22d Iteratioal Cogress o Modellig ad Simulatio, Hobart, Tasmaia, Australia, 3 to 8 December 207 mssaz.org.au/modsim207 O the Set-Uio Budget Sceario Problem J Jagiello ad R Taylor Joit Warfare Mathematical Sciece Joit ad Operatioal Aalysis Divisio Defece Sciece ad Techology Group Caberra ACT Email: erzy.agiello@dst.defece.gov.au Abstract: At MODSIM 205 (Taylor, 205) formalised the Budget Sceario Problem as a simplified mathematical formulatio of the problem preseted by (Order, 2007, 2009). I the Budget Sceario Problem a list of iitiatives is provided each with a aticipated cost. Each iitiative is scored agaist a umber of scearios with a value idicatig how useful the iitiative is agaist that sceario. For a collectio of iitiatives the total score is calculated as a sum of best iitiative scores withi the collectio for each sceario. I this paper we exted the mathematical model to take accout of the depedecy coditios expressed i words (see Table 2) i (Order, 2007, 2009) ad to also accommodate the represetatio of syergies betwee iitiatives. Sice this extesio is aalogous to the Set-Uio Kapsack Problem (Goldsmith, 994) we call this formulatio the Set-Uio Budget Sceario Problem. I mathematical terms the Set-Uio Budget Sceario Problem ca be expressed as: max ( + ), () k = Subect to where, (2) {0,} is the decisio to iclude iitiative or ot, is the probability of sceario i, is the value of score for sceario i ad iitiative, is the measure of idepedece for iitiative, β k is the value of iterdepedecy betwee iitiative ad k, c is the cost of iitiative ad B is the cost boud. The above formulatio was tested o (Order, 2009) data but exteded to differet levels of depedecy with results for depedecy levels ragig from idepedet (Depedecy Level = 0) to fully depedet (Depedecy Level = 4) preseted below i Figure. 9.7 Score 9.2 8.7 8.2 7.7 Score for Dep.Level = 0 7.2 6.7 6.2 Score for Dep.Level = 4 5.7 5.2 4.7 4.2 4 7 0 3 6 9 22 25 28 Cost Boud B Figure. Score vs Cost Boud for Depedecy Levels of 0 ad 4 Results obtaied from formulatio () ad (2) cofirm the results produced by (Order, 2009) with some exceptios where (Order, 2009) appears to have ot take ito accout some depedecies (for example B eeds K ). The proposed formal approach has bee applied to the iterdepedecy cotext successfully ad the obtaied results are very ecouragig ad applicable to the force desig domai. Keywords: Set-uio kapsack problem, budget optimisatio, decisio makig, iterdepedecy 653
. INTRODUCTION I two articles (Order, 2007, 2009), a problem is preseted ad aalysed through a series of fictioal discussios betwee cliets ad aalysts. What appears at first to be a prioritizatio problem is evetually see to be a budget-value problem i which the best value for a rage of budgets is sought. A list of iitiatives is provided each with a aticipated cost. Each iitiative is scored agaist a umber of scearios with a value of -0 idicatig how useful the iitiative is agaist that sceario. As well each sceario is assiged a probability idicatig the likelihood that that sceario is occurrig at ay give time. Table represets the Iitiative Sceario Table (reproduced from (Order, 2009)). Coditios are also specified i the text (Order, 2009) whereby to be effective some iitiatives require other iitiatives to be preset. This is show here i Table 2. Table. Order (2009) Iitiative Sceario Table 2. (Order, 2009) Requiremets for Depedecy For a collectio of iitiatives the total value is calculated by summig the product of each sceario probability by the best value obtaied by ay iitiative for that sceario i the collectio. This iitiative ca be thought of as the best tool i the toolbox (collectio) for the particular ob (sceario), while the total value reflects the expected ability of the toolbox (iitiative collectio) to address ay sigle ob (sceario). Thus the iitiatives O, M, ad N have a total cost of + + 3 = 5 ad a total value of 0.3 * 0 + 0. * 3 + 0.4 * 8 + 0.2 * 6 = 7.7. This situatio ca be modelled as a weighted bipartite graph G(X, Y) with iitiatives X ad scearios Y i which there are weights c (costs) associated with each X, ad weights (values) associated with each edge i; i Y ; X. 2. SET-UNION BUDGET SCENARIO (SUBS) PROBLEM DEFINITION Give a budget boud B the Set-Uio Budget Sceario Problem ca be stated as: Istace: A set X of items with a weight (cost) associated with each item ; a collectio of scearios Y; weights (values) associated with each edge i; i Y ; X ; a cost boud B. B as well as all costs ad weights are o-egative reals. Optimizatio: Fid a subset S of X with maximum total value, whose total cost is bouded by B takig ito accout depedecies. Depedecy is expressed for each sceario by the x depedecy matrix D where is the umber of iitiatives. The value of each elemet dl of the depedecy matrix varies from zero to four, represetig the level of depedecy that iitiative l has o iitiative. A value of zero implies that the iitiative l is etirely 654
idepedet of iitiative. A o-zero value of dl implies that iitiative l has a certai o-zero level of depedecy o iitiative. Table 3 shows the depedecy values described by (Christese, 20), ad matchig idepedet capability values α with liear scalig betwee 0 ad. Table 3. Depedecy ad Idepedet Capability Levels Assumig that a iitiative l ca achieve its idepedet capability at the level α l i the scale betwee 0 to of its capability the the remaiig part of the full capability reflected i this scale is ( - α l ) ad will be shared betwee iterdepedet iitiatives. If we assume proportioal distributio of depedecy for each iitiative from iitiative l tha the sharig formula proposed by (Wag, 207) will be as follows: dl β l = ( αl )( ). (3) d k= If we express depedecy as a cotiuous fuctio istead of a discreet oe tha the sharig formula ca be described as l αl = ( αl )( ) α β (4) k= where α l, α l ad α are values obtaied from a cotiuous fuctio for appropriate d max, dl ad d values. The cotiuous fuctio ca be derived by ay curve fittig method, for example least squares regressio, iterpolatio, Fourier approximatio etc. I mathematical terms the Set-Uio Budget Sceario Problem ca be expressed to make explicit the idepedet ad depedet capabilities of each iitiative as follows: Maximise Subect to m simax i= c = x ν + i α β k xk x (5) k=, (6) where: i is the sceario idex, m is the umber of scearios, is the iitiative idex, is the umber of iitiatives, {0,} is the decisio to iclude iitiative or ot, is the probability of sceario i, is the measure of idepedece for iitiative, is the value of score for sceario i ad iitiative, β k is the value of iterdepedecy betwee iitiative ad k (see Table 2), c is the cost of iitiative (see Table ), B is the cost boud. 655
The implemetatio was based o exhaustive search of the iitiative space supplemeted by the Bellma optimizatio priciple (Mulhollad, 206). This provides exact optimal solutios across the budget rage which was compared to the results of (Order, 2009). As a geeralizatio of the Budget Sceario Problem the Set-Uio Budget Sceario Problem is NP-complete ad so we should ot expect there to be efficiet algorithms to solve this problem exactly for large. This is discussed i more detail i (Taylor, 205) where approximatio results for the Budget Sceario Problem are also provided. 3. SET-UNION BUDGET SCENARIO (SUBS) RESULTS The above formulatio was tested o (Order, 2009) data ad the followig results, showig score vs cost boud were obtaied (see Figure 2). Results obtaied from our formulatio cofirm the results produced by (Order, 2009) with some exceptios where (Order, 2009) has ot take ito accout some depedecies (e.g. B eeds K see Table 2). Sesitivity aalysis shows that there is a big differece betwee optimal solutios for depedecy level 0 (iitiatives are idepedet) ad 4 (depedecy is madatory). Optimal solutios for depedecy levels, 2 ad 3 are the same. Testig the sesitivity of optimal solutios for differet types of cotiuous fuctios (liear, expoetial) did ot alter the results of the optimizatio due to the fact that these fuctios were costructed based o discrete values from Table 3, although this may ot always be the case. Score 9.7 A,B,O for Level = 0 A,B,O B,K,O for Level = 4 9.2 H,D,O B,O F,L or A,L,D 8.7 L,B,O for Level =,2,3 L,D,O J,O L,O 8.2 7.7 7.2 6.7 6.2 5.7 5.2 M,O O N,O M,O D,N,O D,M,O N,M,O N,O B,M,O Score for Depedecy Level = 4 Score for Depedecy Level = 3 Score for Depedecy Level = 2 Score for Depedecy Level = Score for Depedecy Level = 0 4.7 4.2 M Cost Boud B 3 5 7 9 3 5 7 9 2 23 25 27 29 Figure 2. Score vs Cost Boud for Depedecy Levels Ragig from 0 to 4 We ote that there may be may iitiative subsets with the same score withi a give cost boud B i which case we show a subset with the least cost for that score. This subset (or subsets) would correspod to the best choice of iitiatives that achieve a give score. For example the subsets {F,L} ad {A,L,D} produce equal scores of 9. with equal costs of 27, ad there are o other subsets with scores at least 9. ad costs at most 27. 656
If we compare Figures 2 ad 3 we otice that the cost boud rage i Figure 2 is a fractio of that of Figure 3. This is because optimal iitiatives for cost boud 29 could ot be improved further with icreasig cost rage, but the umber of solutios with equal score grew ad declied with icreasig cost rage up to 47. The explaatio of this pheomeo is due to the fact that umber of combiatios is maximized for a cost of aroud a half of the total cost of all iitiatives (i other words choose k is maximized for k=/2). Aalysis of the umber of local optimums as a fuctio of the cost boud B preseted below i Figure 3 shows that the peak is aroud the 70 80 cost boud ad depeds o the depedecy level. Depedecy level 0 ad 4 produced the highest umber of local optimums. We ote that the umber of optimums will grow expoetially with the umber of iitiatives ad depedecy levels which will impact the calculatio time of this approach for very large. Number of Local Optimums 200 50 00 50 Depedecy Level = 4 Depedecy Level = 3 Depedecy Level = 2 Depedecy Level= Depedecy Level = 0 0 2 3 4 5 6 7 8 9 0 2 3 4 Cost Boud B Figure 3. Number Local Optimums vs Cost Boud for Depedecy Levels Ragig from 0 to 4 4. APPLICATION OF SET-UNION BUDGET SCENARIO (SUBS) PROBLEM TO THE SCME DATA The implemetatio of a oit ad itegrated approach to the developmet of future Defece capability has led to a icreasig iterest i the depedecies ad iterdepedecies across proects, products ad programs (defied by sets of proects ad products). Those depedecies ad iterdepedecies should be addressed durig the coceptual desig phase i the Capability Life Cycle (CLC). I particular a proect/product may deped o other proects/products to provide operatioal services. These may be categorised as Sesig(S), Uderstadig (U), Decisio-makig (D), Egagemet (E), Physical Mobility (M), Iformatio Mobility (I), ad Logistics ad support (L), or simply SUDEMIL (Lowe, 205). These services ca also be defied betwee programs from the proects/products they cotai. Based o available depedecy data for the subset of SUDEMIL services for Sesig (S), Commad ad Cotrol (C), Mobility (M) ad Egagemet (E) (SCME) as well as cost ad performace from (Order, 2009) we calculated optimum decisios for each cost boud B. If we compare these results with the origial (Order, 2009) results we ca see substatial differeces due to the impact of the differet depedecy relatioships (see Figure 4). The score for the SCME data is geerally smaller, ad at most equal to the correspodig score for the (Order, 2009) data (depedecy level 0). We also ote the similarity betwee Figure 2 (Order, 2009) data ad Figure 4 SCME data regardig differet depedecy levels. Let the cost boud B rage from B 0 to B m. We ca measure the differece betwee the two depedecy curves as follows B m depedecy = 0 depedecy> 0 B itegral_distace = ( v v ) db (7) 0 657
average_score_distace = itegral_distace/( B B ) 0 (8) The the distace betwee the (Order, 2009) ad SCME curves accordig to (7) ad (8) is as follows: itegral distace = 2.96 i score by cost uits average score distace = 0.4469 i score uits. The itegral distace 2.96 represets the area i Figure 4 betwee the SCME curve ad Order curve ad is quite sigificat. To see this we ote that by compariso both the itegral distace ad the average score distace applied to Figure 2 for depedecy levels equal, 2 ad 3 are 0. Score m 9. 8.6 8. SCME for Depedecy > 0 7.6 Order for Depedecy = 0 7. 6.6 3 5 7 9 3 5 7 9 2 23 25 27 29 Cost Boud B Figure 4. Compariso of SCME ad Order(2009) Results 5. CONCLUSIONS I this paper we exted the mathematical formalizatio give by (Taylor, 205) of the problem described by (Order, 2007, 2009) to also ecompass the depedecies described there i word form. We also show how this allows for more refied depedecy data i the form of depedecy levels proposed by (Christese, 200), as well as service based depedecies i the form of the SUDEMIL framework of (Lowe, 205). The ovelty of this paper is the formal defiitio of the Budget Sceario Problem described by (Order, 2007, 2009) ad the itroductio of differet depedecy levels for both the data provided by (Order, 2009) ad the SCME case study provided here. Based o our experimets the time differeces i obtaiig optimal solutios for the (Order, 2009) ad SCME data were egligible. Further testig might be required for more complex problems to improve the efficiecy of our optimizer. Further research will be carried out to exted the Order idea to more sophisticated optimizatio requiremets as well as to provide alterative optimal solutios. 658
REFERENCES Christese M.B. (20). A Method for Measurig Programmatic Depedecy ad Iterdepedecy betwee DOD Acquisitio Program, Thesis, AFIT/GSE/ENV/-D02DL, Air Force Istitute of Techology, Wright- Patterso Air Force Base, Ohio. Goldsmith O, Nehme D, ad Yu G, (994). Note: O the set-uio kapsack problem, Naval Research Logistics (NRL), 4(6), pp 833-842. Lowe D. (205). A quick overview of SCMILE Services ad their evolutio, persoal correspodece, DSTG, JOAD. Mulhollad M. (206), Applied Process Cotrol, Publisher: Joh Wiley & Sos, ISBN: 97835273484. Order N. (2007). The priority list, PHALANX - The Bulleti of Military Operatios Research, 40 (3), pp. 8-0. Order N. (2009). The priority list - revisited, PHALANX - The Bulleti of Military Operatios Research, 42 (4), pp. 23-34. Taylor R. (205). Algorithmic complexity of two defece budget problems, MODSIM 205, Gold Coast, Queeslad, Australia. Wag Yue-Ji. (207). Applicatio of Mathematical Programmig to Prioritisig Iterdepedet Defece Ivestmet Programs, MODSIM207, Hobart, Tasmaia, Australia. 659