compettve bddng s wdely used to choose contractors under the most favorable condtons. As a typcal case, contractors, who have receved an nvtaton from a potental clent, submt bd prces. If a contractor s bd prce s the lowest among the compettors, s/he wll wn the project contract. The bd prce s then pad to the contractor, and s/he wll execute the project as the clent has requested. If the actual project cost s kept below the bd prce, ths project contract wll be proftable for the contractor. Otherwse, s/he wll suffer a loss on account of cost overruns. In such a process, a contractor s proft s hghly dependent on hs/her bddng strategy. Consequently, snce the semnal work by Fredman (1956), a consderable number of studes have dealt wth compettve bddng strateges (see, e.g., Engelbrecht-Wggans, 1980; Kng and Mercer, 1988; Rothkopf and Harstad, 1994; Stark and Rothkopf, 1979). In compettve bddng, contractors estmate the cost of completng a project and then determne the bd prce. Accordngly, the bd prce s markedly affected by the naccuraces n the estmated cost, and t s crucal for contractors to estmate the project cost accurately. Naert 1

3 and Weverbergh (1978) are the frst to consder the uncertanty about the estmated cost n a compettve bddng model. Ther model shows that the expected proft decreases as the uncertanty about the estmated cost ncreases. Kng and Mercer (1990) derve analytcal solutons to the compettve bddng models for varous dstrbutons of the estmated cost. Takano et al. (2014) establsh sequental compettve bddng strateges consderng naccurate cost estmates, whereas other studes (Kng and Mercer, 1990; Naert and Weverbergh, 1978) dscuss only one-shot bddng. As ponted out by Towler and Snnott (2012), the accuracy of the cost estmaton s postvely correlated wth the man-hours (MH) thrown nto makng the estmate. In addton, Chrstensen and Dysert (1997) devse a cost estmate classfcaton matrx that shows a clear relatonshp between the accuracy of the estmate and the amount of preparaton. These studes ndcate that the contractor can prepare an accurate cost estmate by spendng a large amount of resources; however, n most cases, there s a lmt to the resources avalable for an estmate. It s, therefore, essental to allocate the resources to a number of project contracts n a well-planned manner. Indeed, several studes (Ish et al., 2013, 2014; Takano et al., 2014) have demonstrated that a contractor s proft can be mproved by approprately allocatng the avalable MH for estmatng the costs to each project contract. Resource allocaton has been a subject of actve research over the past half a century (Ibarak and Katoh, 1988; Katoh et al., 2013; Patrksson, 2008), and t has a wde feld of applcaton, e.g., R&D project selecton (Chen and Zhu, 2011; Hedenberger and Stummer, 1999; Taylor III et al., 1982), marketng budget allocaton (Fscher et al., 2011; Soma et al., 2014; Venkatesan and Kumar, 2004), producton plannng (Bretthauer et al., 2006; Ventura and Klen, 1988; Zegler, 1982), project budget allocaton (Sato and Hrao, 2013), sponsored search aucton (Karande et al., 2013; Yang et al., 2012; Zhang et al., 2012) and cloud computng (Beloglazov et al., 2012; We et al., 2010; Xao et al., 2013). Fanran et al. (1999) nvestgate effcent resource allocaton for constructon project plannng, but they do not take account of compettve bddng. L and Womer (2006) solve a resource constraned project schedulng problem to formulate a bddng strategy n consderaton of due date requrements of projects. However, they gnore the relatonshp between the accuracy of the cost estmate and the resources allocated to the estmate. To the best of our knowledge, none of the exstng studes on resource allocaton have addressed the problem of estmatng project costs for compettve bddng. The purpose of the present paper s to establsh a novel method of allocatng lmted resources to the cost estmates of project contracts n a sequental compettve bddng stuaton. For ths purpose, we propose a mult-perod resource allocaton model, n contrast to the standard sngleperod models (Ibarak and Katoh, 1988; Katoh et al., 2013; Patrksson, 2008), to desgn a longterm resource allocaton strategy. Ths model ams at maxmzng the expected proft ganed from compettve bddng and s framed as a mxed nteger lnear programmng (MILP) problem by makng a pecewse lnear approxmaton of the expected proft functon. We demonstrate the effectveness of our method through numercal experments examnng the characterstcs of the optmal resource allocaton. The rest of the paper s organzed as follows. In the next secton, we frst confrm the 2

4 relatonshp between the accuracy of the cost estmaton and the amount of preparaton. We also formulate the expected proft ganed from each project contract on the bass of the effort n preparng estmates. Our mult-perod resource allocaton model s presented n Secton 3. The numercal results are reported n Secton 4. Fnally, conclusons are gven n Secton 5. 2 Cost Estmatng and Compettve dng Model Ths secton explans the correlaton between the cost estmaton accuracy and the effort to prepare an estmate. Then t descrbes the compettve bddng model n order to defne the expected proft from the project contracts. 2.1 Cost estmate classfcaton matrx In compettve bddng, contractors frst estmate the cost of a project and then decde a bd prce on the bass of the estmate. If hs/her estmate s relatvely hgh, t wll be dffcult for hm/her to wn the contract. Conversely, f hs/her estmate s relatvely low, s/he s lkely to wn the contract; however, the project wll eventually produce a loss. Therefore, t s crtcally mportant for contractors to estmate the project cost accurately. Table 1, whch was created from the cost estmate classfcaton matrx (Chrstensen and Dysert, 1997), shows the expected accuracy range of project cost estmates. In ths table, cost estmates fall nto fve classes. A Class 5 estmate s the lowest level of project defnton and s made wth the prmary objectve of screenng or checkng feasblty. By contrast, a Class 1 estmate s the closest to a full project defnton and s made to check an estmate or submt a bd (see, Chrstensen and Dysert, 1997). The Expected Accuracy Range ndcates the degree to whch the actual project cost wll vary from the estmated cost, and Preparaton Effort ndcates the amount of effort needed to prepare a gven estmate. We can see from Table 1 that the estmaton accuracy mproves as the preparaton effort ncreases. For nstance, 0.05% to 0.5% of the project cost s requred to make an estmate wth an accuracy of 5% to +10%. Table 1: Expected accuracy range of project cost estmates (Chrstensen and Dysert, 1997) Expected Accuracy Range Estmate Class Lower Lmt Upper Lmt Preparaton Effort Class 5 100% to 20% +40% to +200% 0.005% Class 4 60% to 15% +30% to +120% 0.01% to 0.02% Class 3 30% to 10% +20% to +60% 0.015% to 0.05% Class 2 15% to 5% +10% to +30% 0.025% to 0.1% Class 1 5% +10% 0.05% to 0.5% Note: Preparaton Effort s expressed as a percentage of the project cost. 3

5 2.2 Compettve bddng model Let us suppose that the contractor plans to bd on contract I = {1, 2,..., I} and estmates ts cost wth a preparaton effort of Class q Q = {1, 2,..., Q}. The estmated cost Ẽ,q s subject to an unavodable estmaton error, and thus, t s reasonable to treat t as a random varable. The contractor determnes a bd prce by puttng a markup m,q on the estmated cost. Therefore, hs/her bd prce s (1 + m,q )Ẽ,q, and f s/he wns the project contract, hs/her eventual proft wll be (1 + m,q )Ẽ,q C, where C s the cost of completng the project I. We defne P [b], the probablty of wnnng a contract I as a functon of the contractor s bd prce, b (see Appendx A for the detals). We wll set the markup to maxmze the expected proft, and thus, the contractor s expected proft obtaned from contract I can be expressed as follows: [ ] R,q = max E P [(1 + m,q )Ẽ,q]((1 + m,q )Ẽ,q C ), m,q where E[ ] s the mathematcal expectaton. The contractor s expected proft s largely dependent on the estmate class,.e., the estmaton accuracy of the project cost. To make the expected proft easer to handle, we wll use the scenaro-based approxmaton as n Takano et al. (2014). Let E (s),q be the estmated cost n scenaro s S and P s be the occurrence probablty of scenaro s S. Accordngly, the expected proft can be rewrtten as follows: R,q = max m,q s S P s P [(1 + m,q )E (s),q ]((1 + m,q)e (s),q C ). (1) 3 Resource Allocaton Model As mentoned n the prevous secton, large profts can be expected from project contracts by nvestng much effort n the cost estmate. It s often the case, however, that contractors are competng on a number of contracts, and the resources avalable for estmatng cost are usually lmted n each perod. In ths secton, we consder a mult-perod resource allocaton model and formulate t as a mxed nteger lnear programmng (MILP) problem. 3.1 Basc optmzaton model Let us suppose that the contractor develops a resource allocaton strategy for plannng perods, t T = {1, 2,..., T }. More precsely, s/he decdes w,t, the preparaton effort for estmatng the cost of project I n each perod t T. It s supposed that the contractor has already started cost estmates of projects I 0 ( I). Let W pre be the preparaton effort that has already been nvested n I; accordngly, W pre = 0 for I \ I 0. Smlarly to Chrstensen and Dysert (1997), the preparaton effort s represented as a percentage of the project cost C. Hence, the sum of preparaton costs n plannng perods t T 4

6 becomes C t T w,t for each I. In addton, the expected proft R for I s defned as a functon of the preparaton effort. Accordngly, the contractor s expected proft ganed from contract I s wrtten as R (W pre + t T w,t). Consequently, our objectve becomes maxmzng the functon, ( R W pre + ) w,t C w,t. (2) I t T I t T The budget for cost estmates s usually lmted n each perod t T. Therefore, we take nto account the followng constrants: C w,t B t ( t T ), (3) I where B t s the total budget avalable to cost estmates n perod t T. Addtonally, let T ( T ) be the perods durng whch the cost estmaton of project s able to be performed. For nstance, t would be useless for contractors to contnue to estmate the cost of a project after they have submtted ther bd prce. Accordngly, we also ncorporate the followng constrants: w,t 0 ( I, t T ), w,t = 0 ( I, t T ). (4) Now, our basc optmzaton model for determnng the allocaton of efforts, w,t, for estmatng the project cost s posed as follows: maxmze (w,t ) subject to ( R W pre + ) w,t C w,t I t T I t T C w,t B t ( t T ), I w,t 0 ( I, t T ), w,t = 0 ( I, t T ). (5) 3.2 Pecewse lnear approxmaton The expected proft functon, R, s approxmated by a pecewse lnear functon n the followng manner. Frst, we calculate the expected proft R,q for each contract I and each estmate class q Q accordng to equaton (1). Next, we make a pecewse lnear functon as shown n Fgure 1, where W,q s the preparaton effort of estmate class q Q for contract I. Specfcally, by ntroducng decson varables e,q correspondng to the nternal dvson rato, we approxmate the expected proft functon, R, as follows: ( R W pre + ) w,t e,q R,q (6) I t T I q Q 5

7 subject to the followng constrants: W pre + w,t = e,q W,q t T q Q e,q = 1 ( I), q Q e,q 0 ( I, q Q), {e,q q Q} = SOS2 ( I), ( I), (7) where the constrant {e,q q Q} = SOS2 s a specal ordered set type two (SOS2) constrant (Beale and Tomln, 1970). The SOS2 constrant mples that at most two consecutve elements of e,q, q Q can have nonzero values, and t s useful for makng pecewse lnear approxmatons of nonlnear functons. Due to ts usefulness, ths constrant s supported by standard mxed nteger programmng (MIP) solvers. Fgure 1: Illustraton of pecewse lnear functon representng the expected proft 3.3 Addtonal constrants Ths subsecton provdes addtonal constrants for the basc optmzaton model (5). Mnmal level of cost estmaton accuracy. If the accuracy of the cost estmaton s very low, the project costs may be underestmated. In ths case, although the contractor s lkely to wn the contracts, they wll cause a large loss. To mtgate the rsk of sufferng such a large loss, t s effectve for the contractor to guarantee a mnmal level of cost estmaton accuracy. To accomplsh ths, we ntroduce 0-1 decson varables, x {0, 1} ( I), (8) 6

8 whch represent whether the contractor wll bd or not on each contract I. followng constrants are mposed on the optmzaton model: W lox W pre + w,t t T x = 1 ( I 0 ), ( I), w,t W up,t x ( I \ I 0, t T ), Next, the (9) where W lo s the lower lmt on the preparaton effort for contract I, and W up,t s the upper lmt on t for contract I n perod t T. If any preparaton effort s nvested n contract, then x = 1 due to the second and the thrd lnes of constrants (9). It follows from the frst lne of constrants (9) that the sum of preparaton efforts nvested n contract s more than or equal to W lo, whch guarantees a mnmal level of cost estmaton accuracy. Monotoncty of preparaton effort. As the date of the compettve bddng approaches, contractors usually ncrease ther effort n makng the cost estmate n order to reflect the latest nformaton. It s also undesrable to dscontnue a cost estmate temporarly. Let s() and e() be the frst and last perods of estmatng the cost of project I, respectvely,.e., T = {s(), s() + 1,..., e()}. In vew of the above facts, we can place a monotoncty constrant on the preparaton effort as follows: W,0 w,s() w,s()+1 w,e() ( I), (10) where W,0 s the preparaton effort that the contractor nvested n the last perod, and accordngly, W,0 = 0 for I \ I MILP formulaton In the basc optmzaton model (5), the expected proft functon s replaced wth a pecewse lnear functon (6) subject to the constrants (7). Moreover, by appendng the constrants (8) (10) to the optmzaton model, our mult-perod resource allocaton model for estmatng the 7

9 project costs can be formulated as an MILP problem, maxmze (x ), (w,t ), (e,q ) subject to e,q R,q C I q Q I t T C w,t B t ( t T ), I w,t 0 ( I, t T ), w,t w,t = 0 ( I, t T ), + w,t = e,q W,q W pre t T q Q e,q = 1 ( I), q Q e,q 0 ( I, q Q), {e,q q Q} = SOS2 ( I), x {0, 1} ( I), W lox W pre + w,t ( I), t T x = 1 ( I 0 ), ( I), w,t W up,t x ( I \ I 0, t T ), W,0 w,s() w,s()+1 w,e() ( I). (11) 4 Numercal Experments Ths secton reports numercal results demonstratng the effectveness of our resource allocaton method. 4.1 Problem settng We supposed that there are 20 project contracts over a plannng horzon of nne perods, as shown n Fgure 2. It was assumed that projects = 1, 2, 6, 9, 12, 15, 18 are small-scale projects; projects = 3, 4, 7, 10, 13, 16, 19 are medum-scale projects; and projects = 5, 8, 11, 14, 17, 20 are large-scale projects. Note that the contractor has already started cost estmates of projects = 1, 2,..., 5,.e., I 0 := {1, 2,..., 5}. The costs of the other projects can only be estmated durng the fve perods pror to the date of bddng. We consdered the sx estmate classes as shown n Table 2. Note that Class 6 means that the contractor cancels bddng and therefore does not estmate the project cost. Moreover, snce the amount of resources requred for estmatng the cost may vary by ndustry sector, we consdered two cases. The preparaton effort of Case A s based on the cost estmate classfcaton matrx (Chrstensen and Dysert, 1997). In Case B, the contractor needs ten tmes as much effort as those of Case A to estmate project costs wth the same accuracy. The cost, C, of each project scale was 100, 300 and 1,000. It was assumed that the average number of compettors for each contract s fve and the mean and standard devaton of each 8

10 Project Small Preparaton 2 Small Preparaton 3 Medum Preparaton 4 Medum Preparaton 5 Large Preparaton Plannng Perods 6 Small Preparaton 7 Medum Preparaton 8 Large Preparaton 9 Small Preparaton 10 Medum Preparaton 11 Large Preparaton 12 Small Preparaton 13 Medum Preparaton 14 Large Preparaton 15 Small Preparaton 16 Medum Preparaton 17 Large Preparaton 18 Small Preparaton 19 Medum Preparaton 20 Large Preparaton Fgure 2: Plannng Perods Table 2: Estmate classes Preparaton Effort Estmate Class Accuracy Range Case A Case B Class 6 0% 0% Class 5 [ 30%, 60%] 0.015% 0.15% Class 4 [ 15%, 30%] 0.025% 0.25% Class 3 [ 10%, 20%] 0.050% 0.50% Class 2 [ 5%, 10%] 0.100% 1.00% Class 1 [ 0.5%, 1.0%] 0.500% 5.00% Note: Preparaton Effort s expressed as a percentage of the project cost. compettor s bd prce are 1.2 C and 0.2 C. To represent ths stuaton, we set the parameters of the probablty of wnnng as κ = 36, θ = C /30 and λ = 5 for all I (see Appendx A for the detals). The values of other parameters n problem (11) are shown n Table 3. To nvestgate how the optmal effort allocaton changes dependng on the avalable budget, we analyzed four values of B t, as shown n Table Expected proft from project contracts Equaton (1) was used to calculate the expected profts wth respect to each estmate class q Q. The estmated costs E (s),q of 10,000 scenaros were randomly generated from a beta dstrbuton, 9

11 Table 3: Parameter settng Case A Case B W up,t B t for all t T 0.1, 0.2, 0.4, 0.8 1, 2, 4, 8 W lo for all I 0.010% 0.10% for all I and t T 0.500% 5.00% W,0 for = 1, 2,..., % 0.05% W pre % 0.15% W pre % 0.05% W pre % 0.15% W pre % 0.05% W pre % 0.10% Table 4: Calculated expected proft from each project scale Optmal Markup Expected Proft Expected Order Estmate Class Small Medum Large Small Medum Large Small Medum Large Class Class Class Class Class Class where the mode of the beta dstrbuton was set to the project cost C, and the support of the dstrbuton was set to the correspondng accuracy range shown n Table 2. The occurrence probablty, P s, of each scenaro s S was 1/10,000. The optmal markup, m,q, was chosen by calculatng the expected proft (1) for m,q = 0.01, 0.02,..., Table 4 shows the optmal markup, expected proft, and expected order of each project scale, where the expected order s the average value of the wnnng bd, s S P s P [(1 + m,q)e (s),q ](1 + m,q)e (s),q. Fgure 3 shows the expected proft functons n Case A, where these functons were created from the expected profts n Table 4. We can see from Fgure 3 that they are all monotoncally ncreasng concave functons. 10

12 20 t f o r P d e t c e p x E Large Medum Small Preparaton Effort (%) Fgure 3: Expected proft functon of each project scale n Case A 4.3 Numercal results for Case A Fgures 4 7 show the optmal effort allocaton, w,t, dependng on the total budget n each perod n Case A. Here, the column labeled Pre s W pre for I, and the column labeled Sum s W pre + t T w,t for I. As mentoned above, the contractor has already started estmatng the costs of projects = 1, 2,..., 5. Hence, a specfc amount of effort must be allocated to these projects on account of the monotoncty constrants (10). For ths reason, most of the effort was allocated to projects = 1, 2,..., 5 n Fgure 4. As for the other projects = 6, 7,..., 20, effort was allocated only to the large-scale ones, = 7, 14, 17, 20. Ths suggests that the contractor should prortze the effort allocaton to large-scale projects when the total budget s lmted n each perod. Snce the total budget was ncreased n Fgure 5, effort was allocated to projects of dfferent scales. Nevertheless, Fgure 5 s smlar to Fgure 4 n that large-scale projects were preferred. More precsely, n Fgure 5, an effort of 0.025% was allocated to each large-scale project, whereas less effort was allocated to many small-scale and medum-scale projects. In Fgures 6 and 7, the effort was nearly evenly allocated to all projects. Moreover, we should notce, n contrast to Fgures 4 and 5, that the amount of effort allocated to several large-scale projects was less than that allocated to the small-scale and medum-scale projects. In other words, the contractor wth a suffcent budget should focus more on evenly allocatng effort to all projects than allocatng most of the effort to large-scale projects. 4.4 Numercal results for Case B Fgures 8 11 show the optmal effort allocaton, w,t, dependng on the total budget n each perod n Case B. We can see that Fgures 8, 9 and 10 are smlar to Fgures 4, 5 and 6, respectvely. However, Fgure 11 s dstnctly dfferent from Fgure Specfcally, an effort of 0.1% (Class 2) was

13 Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 4: Optmal effort allocaton n Case A (B t = 0.1 for all t T ) Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 5: Optmal effort allocaton n Case A (B t = 0.2 for all t T ) allocated to many projects n Fgure 7, whereas 0.5% (Class 3) was allocated to all projects n Fgure 11. Ths means that allocatng an effort of over 0.5% to each project was not worth t n Case B. We should now recall that the cost estmaton was more expensve n Case B than n Case A. Thus, the optmal amount of effort allocated to each contract n Case B was less than that n Case A. Fgure 11 ndcates that the cost estmates for several projects, e.g., = 6, 7, 8, were fnshed n a sngle perod. However, some contractors may prefer estmatng the cost of each project over multple perods. In ths case, t s only necessary to set the upper lmt, W up,t, 12

14 Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 6: Optmal effort allocaton n Case A (B t = 0.4 for all t T ) Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 7: Optmal effort allocaton n Case A (B t = 0.8 for all t T ) to a lower value n our resource allocaton model. 4.5 Expected proft/order and computaton tme Fgures 12 and 13 show the contractor s expected profts n Case A and Case B, respectvely. As explaned above, the cost estmaton was more expensve n Case B than n Case A. Hence, the expected profts n Fgure 13 were lower than those n Fgure 12. These fgures also ndcate a 13

15 Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 8: Optmal effort allocaton n Case B (B t = 1 for all t T ) Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 9: Optmal effort allocaton n Case B (B t = 2 for all t T ) relatonshp between the expected proft and the budget for cost estmates. For nstance n Case A, ncreasng the total budget from 0.1 to 0.2 n each perod produced an addtonal expected proft of about 50. On the other hand, by ncreasng the total budget from 0.4 to 0.8 n each perod, the expected proft mproved by only about 15. Fgures 14 and 15 show the expected orders for all project contracts n Case A and Case B. Snce the expected order represents the project executon cost after wnnng the contracts, the 14

16 Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 10: Optmal effort allocaton n Case B (B t = 4 for all t T ) Plannng Perods Project Pre Sum 1 Small Small Medum Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Small Medum Large Fgure 11: Optmal effort allocaton n Case B (B t = 8 for all t T ) workload requrements for executng the project can be estmated from these fgures. All computatons were conducted on a Wndows 7 personal computer wth a Core 7 Processor (2.80 GHz) and 8 GB memory. FICO Xpress was used to solve the optmzaton problems. The computaton tme of solvng the MILP problem (11) was always less than 0.05 seconds. Thus, even f the number of projects and perods are ncreased, we expect that our resource

17 Expected Proft Expected Proft Total Budget n Each Perod Total Budget n Each Perod Fgure 12: Expected proft n Case A Fgure 13: Expected proft n Case B Expected Order Expected Order Total Budget n Each Perod Total Budget n Each Perod Fgure 14: Expected order n Case A Fgure 15: Expected order n Case B allocaton problem can be solved n a reasonable amount of tme. 5 Concluson Ths paper descrbed a resource allocaton method for estmatng the costs of projects n a sequental compettve bddng stuaton. Moreover, t descrbed numercal experments showng the valdty of the method. In compettve bddng, t s essental for contractors to estmate project costs accurately. Although prevous studes (Chrstensen and Dysert, 1997; Towler and Snnott, 2012) ndcate a close connecton between the accuracy of a cost estmaton and resources allocated to t, a resource allocaton model for estmatng these costs has never been examned. In vew of these facts, we developed a framework for approprately allocatng resources to cost estmates, whch s the major contrbuton of ths research. Specfcally, the expected profts can be calculated from the compettve bddng model. A pecewse lnearzaton technque s used to represent expected proft functons, and the mult-perod resource allocaton model s posed as a mxed nteger lnear programmng (MILP) problem. Ths versatle approach can be appled to resource/budget 16

18 allocaton problems n a varety of areas. The numercal results demonstrated that the resources used to estmate costs should be preferentally allocated to large-scale project contracts when the budget for cost estmates s lmted. In contrast, when there s suffcent budget for estmatng the cost, t s benefcal to evenly allocate the resources to all contracts to be bd on. Moreover, snce our resource allocaton model allows one to analyze the relatonshp between the expected proft and the budgets for the estmates, t wll be of practcal value. We may say that ths research s of essental mportance for generatng profts n compettve bddng; however, there are stll some ssues to be tackled. From a practcal vewpont, the detals of project contracts (.e., date of bddng, project scale, amount of effort necessary for estmatng the cost, and so on) are subject to change. Hence, a drecton of future research s to ncorporate the uncertanty about the detals of the project contracts nto the resource allocaton model. Addtonally, although our resource allocaton model supports the contractor s decsonmakng from a long-term perspectve, more flexble methods mght be needed to determne a resource allocaton dynamcally n response to the announcement of a new compettve bddng. Acknowledgments Ths work was supported by Grant-n-Ad for Scentfc Research (C) by the Japan Socety for the Promoton of Scence. Appendx A Probablty of Wnnng The functon P,.e., the probablty of wnnng a contract s necessary to calculate the expected proft (1). By followng Takano et al. (2014), ths appendx descrbes the formula for computng the probablty of wnnng derved by Fredman (1956), who assumes that the number of compettors follows a Posson dstrbuton and ther bd prces follow dentcal gamma dstrbutons. Let us suppose that the bd prces of the compettors for contract have the same probablty densty functon: F (y) = y κ 1 exp( y/θ ) (κ 1)! θ κ, y 0, (A.1) where y s the bd prce, κ ( 1) s a shape parameter, and θ (> 0) s a scale parameter. The mean and varance of the gamma dstrbuton (A.1) are κ θ and κ θ 2, respectvely. Moreover, suppose that the number of compettors who bd for contract has the followng probablty mass functon: G (k) = λk k! exp( λ ), k = 0, 1, 2,..., (A.2) 17

19 where k s the number of compettors who bd for contract, and λ (> 0) s a parameter whch represents both the mean and varance of the Posson dstrbuton (A.2). It then follows from Fredman (1956) that ( ( k P [b] = G (k) F (y) dy) = exp ( λ b k=0 where b s a contractor s bd prce. 1 exp ) κ ( bθ 1 l=0 ( ))) 1 b l, (A.3) l! θ It s often the case that the results of compettve bddng are always announced. So t s possble to estmate the parameters κ, θ and λ by studyng prevous bddng data of potental compettors. References Beale, E. M. L., & Tomln, J. A. (1970). Specal facltes n a general mathematcal programmng system for non-convex problems usng ordered sets of varables. Proceedngs of the 5th Internatonal Conference on Operatons Research, Beloglazov, A., Abawajy, J., & Buyya, R. (2012). Energy-aware resource allocaton heurstcs for effcent management of data centers for cloud computng. Future Generaton Computer Systems, 28, Bretthauer, K. M., Shetty, B., Syam, S., & Vokurka, R. J. (2006). Producton and nventory management under multple resource constrants. Mathematcal and Computer Modellng, 44, Chen C. M., & Zhu, J. (2011). Effcent resource allocaton va effcency bootstraps: An applcaton to R&D project budgetng. Operatons Research, 59, Chrstensen, P., & Dysert, L. R. (1997). Cost estmate classfcaton system. AACE Internatonal Recommended Practce, 17R-97. Retreved from Analyss/ 17r-97.pdf Engelbrecht-Wggans, R. (1980). Auctons and bddng models: A survey. Management Scence, 26, Fanran, O. O., Love, P. E., & L, H. (1999). Optmal allocaton of constructon plannng resources. Journal of Constructon Engneerng and Management, 125, Fscher, M., Albers, S., Wagner, N., & Fre, M. (2011). Dynamc marketng budget allocaton across countres, products, and marketng actvtes. Marketng Scence, 30, Fredman, L. (1956). A compettve-bddng strategy. Operatons Research, 4, Hedenberger, K., & Stummer, C. (1999). Research and development project selecton and resource allocaton: A revew of quanttatve modellng approaches. Internatonal Journal of Management Revews, 1,

20 Ibarak, T., & Katoh, N. (1988). Resource Allocaton Problems: Algorthmc Approaches. Cambrdge: MIT Press. Ish, N., Takano, Y., & Murak, M. (2013). A two-step bddng prce decson algorthm under lmted man-hours n EPC projects. Proceedngs of the 3rd Internatonal Conference on Smulaton and Modelng Methodologes, Technologes and Applcatons (SIMULTECH), Ish, N., Takano, Y., & Murak, M. (2014). An order acceptance strategy under lmted engneerng man-hours for cost estmaton n engneerng-procurement-constructon projects. Internatonal Journal of Project Management, 32, Karande, C., Mehta, A., & Srkant, R. (2013). Optmzng budget constraned spend n search advertsng. Proceedngs of the 6th ACM Internatonal Conference on Web Search and Data Mnng (WSDM 13), Katoh, N., Shoura, A., & Ibarak, T. (2013). Resource allocaton problems. In P. Pardalos, D.-Z. Du, & R. L. Graham (Eds.), Handbook of Combnatoral Optmzaton (2nd Ed., pp ). New York: Sprnger. Kng, M., & Mercer, A. (1988). Recurrent compettve bddng. European Journal of Operatonal Research, 33, Kng, M., & Mercer, A. (1990). The optmum markup when bddng wth uncertan costs. European Journal of Operatonal Research, 47, L, H., & Womer, K. (2006). Project schedulng n decson-theoretc compettve bddng IEEE Congress on Evolutonary Computaton, Naert, P. A., & Weverbergh, M. (1978). Cost uncertanty n compettve bddng models. Journal of the Operatonal Research Socety, 29, Patrksson, M. (2008). A survey on the contnuous nonlnear resource allocaton problem. European Journal of Operatonal Research, 185, Rothkopf, M. H., & Harstad, R. M. (1994). Modelng compettve bddng: A crtcal essay. Management Scence, 40, Sato, T., & Hrao, M. (2013). Optmum budget allocaton method for projects wth crtcal rsks. Internatonal Journal of Project Management, 31, Soma, T., Kakmura, N., Inaba, K., & Kawarabayash, K. (2014). Optmal budget allocaton: Theoretcal guarantee and effcent algorthm. Proceedngs of the 31st Internatonal Conference on Machne Learnng (ICML 14), Stark, R. M., & Rothkopf, M. H. (1979). Compettve bddng: A comprehensve bblography. Operatons Research, 27,

21 Takano, Y., Ish, N., & Murak, M. (2014). A sequental compettve bddng strategy consderng naccurate cost estmates. OMEGA, The Internatonal Journal of Management Scence, 42, Taylor III, B. W., Moore, L. J., & Clayton, E. R. (1982). R&D project selecton and manpower allocaton wth nteger nonlnear goal programmng. Management Scence, 28, Towler, G., & Snnott, R. K. (2012). Chemcal Engneerng Desgn (2nd Ed.). Waltham: Butterworth-Henemann. Venkatesan, R., & Kumar, V. (2004). A customer lfetme value framework for customer selecton and resource allocaton strategy. Journal of Marketng, 68, Ventura, J. A., & Klen, C. M. (1988). A note on mult-tem nventory systems wth lmted capacty. Operatons Research Letters, 7, We, G., Vaslakos, A. V., Zheng, Y., & Xong, N. (2010). A game-theoretc method of far resource allocaton for cloud computng servces. The Journal of Supercomputng, 54, Xao, Z., Song, W., & Chen, Q. (2013). Dynamc resource allocaton usng vrtual machnes for cloud computng envronment. IEEE Transactons on Parallel and Dstrbuted Systems, 24, Yang, Y., Zhang, J., Qn, R., L, J., Wang, F. Y., & Q, W. (2012). A budget optmzaton framework for search advertsements across markets. IEEE Transactons on Systems, Man and Cybernetcs Part A: Systems and Humans, 42, Zhang, W., Zhang, Y., Gao, B., Yu, Y., Yuan, K., & Lu, T. Y. (2012). Jont optmzaton of bd and budget allocaton n sponsored search. Proceedngs of the 18th ACM SIGKDD Internatonal Conference on Knowledge Dscovery and Data Mnng (KDD 12), Zegler, H. (1982). Solvng certan sngly constraned convex optmzaton problems n producton plannng. Operatons Research Letters, 1,

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