Uncertainty operations with Statool. Jianzhong Zhang. A thesis submitted to the graduate faculty

Transcription

1 Uncertanty oeratons wth Statool by Janzhong Zhang A thess submtted to the graduate faculty n artal fulfllment of the requrements for the degree of MASTER OF SCIECE Maor: Comutng Engneerng Program of Study Commttee: Danel erleant, Maor Professor Gerald Sheblé Soumendra ath Lahr Iowa State Unversty Ames, Iowa 00 Coyrght Janzhong Zhang, 00. All rghts reserved.

2 Graduate College Iowa State Unversty Ths s to certfy that the master s thess of Janzhong Zhang has met the thess requrements of Iowa State Unversty Maor Professor For the Maor Program

3 Table of Contents Lst of Fgures... v Lst of Tables... v Acknowledgements...v Abstract... Introducton.... Interval mathematcs Interval-based analyss... 5 arrowng the enveloes around results usng correlaton.... Facts about correlaton.... Jont dstrbutons....3 Interval-valued correlatons....4 Legal and llegal correlaton values Soluton Aromate soluton Addtonal constrants gotten from correlaton onlnear otmzaton to remove ecess wdth Imrovng results by addng constrants to LP Imroved smle method How to fnd the ntal feasble soluton How to decde the termnaton condton and enterng varable How to determne the leavng varable Decreasng comutng Alyng method An eamle onlnear otmzaton Local and global otmums Classcal theory of unconstraned otmzaton Fndng a soluton teratvely Search methods: drect and gradent Convertng constraned to unconstraned otmzaton Our case Enhancement of functons Transortaton method ackground on the transortaton smle method Ecetons n fndng the ntal soluton Adataton to the unknown deendency case Test result: Cascadng oeratons Soluton Relatonal oeratons Relatonal oeratons on ntervals Relatonal oeratons on random varables Comle eressons... 47

4 v 3.4. Eresson edtor Lmtatons on evaluatng eressons Ecess wdth n eressons Software archtecture Overvew Inut/outut The user nterface Man rmary wndow: the oeraton wndow Other rmary wndows: the data edtor and the vew wndows Wndows management The logcal layer The comutng layer Comonent desgn and mlementaton Overvew Oeraton and roertes wndows Oeraton wndow Proertes wndows of the oeraton wndow Other rmary wndows Lower levels of logcal layer Utlty functons and nternal data structure Access onts to the comutng layer The comutng layer Convertng data General (legacy) smle method Transortaton smle method Incororatng correlaton as a constrant Elanaton of the results Eerments Dscusson References... 98

5 v Lst of Fgures Fgure.. Probablty bounds... 0 Fgure 3.. Convert CDF to IDF Fgure 3.. Result for oeraton Fgure 3.3. Result for +y Fgure 3.4. Result for +y+z Fgure 3.5. Eresson edtor Fgure 3.6. Error nformaton for eersson edtor Fgure 4.. Archtecture Fgure 4.. Oeraton wndow Fgure 4.3. Data edtor Fgure 4.4. Data vew Fgure 4.5. Logcal layer Fgure 4.6. Package vew for the comutng layer Fgure 5.. Oeraton wndow desgn... 6 Fgure 5.. Fle menu... 6 Fgure 5.3. Edt menu Fgure 5.4. Swtch menu Fgure 5.5. Vew menu Fgure 5.6. Otons menu Fgure 5.7. Hel menu Fgure 5.8. Pou menu for oerand X Fgure 5.9. Pou menu for oerand Y Fgure 5.0. Pou menu for result Z Fgure 5.. Correlaton settng Fgure 5.. Oeraton tyes Fgure 5.3. Dslay mode wndow Fgure 5.4. Dslay color wndow Fgure 5.5. Correlaton settng wndow Fgure 5.6. Eresson edtor wndow Fgure 5.7. Class relatonsh for transortaton smle method Fgure 5.8. Class CConfgure Fgure 5.9. Flow of control for transortaton Smle mlementaton Fgure 5.0. Class: COtmalMn... 8 Fgure 5.. Class: CConfgure Fgure 5.. Class: CVarance Fgure 5.3. Class: CSmle Fgure 5.4. Class CMamn Fgure 5.5. Class: CoundCorrelaton Fgure 5.6. Class: CMnCorrelaton Fgure 5.7. Class: CMnCorrelatonEy Fgure 5.8. Class dagram Fgure 5.9. Sequence dagram for Cor_ound

6 v Fgure Sequence dagram for Cor_Mn Fgure 5.3. Sequence dagram for Cor_Mn_Ey Fgure 5.3. Sequence dagram for Cor_Mn_Ey_S... 9 Fgure 6.. X+Y when X and Y are 6 ntervals Fgure 6.. X+Y when X and Y are 3 ntervals Fgure 6.3. X+Y when X and Y have 64 ntervals Fgure 6.4. X*Y for unknown Fgure 6.5. X*Y for correlaton Fgure 6.6. X*Y for correlaton Fgure 6.7. X*Y for correlaton Fgure 6.8. Tmes for oeratons Fgure 6.9. Tmes for multlcaton and dvson

7 v Lst of Tables Table.. Dstrbutons for X and Y Table.. Margnal dstrbuton for X and Y Table.3. Jont dstrbuton for ndeendency Table.4. Jont dstrbuton for secfed value Table.5. Probabltes for result varable... 0 Table.. Jont dstrbuton matr... 5 Table.. Jont dstrbuton for X and Y... 8 Table 3.. Parameter table for transoraton model Table 3.. Margnal dstrbuton Table 3.3. Lower bound... 4 Table 3.4. Uer bound Table 3.5. Dstrbuton for X and Y Table 3.6. Interval value for relatonal oeraton Table 6.. Oeraton evaluaton tme (seconds) for correlaton

8 v Acknowledgements To fnsh ths thess n Englsh was a challengng eerence. I am glad that I dd t. I am grateful to many eole. Frst to my maor rofessor, Dr. Danel erleant. I want to thank hm for gvng me the chance to work wth hm closely on the research roect and for hs great advce on my graduate studes n these two years. Dr. erleant read the drafts of my thess and gave me a lot of constructve feedback even when he was n vacaton wth hs famly n the summer. I was always moved when I got hs comments and correctons on my thess va emal. I arecate deely hs tme and work. It has been a very good luck for me to have worked wth hm. What I have learned from hm wll beneft me n my all lfe. I also want to thank Dr. Gerad Sheble wth whom I have worked on the research roect. Our weekly meetngs have been great oortuntes for me to learn from hm. I have enoyed the meetngs. Thanks also to my frends who gve me suort and understandng. I am heavly ndebted to my arents-n-law. They took good care of my newborn baby and my wfe when I was workng on the fnal drafts of my thess. I thank them for freeng me from the household chores and the moral suort they gave me. I want to say thank you to my dear arents for nstllng n me the value of endless learnng and a thrst for knowledge. Wthout t, I would not have even been able to acheve ths degree. Fnally my heartfelt thanks go to my wfe, De, and to my baby daughter, Pelan for ther wonderful suort, and for not mndng the many lost evenngs and weekends durng my rearaton for ths thess.

9 Abstract Uncertanty ests frequently n our knowledge of the real world. Probablty s a common way to measure uncertanty. Peole sometmes defne random varables whose values are derved from arthmetc oeratons on other random varables. Generally there are two classes of methods to handle ths toc: analytcal and numercal. Analytcal methods are restrcted to secfc classes of nut dstrbutons. umercal methods only gve numercal results and are wdely used n real alcatons f aromate results can be acceted. Monte Carlo smulaton s one of the best-known numercal methods. However the tradtonal aroach of Monte Carlo has some lmtatons. Interval-based deendency analyss (DEnv) was develoed by erleant and Goodman-Strauss. Another aroach s the coula-based aroach. These two methods have been mlemented n software. The coulabased aroach s mlemented n the commercal software RskCalc. DEnv s mlemented n Statool. The current Statool suorts a varety of deendence relatonshs: ndeendence, unknown deendence, and secfc correlaton values. The algorthm etenson to suort correlaton s a sgnfcant mrovement. The current verson of Statool uses the transortaton smle method to seed u comutng. Cascaded oeratons, relatonal oeratons and monotonc bnary functons are newly suorted by the current Statool. These new functons, and usng correlaton as constrants, are the man advances n Statool. Ths software s based on a layer desgn ncludng the user nterface, the logcal layer, and the comutng layer. Ths s sutable for mlementaton and mantenance of dstrbuted and other comutng software. OO methods are adated. Unfed Modelng Language (UML) gves vsualzaton and documentaton suort for the comutng layer. The man algorthms are mlemented n many obects. Currently t s develoed for a Mcrosoft Wndows latform. Vsual C++ and Vsual asc were used for develoment. Dynamcally lnked lbrares are used to contan the comonents of the comutng layer.

10 Introducton Uncertanty ests frequently n our knowledge of the real world. Handlng uncertanty s therefore a common roblem. Probablty s a common way to measure the level of uncertanty. Probablty densty functons (PDF) or ther ntegrals, cumulatve dstrbuton functons (CDF), are often used to model the uncertanty n the value of a quantty. Often, uncertanty can be stated by usng a random varable. ut ths s not enough. Peole some tmes defne random varables, whose samle values are derved from arthmetc oeratons on the values of other random varables. nary oeratons are very basc and common oeratons. When two random varables are oerated on to derve a new random varable, the dstrbuton to descrbe ths random varable s termed a derved dstrbuton (Srnger, 979). Such oeratons are well recognzed n many felds, such as decson analyss and rsk analyss, and many other felds as well. A varety of methods have been develoed to address ths toc. Generally there are two classes of methods to handle t: analytcal and numercal. Analytcal methods are restrcted to secfc classes of nut dstrbuton, under assumtons, such as ndeendence. For eamle, normal dstrbutons are often used. If two random varables are normal and ndeendent, the sum of these two random varables stll s normal. It s also ossble to obtan derved dstrbutons for secfed deendency relatonshs other than ndeendence, such as erfect ostve rank correlaton. However, t s often not easy to fnd analytcal results for random varable oeratons and t s not always reasonable to make convenent assumtons about deendency. Sometmes, we don t have any nformaton about deendency. ut an advantage of analytcal methods s accurate result. Unlke analytcal methods, numercal methods only gve numercal results. ut ths s sutable for a wde class of dstrbutons. umercal methods are wdely used n real alcatons f aromate results can be acceted wthn secfc tolerances. Monte Carlo smulaton s one of the best-known numercal methods. However, the tradtonal aroach of Monte Carlo has some lmtatons. It assumes the dstrbuton of the random varables s known, and ther relatonsh s ndeendent or known (Ferson 996). If ether the robablty dstrbutons or the deendency relatonsh of the random varables are

11 not avalable, some assumtons are usually made to rocess t. If the assumtons don t hold, results can be serously affected. A dscretzed convoluton aroach can be used to calculate the result for the ndeendent stuaton (Ingram et al. 968; Colombo and Jaarsma 980; Kalan 98). Interval analyss can be used to solve ths roblem. (It s obvous that nterval numbers wll be really close to ont values f the nterval s narrow enough.) Interval mathematcs can then be aled (Moore, 966). Intervals have the otental for boundng the result of an oeraton. Dscretzaton error comng from dscretzng dstrbutons may be bounded by nterval based dscretzaton (erleant 993). If the deendency s not secfed, result bounds wll nclude the entre range of ossble deendences. These bounds should be wder than f a artcular deendency s secfed. Interval-based deendency analyss s develoed by erleant and Goodman-Strauss (998). Ths aroach has fundamental smlartes wth the coula-based aroach (Frank et al. 987), whch was sgnfcantly etended by Wllamson and Downs (990). These two methods have been mlemented n software. The coula-based aroach, termed robablstc arthmetc, s mlemented n the commercal software RskCalc (Ferson et al. 998). DEnv s mlemented as Statool (erleant and Goodman-Strauss 998), whch etends the revous tool (erleant and Cheng 998) through elmnatng the ndeendence assumton. Statool can handle the case where a deendency relatonsh s unknown or unsecfed, by not makng any assumton about the deendency relatonsh between oerands. ut artal deendence nformaton mght be avalable n some cases. If we can use ths nformaton n the calculaton, we wll get more accurate results than can be obtaned wthout usng ths nformaton. The current Statool suorts a varety of deendence relatonshs, such as ndeendence, unknown deendence, and correlaton values. The algorthm etenson to suort correlaton s a sgnfcant mrovement. The current verson of Statool uses the transortaton smle method to seed u comutng. Cascaded oeratons and monotonc bnary functons are suorted by the current Statool. These new functons, and usng correlaton as a constrant, are the man advances n Statool. Among the other contrbutons reorted here are addressng eamle alcaton roblems.

12 3. Interval mathematcs Interval mathematcs was develoed by Moore (966). Comared wth the real doman, let us see what nterval arthmetc and analyss are. An nterval value s comosed of real numbers, whch are called the low bound and hgh bound. For eamle, gven nterval value X[a,b], a and b are real numbers, and a s the low bound and b s the hgh bound. Thus, we can see that an nterval value n the nterval system corresonds to an nterval n the real system. If a s equal to b, ths nterval value s the real number a. Or you can use set theory to descrbe the nterval X[a,b]. We can defne t as a set X{: a<<b}. et, we wll defne how to descrbe the relatonsh between two nterval values. If we say [a,b][c,d], t means ac and bd. f [a,b]<[c,d], t means b<c. Interval arthmetc ncludes addton, subtracton, multlcaton and dvson. Here X[a,b] and Y[c,d] are two ntervals. The followng gves the defnton for arthmetc based on the set defnton for ntervals. X Y { y : X, y Y} where s n +,-,*,/. Clearly, X+Y [a+c,b+d] and X-Y [a-d,b-c]. Multlcaton s a lttle more comle. X*Y[mn(ac,ad,bc,bd), ma(ac,ad,bc,bd)]. And dvson s even more comle. Frst, note that Y doesn t nclude zero. /Y [/d,/c] f 0 Y X/Y X* (/Y) f 0 Y If Y s an nterval ncludng zero, X/Y should be [-, ] f the nterval system ncludes nfntes as allowable endonts. Interval arthmetc also ncludes the followng characterstcs: Set Rule (V W) ±Z (V±Z) (W ±Z) Rule for the addton and subtracton of nfnte or sem-nfnte ntervals [a,b]+[-,d] [-,b+d] [a,b]+[c, ] [a+c, ] [a,b] ±[-, ] [-, ] [a,b]- [-,d] [a-d, ]

13 4 [a,b]-[c, ] [-,b-c] Assocatvty and Commutatvty X+(Y+Z) (X+Y)+Z X*(Y*Z) (X*Y)*Z X+Y Y+X X*Y Y*X Unlke n real arthmetc, oeratons are not nvertble, whch means there s no nverse oeraton estng for a gven oeraton. For the real doman, we know + and are nverse oeratons, but n nterval mathematcs, ths s not true. In nterval analyss, nterval functons form a maor toc. An nterval functon F s an nterval-valued functon of one or more nterval arguments. For a real-valued functon f of real varables,,n, f we have an nterval functon F of nterval varables X, Xn, and f F(,,n) f(,,n) for all (,,n) then F s an nterval etenson of f. Interval functons have the followng characterstcs: Incluson monotoncty If X Y (,,n) then F(X,,Xn) F(Y,,Yn). Arthmetc ncluson monotoncty If o denotes +,-,*, or /, then X Y (,) mles (X o X) (Y o Y) Ecess wdth s a bg roblem n nterval mathematcs. Let us use a smle eamle to elan ths roblem. For nterval value X[a,c], what s the result for X-X? In naïve nterval arthmetc, the result s not zero, but [a-c,c-a]. Zero s ust one real number ncluded n ths result. Obvously t s really not our eected result snce t s too wde. For functons, an nterval functon etenson need not be unque, but can deend on the form of the real functon. For eamle, there may be three eressons corresondng to the same real functon: f() * +, f() (-/) + ¾, and f3() *(-) +. The corresondng nterval etensons are: f(x) X*X X +, f(x) (X-/) + ¾, and f3(x) X*(X-) +.

14 5 These don t reresent the same nterval functon, as: f([0,]) [-,5], f([0,]) [3/4,3], and f3([0,]) [-,3]. The true range of f ([0,]) s [3/4,3] comuted by the nterval functon f, because aears only once. Ths s referred to as the deendency roblem or ecess wdth. It enlarges ntervals n the result collecton. The reason why ecess wdth occurs s that a varable occurs more than one tme n eresson. So far, many methods have been develoed to address ths ssue. Some methods are as follows. Varous centered forms: Comutng the range of values (Asathamb, Zuhe, and Moore,98) Enclosure methods (Alefeld, 990) Artfcal ntellgence work (Hyvonen, 99) Comutaton tme tends to be a roblem wth these ecess wdth removal technques. To aly nterval analyss, the followng gudng rncles should be consdered. (Walster 998): Interval algorthms should bound error Interval nut/outut conventons should be consstent wth eole s normal nterretaton of numercal accuracy The alcaton of nterval algorthms should be unversal Where nterval algorthms currently do not est, we should get to work develong them rather than abandonng the rncle of unversal alcablty. Interval-based analyss An nterval can be used to bound the range for a value. If ths nterval s assocated wth a secfed robablty, as when the doman of a random varable s arttoned, we have lost nformaton about robablty dstrbuton n ths nterval. The arttonng of the doman of a random varable nto ntervals and robabltes s the bass for etendng bnary oeratons from ntervals to dstrbutons.

15 6 At ths ont, we only consder the bnary oeratons. We can etend bnary oeratons and later we wll talk about how to do ths. Assumng there are random varables X and Y, to get the eact dstrbuton for the result of oeraton, we must know the ont dstrbuton for random varables X and Y. The ont dstrbuton s related to the correlaton for these two random varables. Let us see an eamle. Consder two random varables X and Y. Ths table shows ther dstrbutons. Table.. Dstrbutons for X and Y. X Y Range [,] [,3] [3,4] [,3] [3,4] [4,5] Probablty We don t have any nformaton about dstrbuton wthn these ranges. And we also don t have any nformaton about the deendency relatonsh between X and Y. Obvously, we don t know the ont dstrbuton for X and Y. Consder addton: ZX+Y. ecause we don t have the ont dstrbuton for X and Y, t s mossble to fnd the eact result for Z. ow we ut these two random varables nto a matr shown n the followng table. Table.. Margnal dstrbuton for X and Y. z [3,5] z [4,6] z [5,7]?? 3? y [,3] 0.5 Y z [4,6] z [5,7] z [6,8]?? 3? y [3,4] 0.3 Y z [5,7] z [6,8] z [7,9] y [4,5] 3? 3? 33? Y 3 0. [,] [,3] [3,4] b 0.5 X 0.5 X X X Y The last row n the table s the dstrbuton for X and last column s the dstrbuton for Y. We don t know the value for cells through 33 because we don t know the ont

16 7 dstrbuton. For the smle case, f X and Y are ndeendent, we can fll n the mssng values as n the followng table. Table.3. Jont dstrbuton for ndeendency. z [3,5] z [4,6] z [5,7] z [4,6] z [5,7] z [6,8] y [,3] 0.5 Y y [3,4] 0.3 Y z [5,7] z [6,8] z [7,9] y [4,5] Y 3 0. [,] [,3] [3,4] b 0.5 X 0.5 X X X Y Thus, we can see that the ont dstrbuton s affected by the deendency relatonsh between X and Y. If we don t know the relatonsh between X and Y, we can t determne the ont dstrbuton n ths matr. ut we can nfer some thngs about the result varable from ths matr. For eamle, consder z5. It only occurs n the followng grey cells. Table.4. Jont dstrbuton for secfed value. z [3,5] z [4,6] z [5,7]?? 3? y [,3] 0.5 Y z [4,6] z [5,7] z [6,8]?? 3? y [3,4] 0.3 Y z [5,7] z [6,8] z [7,9] y [4,5] 3? 3? 33? Y 3 0. [,] [,3] [3,4] b 0.5 X 0.5 X X X Y

17 8 As revous stated, we don t know the eact robablty for z<5. ut we can thnk about what are the ossble robabltes for z<5. As ths matr shows, only grey cells contrbute to the robablty of z<5. We would lke to determne the mamum robablty and the mnmum robablty. To get the mamum value, all cells n whch Z can be < 5 wll have ther robabltes summed. To obtan the mnmum value, only cells, n whch Z must be < 5, wll have ther robabltes summed. For eamle, consderng cell for { Z 5}, when we calculate the mamum value, ths cell must be counted because Z can be < 5 n ths cell. ut for the mnmum value, we don t count ths cell because Z mght not < 5 n ths cell. Ths way, we can fnd the ossble range of cumulatve robabltes for varous values of Z. We can fnd the mamum ossblty and mnmum ossblty for every value of Z and connect all these onts to get curves: a to curve and a bottom curve. All the CDFs that are ossble for Z, must belong between these two curves. In ths eamle, Z s range s from 3 to 9. It s clear that the robablty for Z<3 s zero and for Z>9 s. The followng art dscusses the robablty of Z 4. Mamum: We try to fnd all the cells n whch ths stuaton may occur. From the revous table, these cells are,, and. So the mamum value should be the mamum value for the sum of,, and. Mnmum: To obtan the mnmum, we wll fnd all the cells n whch Z must be < 4. In ths table, there are none. Although,, and may satsfy Z 4, they also mght not. For eamle, the whole robablty for the cell mght be concentrated at the hgh bound of ts range. So there s no cell n whch Z must be < 4. Summarzng the above analyss, we can defne a way to tell whch cells contrbute to the mamum and mnmum robablty values. Mamum: all the cells n whch the low bound s not greater than the value of Z contrbute to the ma value. Mnmum: all the cells n whch the hgh bound s not greater than the value of Z must contrbute to the mn value. After fndng all the cells satsfyng the ma (or mn) condton, we wll calculate the sum of the robabltes of these cells. ased on the revous table, there est constrants for the robabltes P. It s clear that the sum of the P s n a row or column can t go over the

18 9 margnal robablty of that row or column. These constrants can be descrbed as follows: Row Constrants: 3 Y for to 3 Column Constrants: 3 X for to 3 Therefore, the queston becomes: fnd the mamum and mnmum value for the sum of cells under these constrants. For the case Z 4, we can descrbe these questons usng mathematcally: Mamum - make the sum of the secfed cells value bg enough, that s, fnd ma ( + + ) such that: 3 Y for to 3 and 3 X for to 3. Mnmum - make the sum of secfed cells value small enough, that s, fnd 3 mn ( 0* 3 such that: ) 3 Y for to 3 and 3 X for to 3 For these two otmzaton questons, lnear rogrammng s the best tool to fnd the soluton. Ths way, we can fnd the robablty range for the secfed value of Z. The followng table shows the robabltes for varous values of Z.

19 0 Table.5. Probabltes for result varable. Z range Mamum robablty Mnmum robablty Z<3 0 0 Z< Z< Z<5 0 Z<6 0.5 Z< Z<8 0.8 Z<9 Z<0 From ths table, we can draw two curves, a to curve and a bottom curve, usng the mamum and mnmum robabltes shown Z value. These two curves also can be called enveloes for the CDF of derved varable Z because the CDF for derved varable Z must be between these curves whatever the relatonsh between X and Y s. Ths fgure shows the fnal result. Fgure.. Probablty bounds

20 arrowng the enveloes around results usng correlaton In the revous chater we noted an mortant factor: correlaton. If one knows somethng about correlaton, t would be good to be able to use t. We descrbe how net.. Facts about correlaton Correlaton s used to measure the degree of corresondence between random varables. To descrbe ths knd of relatonsh, there are a number of methods. For eamle, we can consder the lnear relatonsh between two random varables, or the square relatonsh. Currently, the most oular correlaton coeffcent s called Pearson correlaton or roduct-moment correlaton. It s used to measure the strength of the lnear relatonsh between two random varables. It s defned as ρ E [( X EX )( Y EY )] D( X ) D( Y ) Here D(X) s X s varance and D(Y) s Y s varance. E means eectaton. It s clear that ρ. Correlatons can be classfed nto 3 tyes: ostve correlaton ( ρ > 0, meanng there s a drect lnear correlaton between the R.V. s), negatve correlaton ( ρ < 0, meanng there s an nverse lnear correlaton between the R.V. s), and no correlaton ( ρ 0, meanng there s no aarent lnear correlaton between the R.V. s). There also are secal cases: erfect ostve correlaton ( ρ ) and erfect negatve correlaton ( ρ ). For erfect ostve correlaton, we can get: P[YaX+b], for some b and some a > 0. When X takes on ts largest value, Y also does. For erfect negatve correlaton, we can get: P[YaX+b], for some b and some a < 0. When X takes on ts largest value, Y has ts smallest value.

21 . Jont dstrbutons A ont dstrbuton s used to descrbe the detaled deendency between two R.V. s. From the ont dstrbuton, we can get the correlaton. ut correlaton doesn t mly a secfc ont dstrbuton, so we can t get the ont dstrbuton from a value of correlaton, n general..3 Interval-valued correlatons When the correlaton s unknown we use lnear rogrammng to fnd CDF enveloes. If we know the correlaton for two oerands, we would lke to use t to determne addtonal constrants for the lnear rogrammng roblem. In another words, we wsh to decrease the feasble soluton sace and get a better soluton. Accordng to the defnton of correlaton, for two random varables and y, the correlaton s ρ E[( E)( y Ey)] D( ) D( y) E[( E)( y Ey)] E[( E) ] E[ y E( y) ] Where E and Ey are the means for varable and y. Usng the followng formulas, we can reduce (): E[( E)( y Ey)] E[ y ye Ey + E * Ey] Ey Ey * E E * Ey + E * Ey Ey E * Ey Also, E[( E) ] E[ E ( E) So revous formula becomes E + E * E] E E * E + E * E ρ E[( E)( y Ey)] E[( E) ] E[ y E( y) ] ( E Ey E * Ey ( E) )( Ey ( Ey) ) Usng the defnton of mean, when varable s dscrete, E * where ( ) When varable s contnuous, and has densty functon f, then E f ( ) d.

22 3 In the DEnv algorthm, we don t care f a random varable s dscrete or contnuous. We use bars to dscretze the dstrbuton. Ths method has the followng characterstcs:. ars may overla.. Hstograms are a secal case of collectons of bars. 3. A bar descrbes the robablty of an nterval contanng the value of a varable. 4. o assumton s made about the dstrbuton over the nterval of bar. We now etend the defnton of mean to ntervals. We can handle t lke the dscrete case. So EX X * P where P ( X ) P. It s clear that the mean of varable must be n EX. When s contnuous, we also can get the mean based on the followng argument. a b Consder some bar n the dscretzaton of varable whose dstrbuton functon s f(). The robablty that t s n [a,b] s the area of f() between a and b. P ( a b) f ( ) d b a We can artton the doman of varable nto many ntervals such as ths one, denotng them X. These ntervals do not overla. They together wll cover the range for varable. So the mean of varable becomes E f ( ) d f ( ) d X Consder one tem n the revous formula, assumng X s [a,b] as n the revous fgure. X f ( ) d f ( ) d af ( ) d a * [ a, b] Smlarly, we also get [ a, b] [ a, b] f ( ) d a *.

23 4 X f ( ) d f ( ) d bf ( ) d b* [ a, b] [ a, b] [ a, b] f ( ) d b*. So, f ( ) d must belong to X*. So, E contans the mean of nterval varable X. X If the ntervals overla, the wdth of the mean s wder than n the non-overlaed case. So the mean of the non-overlaed ntervals s a subset of that of the overlaed ntervals. Thus the E belongs to mean of nterval varable X n ths case too. Here s how we use ths result. If any ntervals overla, t means at least two ntervals overla. We know the left endont for one nterval s not bgger than that of the nonoverlaed condton and the rght endont for one nterval s not less than that of the nonoverlaed condton..4 Legal and llegal correlaton values In the current software, the user can nut any value of correlaton from to. ut n fact, for some margnal dstrbutons, there are correlaton values whch wll not be ehbted by any ont dstrbuton. In fact, the constrants comng from settng correlaton to an mossble value should be conflct wth the constrants comng from the margnals of the ont dstrbuton matr. From the defnton of correlaton, we can get ths formula for Ey: Ey E * Ey + ρ ( E ( E) )( Ey ( Ey) ) Let f(,y)ey, so f(,y) s a real functon of and y. We can rewrte Ey wth ntervals X and Y. f ( X, Y ) EX * EY + ρ ( EX X P Y P y + ρ ( X ( EX ) )( EY P ( X P ) ( EY ) ( Y ) P y ( Y P y ). The corresondng real functon s (, y) + ( ( ) ( ( y y y y f ρ y y ) where X and y Y. If ρ s an nterval, t becomes another varable for functon f..4. Soluton The software should rovde a way to hel the user to set a reasonable correlaton. To do ths, frst, the software must fgure out the range of ossble correlatons for the current

24 5 random varables. Then the software can dslay ths nformaton. It then only accets values ntersectng wth ths range. As mentoned before, there are knds of constrants, one comng from the margnals of the ont dstrbutons matr and another comng from the correlaton settng. The ont dstrbuton matr margnals are assumed correct. So, the constrants comng from t are a gven. If constrants comng from a correlaton settng conflct wth them, they must be n error. Constrants comng from the matr are rmary and constrants comng from correlaton should be consdered secondary. Consder a ont dstrbuton matr for an oeraton. Table.. Jont dstrbuton matr. Y Ym X m Xn n nm n y ym We can get Ey as follows: Ey n m X Y where X and Y are nterval values. s the robablty assgned to cell. We use underlnng to ndcate the low bound of an nterval and overlnng to ndcate the hgh bound of an nterval. We can get the bounds of Ey as follows: Ey y + y + y y + y + y n ym nm Ey y + y + y y + y + y From ths, we get two lnear rogrammng roblems: Mn Ey y + y + y y + y + y n y subect to: row Constrants: column Constrants: m n for to n; y for to m. n y m nm m nm

25 6 Ma Ey y + y + y y + y + y n y subect to: row Constrants: column Constrants: m n for to n; y for to m. Solvng these two lnear rogrammng roblems, we can get the bounds of Ey, call these numbers k and k. We also know Ey y + ρ y ( ( ) ( ( y y y y ) m nm where X and y Y. In ths formula, only ρ s an unknown range. ow the roblem becomes solvng for Ey. The mnmum should be the mnmum value of ρ. The mamum should be the mamum value of ρ. So the roblem s transformed nto fndng the root range of a nonlnear functon..4. Aromate soluton ( From f(,y) y + ρ ( )( y y ), n most cases, y s greater than )( y y ). So we ust consder y. It s obvous that t s an ncreasng functon of and y. Assgnng the mnmum values to and y, and the mamum value ossble for f(,y), we can obtan the mamum value of ρ. Assgnng the mamum values to and y, and the mnmum value to f(,y), we can get mnmum value of ρ..5 Addtonal constrants gotten from correlaton When the user sets the correlaton range, we know the range of every varable n formula f(,y). Under ths stuaton, we can get the range of f(,y). Ths range of f(,y) s thus controlled by the user. At the same tme, we know another range for f(,y) whch s derved from the ont dstrbuton matr. As revous noted, the range derved from the ont dstrbuton matr s consdered gven. So t s always correct. The range comng from the

26 7 user must be ntersected wth ths range. From ths restrcton, we can get addtonal constrants for lnear rogrammng. Obvously, formula f(,y) s non-lnear, so we use non-lnear otmzaton to do mnmzaton and mamzaton on t. Usng a enalty functon transforms a constraned otmzaton roblem to a non-constraned roblem. We also can get the frst and second dervatve for ths functon. Call the values obtaned fmn and fma. So we get f(,y)[fmn,fma]. From the revous secton, we know another range for f(,y), [ k, k], from the ont dstrbuton matr. It s obvous that these two ranges must ntersect; otherwse, the user nut s not ossble. These two ranges both are ntervals. If the followng condtons are satsfed, these two ntervals must be ntersected: f ma k and f mn k. Snce k y + y + y y + y + y k y + y + y y + y + y we know fmn and fma. So we get an addtonal two lnear constrants for the lnear rogrammng roblems based on correlaton: n n y y m m nm nm y + y + y y + y + y n ym nm f y + y + y y + y + y n ym nm f ma mn.6 onlnear otmzaton to remove ecess wdth From the revous secton, we saw that f(x,y) s an nterval, not a real number. In nterval mathematcs, t s called an nterval functon (Ramon E.Moore, 966). In evaluatng an nterval functon, ecess wdth may haen. Dfferent functon formats wll result the dfferent values for functon although they are the same functon n the real doman. From the term ρ * D ( X ) * D( Y ) + E( X ) E( Y ), we defned the corresondng functon f(,y): f (, y) E * Ey + ρ ( E ( E) )( Ey ( Ey) ). Frst, we can consder f(,y) as a real functon of varables and y. If we relace and y wth ntervals X and Y, t becomes an nterval functon.

27 8 ased on the rule cancellaton or reducton of the number of occurrences of a varable before nterval evaluaton, f the number of occurrences of each varable s only one, evaluatng an nterval functon cannot result n ecess wdth. However t s mossble to use ths rule for ths functon. Instead, we can avod ths roblem by evaluatng ths functon n the real doman usng real numbers belongng to nterval X. So we can use the mnmum value and the mamum value of ths real functon as the way to get bounds on the nterval. ow we rewrte the formula wth ntervals X and Y. f ( X, Y ) EX * EY + ρ ( EX X P Y P y + ρ ( X ( EX ) )( EY P ( X P ) ( EY ) The corresondng real functon s f (, y) + ( ( ) ( ( y y ρ y y y Here X and y Y. If ρ s an nterval number, t becomes another varable for functon f. Obvously, f(,y) s a non-lnear functon. We use non-lnear otmzaton to fgure out the mnmum and mamum. ut ths otmzaton queston s restrcted to a secal regon, the ntervals for the s and y s..7 Imrovng results by addng constrants to LP ased on the above dscusson, we get another two constrants for LP after calculatng the nterval k. From the ont dstrbuton matr, Table.. Jont dstrbuton for X and Y. [ ] [ ] X [ ] n [ ] m mn m Y y. y n ( Y ) P y ( Y P y ) y ) we get the LP model: Mnmze Z, Ω

28 9,... m subect to: y,... n 0, 0, y 0,, y To these we add the two constrants mled by the correlaton. When the two constrants, a k and a k, are added to the LP, the transortaton smle method can t handle ths augmented model because we can t ut these two constrants nto the balanced transortaton tableau. So we use the smle method to solve the roblem. The seed of calculaton s very mortant. Ths s dscussed later..8 Imroved smle method Consder the standard LP queston: Mn Z CX Subect to: AX b, 0 and b 0 for to n. Here C c,..., c ) s a row vector, ( n X... n s a column vector. A P,..., P ) ( n and a P.... So, A s an m*n matr and am b b... b m s a column vector. We can transform the mamzaton roblem to a mnmzaton roblem through the followng aroach. Ma Z CX Mn Y Z CX The constrants are unchanged.

29 0 ased on the smle method, A s slt nto ( ) A A. A has the coeffcents for the basc varables (assumng there are m basc varable from to m ), and A has the coeffcents for the non-basc varables (from m+ to n ). X s also searated nto X X. So AXb becomes ( ) b X X A A. b X A X A + * * ) * * ( X A b A X Here A means the nverse matr of A. In other word, I A A * where I s the unt matr. For eamle, s a 3*3 unt matr. So, the obectve functon becomes X A A C C b A C X C X A b A C X C X C X X C C X C Z ) ( * ) * ( * * * ) * ( * Let us see an eamle. Mnmze z + + Subect to: , Here ( ) C, n X..., A and 8 0 b.

30 If we assume and are the basc varables, we get C 5 4 ( 3 ), C ( 7 3 ), A, A 3 5 X. We also get 3, X 4, 5 / 3 / 3 A, X A ( b A X ). / 6 5 / 6 The followng dscusson wll be based on the revous defnton and equatons, and also n art on Qan and Murty (985)..8. How to fnd the ntal feasble soluton For the standard LP queston, f you can fnd a unt matr be needed. Set m m matr n A, you let ths A by multlyng one row by a constant and addng t to another row, reeatng as X (non-basc varables) to zero (that s m, m+,..., n + all equal 0). Then X A ( b A X ) A * b I * b b because snce A s a unt matr, so s A. Then, b 0,(... m) s a feasble soluton although t s robably not the otmal soluton. If you can t fnd a unt matr, you can choose a sub-matr ( m m ) of A whch s nonsngular (meanng that the determnant of the matr doesn t equal zero and the rank of the matr s m), and every of X A b s not less than 0. Under ths condton, t s a * feasble soluton. ut frequently, t s not so easy. Therefore artfcal varables are ntroduced. y To AX b, we add the artfcal varables Y... 0, and revse the equaton to y m AX + IY b. In the obectve functon, the coeffcents of Y should be very large ostve real numbers. Through ths way, the mnmzng obectve functon wll be unaffected by artfcal varables Y. Stll we can use Y as the ntal feasble soluton. Imortng the artfcal varables ust rovdes an easy way to get an ntal feasble soluton.

31 .8. How to decde the termnaton condton and enterng varable ow consder otmzaton of Z. Let ) (,..., m n w w A A C C W. Here w s the coeffcent of m + and descrbes the coeffcent of a non-basc varable n the obectve functon. If we want to make WX b A C X A A C C b A C X C Z + + ) ( * smaller, we must hoe to fnd the negatve elements of W because all elements of X are ostve. From ths dscusson, we can derve the termnaton rule for an teratve otmzaton rocess.. If every element w of W s not less than 0, then the current soluton s otmal.. If at least one element of W s negatve, we contnue to search for the otmal soluton. Let ) 0 mn( < k w w w. Ths means f every non-basc varable changes by the same factor, value k k m w + * wll have the mamum effect n mnmzng the value of Z. So let non-basc varable k m + be the enterng varable (enterng the basc varable set from non-basc varable set). 3. If 0 + k P m A, there s no soluton (k s the enterng varable nde, and k m P +, belongng to ),..., ( n m P P A +, s the coeffcent for non-basc varable k m + ). Proof: From ) * * ( X A b A X, assumng the enterng varable k m + does not equal 0 and other non-basc varables stll equal 0, let k m + equal α and be greater than 0. Then k m k m k m n m n m P A b A P A b A P P A b A X A A b A X * *... ),..., ( * * * α ecause 0 + k P m A, X stll are greater than 0, and 0 X ecet for α m+k. So t s a feasble soluton. Consder the obectve functon: *α... ),..., ( ) ( k b n m m n b n b n b w b A C w w b A C X A A C C b A C Z

32 3 ecause w k s less than 0, f α +, Z. So, there s no mnmum value for the obectve functon. Let us see an eamle: Mnmze z Subect to: + 4 0, We choose and 4 as basc varables. Then A, A, C 0 ( 0) 0 and C ( 0). A, so W C C A A ( 3 ). We can choose as 0 the enterng varable. We get A P < 0. ow let equal β > 0. So X A b A A * X Here X s a feasble soluton f 0. ut Z ( ) (4 + ) 4 3. If β, then Z. So there s no mnmum value for Z. ased on the revous dscusson, there are three condtons that can occur durng the teratve rocedure.. Fndng the soluton. Contnung to try mnmzng Z 3. o mnmzaton soluton.8.3 How to determne the leavng varable Let X be a feasble soluton. So A X b *. Here A ( P... ). We know P m A s nonsngular, so P to P m are the ndeendent vectors. The other vectors P m+ to P n are lnearly deendent on P to P m. Therefore we can get + m P m α P, m+ *

33 4 α, m+ > Pm + ( P,..., Pm ) *... 0 αm, m+ From A * X b, we get ( P... Pm )* X b. Let β a ostve real number. Then ' ( P,..., Pm ) * X + β ( Pm + ( P,..., Pm ) * ( α, m+,..., αm, m+ ) ) 0 ' > ( P,..., Pm ) * ( X β ( α, m+,..., α m, m+ ) ) + βpm + 0. Let m + relace a varable n X. We can get a new feasble soluton f we set sutable values for X and make sure 0. We can get a sutable soluton from the revous formulaton through settng the new X to equal β α α ' X (, m+,..., m, m+ ). We wll let one element that equals 0 to be relaced by +. To assure the other varables n X stay m ostve, we can choose a sutable β. Let l β mn( α, m+ > 0). α, m+ αl, m+ Ths mles that l s the leavng varable and enterng varable ow we can aly ths result. ased on X A * ( b A * X ) A A we know b A b m+ m + A * A * X P m+ m+ l α l, m+ s the enterng varable. We can determne the leavng varable by choosng the mnmum β usng the equaton ( A b) ( A b) l mn( ( A Pm ) 0) + > ( A Pm + ) ( A Pm + ) l β. Ths mles that the leavng varable s l Decreasng comutng The smle method s a good way to solve lnear rogrammng. ut t can have comutatonal comlety roblems.

34 5 From the revous dscusson, we can see the man comlety roblem focuses on the nverse matr A. If we can fnd a better way to comute t, we can get better effcency. A smle aroach s to fnd the relatonsh between the two can use the revous orgnal A ( P... ) P m A to seed comutng the net A, t wll hel. If the, the new A n the closng stes. If we A s A ( P P P P... P ). There... l m+ k l+ s only one dfferent column. So the coeffcent of the leavng varable s relaced wth that of the enterng varable. We can guess there s a relatonsh between these two m A. From X old old new old old new A b, X A b m+ k * A Pm + k A b, and basc varable l s relaced wth +, m k new old A A C. Then A new old old C A DA So f we can fnd D, the nverse of C, we wll seed comutng the nverse of From the relatonsh of the orgnal and new / a new old m+ k l lk. Here k P m + k P m + k ). We can see D ( l l+ m e,..., e, Ek, e,.., e ) and are 0. Ek X, new A. old new + a,... m, l and a ( ), m ( refers to the th element of the vector e 0..., and only element of row s, whle the others... 0 ',/ a, a / a,.., a / a ( a k / alk,..., a( l ) k / alk lk ( l+ ) k lk mk lk ). Ths way, we can use the revous nverse matr to calculate the new nverse matr. Sosto (989) gves a smlar descrton of ths method..8.5 Alyng method For our case: Mn Z CX subect to: AX b and X>0. m k k

35 6 n+ Usng artfcal varables X av... n+ m, the equaton becomes AX + IX av b. Let Con be a very bg ostve real number based on g-m method. Then the obectve functon becomes Z CX + Con *(... ) X av. ased on the revous dscusson, X are non-basc varables. A equals I. It s easy to comute. It s not needed to calculate the nverse matr. ut artfcal varables are n the obectve functon. We must remove them from the obectve functon. If an artfcal varable s removed from the basc varables, t wll be removed from the obectve functon. Ths means the coeffcent of the artfcal varable becomes 0, not. After changng the coeffcent of an artfcal varable to 0, the artfcal varable s n effect not resent. When the otmum s reached, the coeffcents of the artfcal varables must be zero. Otherwse, there s no otmum..8.6 An eamle Mnmze Z subect to: + 4 0,,,3 Soluton: usng artfcal varables 4 and 5, we can get an ntal feasble soluton. The queston changes to: mnmze Z M *( ) subect to: ,,...,5 To remove the effects of the artfcal varables, we set the coeffcent M of the artfcal varables n the obecton functon to a bg real number, for eamle Iteraton : 7 C ( ), 4 and 5 are the basc varables. A A A I. 4 0 b. A ( P P P )

36 7 W C C A A ( 3 ) ( ) 3 I 4 ( 3 ) ( ) ( ) So s the enterng varable. In the net ste we wll decde on the leavng varable. 7 A b, A P. So the leavng varable s 5. a lk / 4 So E k (/ 4,/ 4). We get A EA. 0 / 4 Iteraton : ow 4 and are basc varables, and 5 A W C 3 0 C A A ( ) ( ) ( ) ( ) ( ) s dscarded. ( ) 0 / 4 / C. So s the enterng varable. 0.5 A b, A P. So the leavng varable s 4. a lk / 5 /0 So E k ( / 5,/ 5). We get A EA. / 5 3/0 Iteraton 3: ow and are basc varables, and 4 s dscarded. C ( 3 ). A. 0 / 5 /0 W C C A A ( ) ( 3) / 5 3/0 0 ( / 5).4. So the otmal soluton becomes: 4 4 X A b, Z C X ( 3) onlnear otmzaton For most cases, there s a functon f(), called the obectve functon, whch belongs toc, meanng that the functon f() has a second dervatve. We want to fnd the mnmum or mamum value of f(). We can descrbe ths queston as follows: mn f ( ) 0

37 8 Subect to: n R n where R s the n-dmenson real doman. For the mamzaton queston, we convert t to the mnmzaton roblem accordng to the followng formulaton: ma f ( ) mn( f ( )) So we only need to solve the mnmzaton queston. In ths case, the varable belongs to the n-dmenson real doman. The number of dmensons may vary from to n. Ths knd of mnmzaton roblem s called unconstraned otmzaton. If any constrants are aled to the varable, we have the followng stuaton: mn f ( ) subect to: The equalty constrants are: e ( ) 0 for, and the q nequalty constrants are: w ( ) 0 for,, q. Ths knd of roblem s called constraned otmzaton. All onts satsfyng all the constrants are feasble and all others are non-feasble. All feasble form the feasble regon. All non-feasble form the non-feasble regon. For unconstraned otmzaton, the feasble regon s the real doman..9. Local and global otmums A local mamum s a ont n the feasble regon whch s hgher than all other onts wthn ts mmedate vcnty, but not necessarly the whole feasble regon. The global mamum s the mamum for the whole feasble regon. The followng fgure llustrates local otmums: Local mamum Local mnmum

38 9 From ths fgure, we can see the followng onts about the global and local otmums. There may be more than one local otmum for the functon and ther values erhas are not the same. The global otmum must be a local otmum. A local otmum may be the global otmum. It s ossble that there s more than one global mnmum or mamum, f the functon values are be same. The global otmum s the best of all the local otmums and s the soluton for our roblem..9. Classcal theory of unconstraned otmzaton Gven a functon f(), for vector, assume all the frst dervatves f est at all onts n the doman of f. A necessary statement for a mnmum of f() s: f f f... n 0. The condton necessary means that where the functon s at a mnmum, the equaton holds. ut ths equaton s not a suffcent condton. A suffcent condton for a ont to be a mnmum of f() s that the second dervatves of functon f() est at the otmum ont and D > 0. D f... f f... f ote: when the dervatves of the functon f() are dscontnuous, the classcal theory s not fully alcable..9.3 Fndng a soluton teratvely Almost all numercal otmzatons methods use teratve technques. They start at an ntal ont 0 and roceed by generatng a sequence of onts, m (each s an n-

39 30 dmenson vector). Let f + ) f ( ). Then, the mnmum of f() wll be aroached more ( closely wth each teraton. Clearly, the choce of s very mortant. Defned by + + d s, d s an drecton vector for fndng the net and s s the ste sze or dstance to move. Here, a sutable choce of drecton d s very mortant. How to search for the net s an mortant ssue. Tycally, methods are classfed nto two classes: drect search and gradent methods..9.4 Search methods: drect and gradent Drect search methods don t requre the elct evaluaton of any dervatves of the functon, but rely solely on values of the obectve functon f() and nformaton ganed from earler teratons. Some use functon values to obtan numercal aromatons of the dervatves. Gradent methods select the drecton usng the values of the dervatves of the functon f(). Usually, the frst order dervatves are used by these methods..9.5 Convertng constraned to unconstraned otmzaton For constraned otmzaton roblems, t can be useful to make use of unconstraned otmzaton methods. So convertng to an unconstraned otmzaton roblem s the frst task. Many methods have been develoed for transformng the otmzaton roblem. The followng methods are wdely used:. Transfer functons. Lagrangan multlers 3. Penalty functons.9.5. Transfer functons Its basc dea s to etend the restrcted feasble regon to the whole real doman. For eamle, to mnmze f(), subect to >a, we can defne a new varable y. Let a + y Usng ths equaton, we can convert f() to f(y), and then mnmze f(y). Here varable y doesn t have any restrcton. So ths s now an unconstraned otmzaton roblem.

40 Lagrangan multlers Ths s a very common method for transformng otmzaton roblems. If a mnmzaton roblem has many equalty constrants e ( ) 0 for, defne a new obectve functon to mnmze wth a new varable λ h(, λ ) f ( ) + λ e ( ). For the frst dervatves of ths functon, h(, λ) f ( ) + h(, λ) e λ ( ) 0 e λ ( ) 0 The soluton wll satsfy the constrants e ( ) 0. For the nequalty constrants w ( ) 0 for,, q we can ntroduce new varables called slack varables, n+... n + q. Let w ( ) n + 0. ow we can transform the nequalty nto equalty: w ( ) n+ 0 So usng ths method, we can handle the constraned otmzaton roblem Penalty functons The bass for the enalty functon method s to defne a new obectve functon lke the followng: h ( ) f ( ) + ( c( )), where f() s the orgnal obectve functon, and (c()) s the enalty functon based on the equalty and nequalty constrants. For a mnmzaton roblem, the man ont s to choose the enalty functon to make sure that t s zero for all feasble onts and s very hgh for all non-feasble onts. Then, the mnmum of h() s equvalent to the mnmum of f().

41 3.9.6 Our case For our roblems, the otmzaton queston s defned as follows: Fnd the mnmum and mamum of functon (, y) + ( ( ) ( ( y y y y f ρ subect to: l u s y where l l... l ), u u... u ), s s... s ) and... ) ( n ( n ( m ( m y y ). l, u, s and are real numbers, not nfnty. Ths knd of queston s called bo-constraned otmzaton or boundconstraned otmzaton. We only dscuss the mnmzaton roblem. For mamum roblems, we can use the revous formulaton to convert them to mnmzaton roblems. et, we need to convert the roblem to an unconstraned otmzaton. Let use the three methods ntroduced revously Transfer functon The constrants for varable and y are l u, and s y. Ths means that les between l and u and y les between s and. so we need to ntroduce a new varable to relace and make sure satsfes the constrant. Defnng l + ( u l)sn u, and y s + ( s)sn v we wll get the new obectve functon f ( u, v). For ths functon, y belongs to the whole real doman, so t s unconstraned. ut ths functon s very comlcated. If you want to use the frst dervatve to get the soluton, t s trcky because y has many solutons Lagrangan multlers For l u, s y, we can convert to: l 0, u 0, y s 0, and y 0. Usng the revous methods, we can get a new obectve functon. ut ths method ntroduces many slack varables and equaltes. To solve these equaltes s not easy work. It needs much CPU tme to comute and t s also very comlcated.

42 Penalty functon We wll desgn a sutable enalty functon. ased on the constrants, we ntroduce ths enalty functon: ( ) λ * ma(0, l, µ,...,..., ln n, n µ n, s y, y,...,..., s y m m m m Here, λ wll be chosen as a very large ostve real number. So the new obectve functon s h (, y) f (, y) + (, y)., y ) From ths functon, we can see that f l u, and s y, then and y belong to the feasble regon and h(,y) equals f(,y), but f constrants are volated, h(,y) wll become very large, clearly far from the mnmum value Search method Our obectve functon has a good attrbute; both the frst dervatves and second dervates est. So gradent search (Luenberger, 984,. 384) can be freely aled to our case. And generally seakng, gradent searchng methods rovde effcent drecton nformaton n searchng for the net. In vew of the revous dscusson, gradent search s used to our case Soluton Fnd the mnmum value of functon f(), stated by Mn f (, y) + ( ( ) ( ( y y ρ y y y subect to: l u s y where l l... l ), u u... u ), s s... s ) and... ) ( n ( n ( m ( m. l, u, s and y ) are real numbers, not nfnty. We use a enalty functon to convert ths roblem to an unconstraned roblem. The new obectve functon h(,y) s constructed as: h(, y) f (, y) + λ *ma(0, l, µ,...,..., ln n, n µ n, s y, y,...,..., s y m m m m o the roblem s to fnd the mnmum value for functon h(,y). For ths unconstraned, y ) S

43 34 otmzaton roblem, the teratve technque s adoted. Frst, we defne some terms: n m y y y ) ( ( D ) ( ( y y y y y D y y D D D *. ow we get the frst dervatve of h(,y) through f(,y) and enalty functon (,y). ) *( * * * / y y D D y f + ρ ) *( * * * / y y y y y y D D y f + ρ / D y f ρ ),,...,...,,,,,...,...,, 0, ma( ),,...,...,,,,,...,...,, 0, ma( 0 m m m m n n n n m m m m n n n n y y s y y s l l l y y s y y s l l u others µ µ λ µ µ λ ),,...,...,,,,,...,...,, ma(0, ),,...,...,,,,,...,...,, ma(0, 0 m m m m n n n n m m m m n n n n y y s y y s l l y s y y s y y s l l y others y µ µ λ µ µ λ Along the drecton determned by the dervatves, the net and y are defned. Through teraton, the numercal soluton can be found. et, fndng the mamum value of functon f(), Ma ), ( y f subect to: u l, y s. ased on the formulaton )), ( mn( ), ( ma y f y f, we can transform ths roblem to:

44 35 Mn f (, y) subect to: l u, s y. Usng the revous method, we can get the mnmum value fmn, and negate to get the mamum value of f(), fmn.

45 36 3 Enhancement of functons Ths verson of Statool removes certan mortant lmtatons estng n the revous verson. The followng etensons had to be develoed n order to aly Statool to the roblems we wanted to solve. Use of the transortaton method to seed lnear rogrammng Cascadng oeratons to suort more than two varables Relatonal oeratons Evaluaton of f(,y) for monotonc functons f 3. Transortaton method In the revous verson, only the standard smle method s rovded to solve lnear rogrammng. The seed of ths method s slower than that of the transortaton smle method. 3.. ackground on the transortaton smle method Many comanes need to determne how to otmally transort goods from dfferent warehouses to dfferent destnatons. Isomorhc roblems are found n other stuatons unrelated to transortaton, such as the assgnment roblem and roducton schedulng. Hller (00) gave detaled nformaton about such alcatons Model In general, ths knd of roblem nvolves dfferent tyes of locaton: sources and destnatons. Sources suly somethng and destnatons accet resource. Costs for transferrng resources between each source and destnaton may be dfferent. The am s to mnmze the total cost to transfer resource from these sources to those destnatons. In most cases, the total suly for all sources s equal to the total demand for all destnatons. If we have M sources, destnatons, the suly at source s S, and the demand at destnaton s D, we get the equaton M S D. Let C be the unt cost of movng resources from

46 37 source to destnaton. Ths table dslays the relatonsh between sources and destnatons. Table 3.. Parameter table for transoraton model. Cost er unt dstrbuted Destnaton 3 Source Suly C C C3 C S C C C3 C S M C M C M C M 3 C M S M Demand D D D3 D We can descrbe ths mode as a standard lnear rogrammng roblem. mn Z M C Subect to: S for M M D for and 0 for all and If total suly s not equal to total demand, t s called an unbalanced model. For these cases, we can use dummy sources or destnatons to make the model balance. If total suly s greater than total demand, we can make u dummy destnatons to demand etra resources and set the unt cost from each source to any dummy destnatons to be very small. Ths way, etra resources wll be transferred to dummy destnatons. If total suly s less than total

47 38 demand, we make u some dummy sources and set unt cost from each dummy source to any destnatons very large. If these unt costs are really large, no destnaton wll want to get resources from these dummy sources. So the soluton wll be for resources from actual sources rather than dummy sources Soluton The Transortaton roblem s a secal tye of lnear rogrammng. We can use general methods for lnear rogrammng such as the smle method. If the smle method s used, the smle tableau wll be comle and conssts of M++ rows and (M+)(+) columns. To handle ths bg table, you wll need a lot of comutaton. As a secal tye of lnear rogrammng roblem, there s an effcent method called the transortaton smle method to handle t. Ths method uses a tableau, but t only has M rows and columns. You don t need to use artfcal varables to get an ntal soluton. It has ust M+- basc varables (not M+), so a degree of freedom wll be removed. To solve transortaton roblems, generally two stes are necessary. Ste One: Intalzaton to get an ntal basc feasble (F) soluton. There are 3 common methods for ths ste. orthwest corner rule Russell s aromaton method Vogel s aromaton method Russell and Vogel s methods consder costs n generatng an ntal soluton. The solutons are better than for the orthwest corner method. Hller and Leberman (00) clearly comares these three methods. Ste two: Otmalty testng. In ths ste, every soluton s a feasble soluton. Our am s to fnd the best soluton. It has a loo to do the followng work. Get the two varables u and v from each basc varable s equaton (Cu+v). Calculate the related cost CC of each non-basc varable accordng to CCCu-v. Get the enterng non-basc varable, the one wth the mnmum CC of all nonbasc varables wth negatve CC.

48 39 Determne whether the soluton s otmal. If all CC are not less than 0, the soluton s otmal. Get the leavng basc varable. Ths s done n a loo whose calculatons use the enterng non-basc varable and other basc varables. Ths loo dentfes the cell whose assgned flow s the mnmum and whose order to the enterng cell s odd. Ths cell wll be the leavng varable. Adust the flow of the loo. For all the cells adacent to the enterng cell or another odd dstance from t n the loo, subtract the mnmum flow and for all cells an even dstance, add the mnmum. Get the new basc varable set. Markng the enterng cell basc varable and the leavng cell non-basc varable. egn the loo agan from ste. 3.. Ecetons n fndng the ntal soluton Handlng the eceton of degeneracy can be very mortant n fndng the ntal soluton n a transortaton smle roblem. Degeneracy means there are not enough basc varables n the ntal feasble soluton. For eamle, there are 5 basc varables for *3 tables. In fact, maybe only 4 varables are found for some ntalzaton methods for some roblems. Ths stuaton occurs where there are too many choces for whch ones are basc. In the revous ntalzaton methods, the northwest corner method doesn t have ths knd of roblem. Ths method always can fnd enough basc varables although values of some of them may be zero. ut Russell s method wll have ths knd of roblem for some cases. Usually, the ntal soluton found by Russell s method s closer to the otmal soluton than that found by the northwest corner method. So comutng tme s less for Russell s method. Therefore, there s a tradeoff Adataton to the unknown deendency case For the unknown deendency case, the margnal dstrbuton table for varables X and Y s really a transortaton tableau. Here you can consder X as the sources and Y as the destnatons. The total suly s and total demand s also. et we use an eamle to llustrate ths stuaton. Eamle:

49 40 X dstrbuton: P([0,]) 0., P([,]) 0., P([,3]) 0., P([3,4]) 0.4. Y dstrbuton: P([,] 0.5, P([,3]) 0.5, P([3,4]) 0., P([4,5]) 0.3. Consder X+Y for the case of unknown deendency. We get the margnal dstrbuton table net: Table 3.. Margnal dstrbuton X [0,] [,] [,3] [3,4] Prob. Y [,] [,3] [,4] [3,5] [4,6] [,3] [,4] [3,5] [4,6] 3 [5,7] [3,4] [3,5] [4,6] [5,7] [6,8] [4,5] [4,6] 4 [5,7] 4 [6,8] 43 [7,9] Prob Our queston s how to assgn the dstrbuton to to 44 to gve some subset a mamzed robablty. E.g. to fnd the uer bound for X+Y at (the revous chater dscussed fndng the subset), we get the lnear rogrammng roblem Ma f subect to: ++3+P4 0.5 P To fnd the uer bound for the CDF at, we get the roblem Ma f++ subect to: ++3+P4 0.5 P For every ont n the suort of the result dstrbuton, we wll get a lnear rogrammng roblem. Through solvng these roblems, the uer bound of the CDF wll be gotten.

50 4 The low bound of the CDF s found smlarly to the uer bound. To seed the solutons, we use the transortaton method to solve these lnear rogrammng roblems. From the revous eamle, we can see these lnear rogrammng roblems use transortaton tables. The man dfference s to mamze the value of the obectve functon rather than the mnmze as n real transortaton roblems. To solve these roblems, we can use negaton to transform Ma to Mn. The C are very mortant n transformng the roblems. For the obectve functon, we have to let C be,0, or -. We need to transform Ma to Mn, so we must use C- for all tems that wll contrbute to the obectve functon, wth others zero. For the revous cases we wll get: Mn -f- subect to: ++3+P C -, other C0 Mn -f--- subect to: ++3+P CCC-, other C0 Thus we have a way to transform an unknown deendency case to a transortaton roblem. It ncludes two stes: Get the transortaton table from the margnal dstrbuton table Set the cost attrbute for cells contrbutng to the obectve functon to, and other cells cost to zero. ecause the balance of suly and demand s a basc requrement for the transortaton roblem, we must kee margnal sum of X and Y equal to and the same for Y Test result: Consder an eamle: X, ([0,0.333])0., ([0.333,0.667])0.4, ([0.667,0.999])0.4 Y, ([0,0.5]) , ([0.5,])

51 4 Consder X+Y under the unknown deendency condton. Table 3.3. Lower bound. Result nterval Smle Transortaton [-0.5,0.833] E-08 0 [0.833,.67] E-08 0 [.67,.333] [.333,.5] [.5,.667] [.667,] [,.5] Table 3.4. Uer bound. Result nterval Smle Transortaton [-0.5,0] 0 0 [0,0.333] [0.333,0.5] [0.5,0.667] [0.667,0.833] [0.833,.67] [.67,.5] For ths eamle, we got almost the same answer for both methods. 3. Cascadng oeratons Prevous Statool software only suorted bnary oeratons (two oerands). ut n real alcatons, there are often over oerands to be calculated, for eamle, +y+z, Ma(,y,z), etc. Assocaton s how to solve ths queston. E.g. for +y+z, we can frst calculate +y, and save the result to temorary varable w+y, then calculate w+z. Ths way, we can get +y+z. ut there s a constrant that ths knd of oeraton must suort assocaton and commutaton. For eamle, you can frst calculate +y or y+z, for +y+z.

52 43 In Statool, we would get the CDF enveloes for the result of two varables oeraton under unknown deendence. So we can solve ths roblem f we can convert CDF envelos to a set of ntervals and assocated robabltes. 3.. Soluton We can transform uer and lower envelos nto a set of ntervals and assocated robabltes. The robablty of each enveloe s ts to-to-bottom heght. For eamle, four ntervals wll be gotten from the followng CDF enveloes. Fgure 3.. Convert CDF to IDF. Ths method s mlemented n Statool usng V. ow both CDF and IDF data format can be saved or dslayed. From the followng fgures, we can now use the results of one oeraton as nut to the net. Fgure 3.. Result for oeraton. The above fgure shows the CDF enveloes resultng from an oeraton on two varables.

53 44 The followng fgures show the rocedure to calculate usng multle oerands (e.g. +y+z). Fgure 3.3. Result for +y. Frst, we get the result of +y. Result s shown n the 3 rd anel. Fgure 3.4. Result for +y+z. Then we load the result of +y as a new oerand (to anel above) and oerate on t and z, whch s shown n the mddle anel. The bottom anel shows +y+z.

54 Relatonal oeratons Relatonal oeratons are used to descrbe the relatonsh between two oerands. Ths verson of Statool suorts 4 relatonal oeratons: >, >, <, and < Relatonal oeratons on ntervals Consder two real numbers and y. We defne the nterval value to descrbe the relatonsh between and y. The value [0,0] ndcates the relatonsh s false. The value [,] ndcates the relatonal oeraton s true. The value [0,] means the value of the relatonal oeraton s not determned or t s uncertan. For nterval number A, A-left means the left (or low) bound of nterval value A, and A-rght means the rght (or hgh) bound of t. ow consder two nterval numbers A and. [,] A > [0,0] [0,] A left > rght A rght left otherwse [,] A [0,0] [0,] [,] A < [0,0] [0,] [,] A [0,0] [0,] A left rght A rght < left otherwse A rght < left A left rght otherwse A rght left A left > rght otherwse 3.3. Relatonal oeratons on random varables Consder random varables X and Y. We consder the robablty of X>Y, P{X>Y}. Accordng to the DEnv algorthm, random varables X and Y are slt nto ntervals whch are assgned robabltes. Therefore, oeraton X>Y s transformed nto a seres of nterval oeratons. Here s an eamle to show how to handle ths.

55 46 Table 3.5. Dstrbuton for X and Y. X [0,] [,] [,3] Prob. Y [,] [,3] [3,4] 3 P Prob Consder the relatonal oeraton X>Y. It s transformed nto an nterval relatonal oeraton between ntervals of X and ntervals of Y. for eamle, the result of [0,] > [,] s [0,0], so [0,0] wll be ut nto cell. smlarly, [0,] wll be ut nto cell. Fnally, we get the followng result: Table 3.6. Interval value for relatonal oeraton. X [0,] [,] [,3] Prob. Y [,] [0,0] [0,] [0,] [,3] [0,0] [0,0] [0,] [3,4] [0,0] 3 [0,0] P3 [0,0] Prob ased on the DEnv algorthm, we can now get the robablty for X>Y. It s clear that all cells whose nterval bounds nclude are consstent wth X>Y. To get the mamum value of P{ >y}, the sum of all cells ncludng wll be mamzed. For ths case, mamze (+3+3). To get the mnmum value of P{>y}, the sum of all cells wth the value [,] wll be mnmzed. All cells whose value s [0,] wll be dscarded snce for them, maybe <y. In summary, the value of each cell should be one of [0,0], [0,], and [,]. Here [0,0] means ths relatonsh doesn t hold. The value [0,] means ths relatonsh s not certan. The value [,] ndcates ths relatonsh must hold. To get the mamum value, mamze

56 47 the sum of all cells whose bound nclude. To get the mnmum value, mnmze the sum of all cells whose values are [,]. 3.4 Comle eressons The revous Statool only suorted the basc arthmetc oeratons +,-,*, and /. It s more useful to be able to calculate any arthmetc eresson. User should be able to nut the eresson desred. To solve ths roblem, Statool needed an arthmetc eresson edtor to rovde the functonalty to nut an arthmetc formula. We decded to suort eressons usng arthmetc oerators +,-,*, and / and also to suort assocaton through usng ( ) Eresson edtor To mlement ths eresson edtor, frst, a grammar defnton of allowed eressons was wrtten. Ths grammar s contet-free. The followng grammar descrbes arthmetc eressons that Statool suorts: <eresson>::<term> <term> + <eresson> <term> - <eresson> <term>::<factor> <factor> * <term> <factor> / <term> <factor>::(<eresson>) <number> <varable> <number>:: <nteger> <nteger>.<nteger> <nteger>::<nteger> v <varable>:: y Here v ndcates the numbers from 0 to 9. ased on ths grammar, arthmetc eressons such as (a*x+b*y)/(c*x+d*y) are allowed. Parsng s a good method to generate a arse tree: a dagram of the comlete grammatcal structure of the strng beng arsed. For ths case, t s not very comle. Every oerator needs two oerands. ( ) wll ncrease the rorty of oeraton. So, eresson tables could be used. In such a table, the oerands and oerators wll be recorded. Every row descrbes an oerator. Software wll analyze the nut strng, and generate the eresson table accordng to the order of calculatons. Then usng ths table, the result can be calculated. Statool software now only suorts random varables, X and Y. When an eresson s nut, varable names must use the symbols X and Y. The eresson edtor wll check the nut eresson after the user confrms the nut. If ths eresson s not allowed,

57 48 error nformaton wll be dslayed and reason also wll be lsted. Here s a fgure showng the eresson edtor. Fgure 3.5. Eresson edtor. From the followng fgure, we can see the error nformaton and reason for a bad nut eresson. Fgure 3.6. Error nformaton for eersson edtor Lmtatons on evaluatng eressons Ths eresson edtor can t handle dvson by 0 snce the eresson evaluator doesn t know how to evaluate the value of such eressons. Therefore, oerands X and Y can t nclude zero n ther suort f the user want to use the eresson edtor.

58 Ecess wdth n eressons A tycal eresson s P(,y) f(,y)/g(,y) (ax+by)/(cx+dy). For ths knd eresson, there s ecess wdth n nterval calculatons. The reasons nclude that a random varable s cted more than one tme n the eresson. To solve the roblem, t s necessary to remove ecess wdth n calculatng ths tye of eresson Removng ecess wdth The easest way to handle t s to smlfy the eresson such that each random varable s cted only one tme. It s a quck way to handle ths queston. ut t s a very restrctve constrant. Many eressons can t be smlfed to meet ths knd of condton. For some eressons, we can use another way to remove ecess wdth. It s to use the low and hgh bounds of the nterval oerands to calculate the eresson. Then from these calculated values, the result bound s determned. For two varables, there are four combnatons of bounds. So 4 canddate result values are obtaned. We select the mnmum of the 4 as the low bound of the result nterval, and the mamum of the 4 as the hgh bound of the result nterval. Let us see an eamle: Suose: [,], y [,3] F(,y) ( y)/( y). Frst: let, y, calculate F(,y), and we get the value 85. Second: let, y3, calculate F(,y), and we get the value 300. Thrd let, y, calculate F(,y), and we get the value 60. Fnally, let, y3, calculate F(,y), and we get the value So, the nterval for F(,y) s [60,300]. If we calculate the eresson based on the ntervals for and y, we wll get the nterval [6.9, 480]. It s obvous that result nterval has ecess wdth. So we can use ths method to remove ecess wdth for ths eresson Lmtaton of the method for removng ecess wdth Although the method of selectng the mn and ma value to get the result bound works for our case, there are lmtatons to ths method. When the eresson s monotonc, the method can handle ecess wdth. In addton, the denomnator of the eresson can not

59 50 nclude the zero, that s, zero s not n the suort of g(,y), otherwse ths eresson can t be calculated.

60 5 4 Software archtecture Ths chater descrbes the archtecture of Statool. Frst an overvew of ths software s gven. Then the comonents of Statool are descrbed. 4. Overvew Layer desgn s wdely used for software develoment. It rovdes a clear descrton of software archtecture and makes t easly understood. It s also sutable for mlementaton and mantenance of dstrbuted comutng software. Fgure 4. shows the general layer archtecture of Statool. Ths verson was develoed for a Mcrosoft Wndows latform. Layer desgn s helful for ortng to other latforms. Fgure 4.. Archtecture. Statool conssts of 3 levels layers as shown n the revous fgure. The frst level, the alcaton layer, s the user nterface layer, whch s n charge of nteracton wth the user such as recevng the user settngs and dslayng the results of oeratons. The mddle level transforms user nuts to ft the underlyng algorthms. Ths layer can be called the logcal layer. The low level, the comutng layer, mlements the secfed algorthms. It can run n the background and be rovded as dynamcally lnked lbrary. Snce Mcrosoft oeratng systems are wdely used n the world, the latform for ths software s the Mcrosoft wndows seres, such as Wndows 98 and Wndows 000. ut the

61 5 rmary latform s the Wndows 000 famly. The software wll be fully tested on ths latform. For other wndows latforms, t won t be fully tested. For the current software, the user nterface and the logcal layer were develoed usng Vsual asc. Vsual asc s a very good rad develoment tool. Snce the user nterface s not desgned to run n a Web browser currently, t was develoed together wth the logcal layer comonent usng Vsual asc. It may be useful to move the user nterface to a Web browser latform n the future because the browser / server archtecture s so oular. The comutng layer nvolves a lot of comutng, so seed s a key ssue. Therefore, ths layer was develoed usng Vsual C++. Ths software can be run on any machne n whch wndows 000 s nstalled. There s no other requrement for hardware. ut as revous noted, ths software wll consume a lot of comutng and memory resource. If you solve bgger roblems, a confguraton should have at least 56M memory and a Pentum III at 000 Mhz. 4. Inut/outut The fle s the rmary way to echange and save the data for ths software. Grahs are used to show robablty dstrbuton functons (PDFs) and cumulatve dstrbuton functons (CDFs) of the data. Data lsts can show the eact nformaton reresentng a random varable. In ths software, 4 knds of fles are used to descrbe random varables. One s called a robablty dstrbuton fle (PDF), whch s a lst of ntervals and robabltes and, and dscretzes a robablty densty functon. The second s called a cumulatve dstrbuton fle (CDF), whch s used to descrbe the enveloes of the cumulatve dstrbuton. The thrd knd s called an ntermedate dstrbuton fle (IDF), whch s lke a.pdf fle but the ntervals can overla. Although ts nterval can be overlaed, the sum of ther robabltes should be equal to. Otherwse, ths fle s nvald. The last one s the samle data fle (SMP). It s a secal tye of fle. It s used to show the robablty dstrbuton of samle data. Only IDF and SMP fles are used to nut or outut dstrbutons. PDF and CDF fles are manly used nternally. Outut dslays are lotted based on these tyes of fles. oth IDF and SMP are tet fles. They are comosed of arts: a control lne and data lnes. The control lne s the frst lne and ncludes tems: the number of ntervals and

62 53 whether t s an alcaton fle (ths arameter s not used for most cases). The comma s used as a searator. The number of data lnes s secfed by the number of ntervals art of the control lne. Every data lne ncludes 3 tems: low bound of an nterval, hgh bound, and robablty for ths nterval. The followng fle gves an eamle IDF fle: 4,0 0,0.5,0. 0.5,0.50, ,0.75, ,.0,0.5 Ths fle descrbes the dstrbuton of a varable consstng of 4 bars from 0 to. A samle fle s a secal.idf. In ths tye of fle, the low bound of every nterval s set to equal the hgh bound. It ust s used to say how much the robablty at ths ont s. Two dfferent ways are used to nut data n Statool. One s to drectly load the dstrbuton of a random varable from a fle. The other s to edt the dstrbuton of a varable. To use the edtor to nut the dstrbuton, Xe (998) gave a detaled descrton. There are two ways to outut the result of an oeraton. One s to draw the grah of the varable. It s convenent for seeng the result mmedately. The other way s to save data as an IDF or SMP fle. These two fles are dscussed n the revous aragrah. The user can ermanently kee data n and use these fles n the future. There are tyes of grah to show the data: robablty bars for PDF format and robablty bound curves for CDF format. From the robablty bars of PDF format, the user can see the hstogram of random varable. Probablty bound curves show envelos boundng the sace of ossble cumulatve robablty functons of a random varable. 4.3 The user nterface "A user nterface s an nterface that enables nformaton to be assed between a human user and hardware or software comonents of a comuter system." [IEEE, Std ]. ased on the Mcrosoft latform system, wndows are used to mlement the user nterface. Generally, the user nterface conssts of varous wndows. They can be moved on the screen, overla each other and mnmzed nto cons on the task bar of MS oeratng system.

63 54 In general, a rogram runnng on an MS latform ncludes the rmary wndows and the secondary wndows. The rmary wndows handle maor nteractons wth the user, and often contan an arbtrary number of obects. Secondary wndows are used to suort the nteractons wth rmary wndows by rovdng detals about the obects of the rmary wndow and oeratons on those obects. Statool s comrsed of these two tyes of wndows. There are three rmary wndows and there are secondary wndows for each of the rmary wndows. The three rmary wndows are the oeraton wndow, the data edtor wndow, and the data vew wndow. The oeraton wndow s the man rmary wndow. The other two rmary wndows can be accessed from ths man rmary wndow. To decrease nteracton overhead, wndow navgaton aths are restrcted to three levels. Otherwse the user wll be lkely to get lost n the system. The oeraton wndow s oened when the user starts the rogram and s always oen as long as the rogram s runnng Man rmary wndow: the oeraton wndow The oeraton wndow s the lace to erform the bnary oeratons. It also ncludes some assocated functons such as fle oeratons and settng otons. The user can use ths wndow to erform the followng functons: Data mantenance ( ncludng load, save, vew, and edt data) Data oeraton ( choosng the oeraton tye) Otons for data oeraton Hel and addtonal nformaton about ths software Control software (ncludng rogram termnaton and otons to run ths software)

64 55 ased on these functons requrement, ths fgure gves a rototye nterface for t. Fgure 4.. Oeraton wndow. Ths wndow conssts of fve arts: menu bar, data dslay anels, oton area for correlaton, oeraton area, and et button. The menu bar s used to contan all functonaltes entres. Data dslay anels nclude 3 anels. Two anels are used to store oerand X and Y, and a thrd one to show the oeraton result Z. The oton area s used to choose the correlaton for oerands X and Y. Currently, 3 tyes of correlaton are suorted for ths software. They are ndeendent, unknown, and known correlaton. the oeraton area contans all the oeratons. The user can choose the secfc oeraton to calculate. The et button s a convenent way to et the rogram although you can also do ths by selectng the et oton on the menu bar. The oeraton wndow ncludes 5 roerty wndows, whch are the secondary wndows. They are the dslay mode wndow, dslay color settng wndow, about wndow, correlaton value settng wndow, and eresson edtor wndow. The dslay mode wndow s used to choose the data dslay mode: PDF or CDF. The dslay color settng wndow s used to choose the colors for the 3 dslay anels. The about wndow lsts nformaton about

65 56 ths software. The correlaton value settng wndow enables the user to nut nformaton about correlaton when the user selects the known correlaton rado button Other rmary wndows: the data edtor and the vew wndows The data edtor wndow s used to edt the nut data, whch s shown n the followng fgure: Fgure 4.3. Data edtor. Ths wndow ncludes a menu bar, a dslay anel for data, oeratons for edtng data, and control buttons for the wndow. The data edtor wndow ncludes 4 roerty wndows: ar number, Value range, Value for sngle bar and About. The bar number wndow s used to set the total number of bars. The value range wndow s used to set the range for the oerand. The value for sngle bar wndow controls the robablty and the wdth for ths bar. The about wndow s the same as the oeraton wndow. The data vew wndow s used to show the eact values for data. It s shown n the followng fgure:

66 57 Fgure 4.4. Data vew. Ths wndow conssts of arts: menu bar and dslay table for data. The user can choose the eected data format to show n the table. The data vew wndow does not nclude any roerty wndows Wndows management All wndows of Statool are comatble wth the standard of MS wndows. Every wndow ncludes the standard functons: move, sze, mn, ma, restore and close. The followng vsual dmensons wll be consdered: oston, sze, shae and color. Poston: every wndow wll be laced n the central of the current screen. Sze: the man rmary wndow wll not eceed els to make sure t can be shown on any montors. Other rmary wndows wll not eceed the sze of the man rmary wndow. Proerty wndows can t eceed the sze of ther rmary wndows. Shae: set dfferent shaes for dfferent obects. Color: kee the colors consstent for the same arts of dfferent wndows, such as background and foreground color.

67 58 The length of the wndows navgaton ath s three. The followng fgure shows the wndows navgaton ath: Oeraton wndow Edtor Wndow ar number Value range Value for sngle bar Data Vew About Dslay mode Dslay Color About Correlaton value Eresson edtor 4.4 The logcal layer Ths layer les between the user nterface and the comutng layer. It contans access onts from the user nterface, all logcal controls and nternal data, utlty functons and access onts to call the net layer. Ths layer ust rovdes the subsystems to servce the user nterface and doesn t erform any actons related to the real functons: uncertanty oeratons, whch are the real work of ths software. Therefore, t makes sense to call t the busness-secfc layer. The fgure shows the structure of ths layer:

68 59 Fgure 4.5. Logcal layer. The maor functon of the logcal layer s to ma user actons to the logcal functons vew of the software. For eamle, the user mght clck to load a fle from storage nto memory n the logcal vew. Ths layer wll do the real work n resonse to an acton haenng n the user nterface. Ths art manly ncludes the resonse functons for the menu bar and resonse functons for the rado buttons dslayed n the user nterface. Ths layer wll also rovde the many utlty functons and classes and mantenance of nternal data structures to kee the mortant state nformaton. Fnally, ths layer rovdes a unform nterface to the comutng layer. All the comutng senstve work s ut nto the comutng layer. To use and manage these modules effcently, the standard nterface s necessary. It also ncreases the reusablty of these modules. In summary, ths layer s used to formulate the user roblem to ft the develoed uncertanty algorthm. Three man functons are rovded n ths layer: logcal functons to resond to the user actons, mantenance state nformaton to control uncertanty oeraton, and effcent access onts to call the comutng layer. 4.5 The comutng layer Ths layer s the comutng subsystem. It mlements almost all the algorthms of the uncertanty oeratons. It s obvous that ths layer s comutng senstve and wll consume a lot of comutng and memory resources.

69 60 Ths layer was develoed searately from the other two layers snce t does not really deend on the other two layers. A beneft s that ths layer can be etended to run on other comutng resources to decrease ts deendency on comutng resources at the clent sde. ased on the MS latform, a dynamcally lnked lbrary (DLL) was used. Callng a DLL s language ndeendent. Any develoment languages can call ths tye of lbrary by followng the callng conventons. Ths layer ncludes manly 4 ackages: convertng data, general smle method, transortaton smle method, and etended smle method. These 4 ackages are ndeendent of each other. There s no callng relatonsh among them. Every ackage ust fnshes the secfc functon usng the corresondng algorthm. mydfcdf ma t_ma cor_mn Fgure 4.6. Package vew for the comutng layer.

70 6 5 Comonent desgn and mlementaton Ths chater lsts detaled nformaton about comonents belongng to dfferent layers. At the same tme, mlementaton ssues also are ncluded wth comonents desgn. 5. Overvew Comonents desgn s closely related to the develoment language. For the user nterface and the logcal layer, MS Vsual asc s used snce t s a very oular rad develoment tool. Each wndow mas to a form. Resonse actons ma to a functon or routne of ths form. So these two arts are ntegrated together. Ths s a convenent way for mlementng although searated from the logcal vew. Internal data structures and utlty functons and classes wll be defned n module fles of vsual basc. For the comutng layer, Vsual C++ s used snce DLLs develoed n t rovde accetable comutng seed. 5. Oeraton and roertes wndows The oeraton wndow s the rmary wndow for ths rogram. So t s the man form (also the starter form) for the vsual basc mlementaton. The comled rogram, Statool wll start from ths form. Ths wndow also ncludes 5 roertes wndows: dslay mode, dslay color, about, correlaton value, and eresson edtor. 5.. Oeraton wndow Ths form s called frmman n the roect form vew. The followng fgure ndcates ths form:

71 6 Fgure 5.. Oeraton wndow desgn Menu bar There are s tems n the menu bar. They are Fle, Edt, Swtch, Vew, Otons, and Hel. Fle: manage fles and control rogram runnng. Ths fgure shows ts submenu: Fgure 5.. Fle menu. It ncludes 6 submenus: ew, Oen, Save, Save As, Prnt and Et. ew: create a new IDF fle for oerand X or Y. Resonse subroutne s mnuewx_clck() for new X and mnuewy_clck() for new Y n form: frmman.

72 63 Oen: oen the estng fle for oerand X or Y. Resonse subroutne s mnuoenx_clck() for oen X and mnuoenx_clck() for oen Y. Save: save the current result Z. Resonse subroutne s mnusave_clck(). Save As: save the current result Z as a secfc fle. Resonse subroutne s mnusaveas_clck(). Prnt: rnt the current oeraton wndow. Resonse subroutne s mnuprnt_clck(). Et: termnate the current rogram. Resonse subroutne s mnuet_clck. Edt: edt the secfc oerand. Ths fgure shows ts submenu: Fgure 5.3. Edt menu. Edt PDF: actvate the edtor wndow to edt the PDF fle chosen from X, Y and Z. Resonse subroutne s mnuedtx_clck() for oerand X, mnuedty_clck() for oerand Y, and mnuedtz_clck() for oerand Z. Clear Wndow: clear the secfc dslay anel. Resonse subroutne s mnuclearwn_clck(). Swtch: swtch the fles between the 3 man wndow anels. It rovdes a convenent way to echange dslays among the three dslay anels. Ths fgure shows ts submenu: Fgure 5.4. Swtch menu. Interchange X Y: swtch oerands X and Y. Resonse subroutne s mnuinterxy_clck(). Interchange X Z: swtch oerands X and Z. Resonse subroutne s mnuinterxz_clck(). Interchange Y Z: swtch oerands Y and Z. Resonse subroutne s mnuinteryz_clck().

73 64 Vew: rovde the entres to set the dslay mode for anels and actvate the data vew wndow. Ths fgure shows ts submenu: Fgure 5.5. Vew menu. Dslay Mode: actvate the roerty wndow to set the dslay mode for oerands. Resonse subroutne s mnudslaymode_clck. Data: actvate data vew rmary wndow. Resonse subroutne s mnudata_clck. Otons: set otons controllng the rogram and ts oeraton. Ths fgure shows ts submenu: Fgure 5.6. Otons menu. ColorSet: actvate the roerty wndow to set the dslay color for anels. Resonse subroutne s mnucolorset_clck. Algorthm: set the referred algorthm to handle the lnear rogrammng roblems. There are two choces: Smle method and Transortaton. Resonse subroutne s mnusmle_clck for Smle method, and mnutransortaton_clck for Transortaton. Hel: dslay hel nformaton and about nformaton for ths software. Ths fgure shows ts submenu: Fgure 5.7. Hel menu.

74 65 Contents: actvate the hel nformaton. Resonse subroutne s mnucontent_clck. About Statool: actvate the roerty wndow to show the about nformaton. Resonse subroutne s mnuabout_clck Dslay anels There are three dslay anels to contan oerands X and Y, and result Z. These dslay anels use Pctureo (one tye of Vsual asc vsual comonent). Ther names are cx, cy, and cz. They are used to show the grahcal reresentaton of PDF bars or CDF bound curves accordng to the dslay mode for X, Y and Z. Every anel rovdes a ou menu to suort mouse functons. When the rght button of the mouse s clcked, a ou menu wll be shown. Resonse subroutne s cx_clck for oerand X, cy_clck for Y, and cz_clck for Z. The followng fgures show 3 ou menus for 3 dslay anels. Fgure 5.8. Pou menu for oerand X. Fgure 5.9. Pou menu for oerand Y. Fgure 5.0. Pou menu for result Z. These ou menus rovde the same functonaltes as the menu bar. ut ts mlementaton s dfferent from that of the menu bar. Oerands X and Y use the same ou

75 66 tem obect, whose name s Poutem. Resonse subroutne for choosng ou menu s Potem_Clck. Result Z uses ts own ou tem obect, called PoZtem. Resonse subroutne s PoZtem_Clck to handle menu commands Correlaton settng Ths art conssts of 3 oton buttons (a tye of vsual comonent n V). They corresond to 3 tyes of relatonshs between X and Y. They are unknown deendence, known deendence and ndeendence. These 3 oton buttons are eclusve snce only one stuaton s true for the current oerands X and Y. Ths fgure shows ths art. Fgure 5.. Correlaton settng. Resonse subroutne s otde_clck for unknown de., otind_clck for ndeendent, and OtCorrelaton_Clck for known de Oeraton tye Ths art conssts of seven command buttons and one lstbo (both are the basc vsual comonents for vsual asc), whose names are cmdo to cmdo8. y choosng these buttons and lstbo, the user can choose the dfferent oeratons for oerands X and Y. Ths fgure shows ths art.

76 67 Fgure 5.. Oeraton tyes. utton X+Y erforms the addton oeraton between X and Y. Resonse subroutne s cmdo_clck. utton X-Y erforms the subtracton oeraton between X and Y. Resonse subroutne s cmdo_clck. utton X*Y erforms the multlcaton oeraton between X and Y. Resonse subroutne s cmdo3_clck. utton X/Y erforms the dvson oeraton between X and Y. Resonse subroutne s cmdo4_clck. utton ma(,y) fnds mamum value dstrbuton enveloes from X and Y. Resonse subroutne s cmdo5_clck. utton mn(,y) fnds mnmum value dstrbuton from X and Y. Resonse subroutne s cmdo6_clck. utton Parsng actvates the roerty wndow to nut eresson consstng of X and Y. Then t evaluates ths eresson usng X and Y. Resonse subroutne s cmdo7_clck. Lstbo erforms relatonal oeratons between X and Y, whch nclude 4 tyes of relatonsh: greater than, not less than, less than, and not greater than. Resonse subroutne s cmdo8_clck Et button Ths button rovdes a convenent way to termnate ths rogram. If the user wants to et ths rogram, by clckng ths button, ths rogram wll et. Resonse subroutne s cmdet_clck.

77 Proertes wndows of the oeraton wndow For the oeraton wndow, there are 5 roertes wndows. Ther wndows are used to set the arameters and control nformaton for the oeraton wndow. They are Dslay mode, Dslay color, About, Correlaton Value, and Eresson edtor Dslay mode Ths roerty wndow s used to set the dslay mode for oerands X/Y and result Z. There are tyes of dslay modes: PDF bars and CDF enveloe curves. Resonse form s called frmdslay. Ths fgure shows t: Fgure 5.3. Dslay mode wndow. Choosng the dfferent dslay mode for X, Y and Z wll change the nternal control varables da_dfmode, db_dfmode, and db_dfmode. When values for these varables are, PDF mode s chosen. Otherwse CDF mode s chosen Dslay Color Ths wndow controls the color for three dslay anels. Resonse form s frmcolorset. The followng fgure shows t.

78 69 Fgure 5.4. Dslay color wndow. Ths wndow controls the color varables for X, Y and Z. These varables are XManColor, XSecondColor, YManColor, YSecondColor, ZManColor, and ZSecondColor About Wndow Ths wndow dslays the coyrght nformaton for ths software Correlaton Value Wndow Ths wndow s used to set any known nformaton about correlaton. Resonse form s DlgCorSet. When user choose the oton Known de.. from the oeraton wndow, ths wndow wll be actvated. Through ths wndow, the user can nut any known correlaton nformaton that may narrow the CDF enveloe curves. Ths fgure shows ths wndow: Fgure 5.5. Correlaton settng wndow.

79 70 From ths wndow, there are 3 ways to nut nformaton about oerand X and Y: known correlaton range or eact value, known eectaton range for XY, and known eectaton and varance for X and Y Eresson edtor wndow User can nut eressons consstng of two varables X and Y. Resonse form s frmeresson. When user clcks the Parsng button on the man wndow, ths wndow wll be actvated so that the user can nut and edt the eresson desred. Ths fgure shows ths wndow: Fgure 5.6. Eresson edtor wndow. The eresson nut by the user wll be evaluated usng ntervals n the dscretzatons of X and Y. Ths form wll call the utlty functons n the modules fle, named MathParserRoutnes. All the functons related to eresson arsng are saved n ths modules fle. 5.3 Other rmary wndows The Data edtor wndow rovdes the data edtng functon for the user. The user can use ths edtor to create and modfy oerands X and Y. Resonse form s frmhstedt. The Data Vew wndow s used to dslay the data of X, Y and Z when user needs numercal outut nstead of grahs. Resonse form s frmvewer. 5.4 Lower levels of logcal layer From the structure of the logcal layer, the lower levels consst of utlty functons, nternal data structures, and access onts to the net layer.