OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming

Transcription

1 OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng

2 Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or mamum value, has to be obtaned. () Global mamum Local mamum Global mnmum Local mnmum FIGURE. Etremtes or a sngle varable uncton Etremt value can be obtaned va optmsaton, whch s dvded nto:. Unconstraned optmsaton.. Constraned optmsaton lnear/non-lnear programmng.

3 Chapter Optmsaton /. ngle Varable Unconstraned Optmsaton Etremtes, an, can be evaluated usng ether the quadratc nterpolaton method or the Newton method usng the condton o. For the quadratc nterpolaton method, consder a second order Lagrange nterpolaton equaton as ollowed: (.) () Actual mamum Appromated mamum FIGURE. Evaluaton o etremtes usng a quadratc uncton Eq. (.) s derentated to eld: Thus, t can be rearranged to get an optmsed value o : (.) Eq. (.) can be repeated untl converged.

4 Chapter Optmsaton / Eample. Use the quadratc nterpolaton method to obtan a mamum value o the ollowng uncton accurate to our decmal places: sn. usng ntal values o, dan. oluton From the gven uncton: sn. () sn().().65 sn.. 9 Usng Eq. (.), the value o can be estmated as ollowed: ( ) (.65)( ) (.9)( ) ( ) (.65)( ) (.9)( ) ( ) sn(.66).(.66). 666 The overal process s as ollowed:.66 ( ) ( ) ( ) ( ) Hence, the mamum value s ().69 at.5. An etremt can ether be a mnmum or mamum value, or otherwse, : dependng on the second dervatve. > () s mnmum,. < () s mamum,. the coordnate, s an nlecton pont. [ ]

5 Chapter Optmsaton / 5 For the Newton method, consder an equaton smlar to the Newton- Raphson ormula (requrng onl one ntal value): g g ( ) ( ) I () s the rst dervatve o g(),.e. g g() s an etremt or (), or ( ) ( ), ths the root o (.) Eample. Use the Newton method to obtan the mamum value o the uncton: sn. usng an ntal value o. Use the convergence crteron o an appromated error o less than.5%. oluton From the gven uncton: cos. sn. Usng Eq. (.), the teraton ormula s: whch produces cos. sn. ( ) ( ) ( ) ε a (%) E E.959. Hence, the mamum value s ().698 at.5.

6 Chapter Optmsaton / 6. Multvarable Unconstraned Optmsaton For a multvarable case, etremtes can be evaluated usng the gradent method va the steepest slope condton. For a multvarable case, the gradent vector can be wrtten as, K,, n T z (,) FIGURE. Optmsaton or the -D case z (, ) Consder the equaton o two varables: ( ) The objectve s to obtan a condton where z, (.), and or ths case: j Ths vector wll gude the soluton towards a normal drecton (or orthogonal) to a contour lne o constant (,).

7 Chapter Optmsaton / 7 I h s the dstance needed to reach the etremt, the net appromaton to and are h, h (.5) Thus a uncton g(h) can be ormed such that, (.6) ( h) h h g and the relaton g h values o and. gves the optmsed h and hence the optmsed For the multvarable cases, the tpe o etremtes s determned usng the Hessan H parameter, whch has been dened as The parameter H s equvalent to H (.7) or a sngle varable case, where:. H > and > (,) has a local mnmum,. H > dan < (,) has a local mamum,. H < (,) has a plateau. Eample. Mamse the ollowng uncton: (, ) usng the gradent method o the steepest slope usng an ntal values o dan. Get the answer accurate to three decmal places. oluton In the rst teraton: () 6 6

8 Chapter Optmsaton / 8 g ( h) ( 6h, 6h) ( 6h)( 6h) ( 6h) ( 6h) ( 8h 7h 7 Thus ater the rst teraton: In the second teraton: g ( h) 6h 7. g h 6 6(.). (.). (.) (.) (.) (.).. ( h) (..h,..h).h g ( h).88h.88 h.....() The overall process s as ollowed: ) 6h.88h. h Fnall, the soluton converges at the -th teraton where dan,,. resultng n a mamum value o

9 Chapter Optmsaton / Mamum 5 6 FIGURE. Propagaton o estmated ponts o Eample.

10 Chapter Optmsaton /. Lnear Programmng In ths topc, onl the lnear case s consdered. The objectve o lnear programmng s to mnmse or mamse an objectve uncton Z,.e., Maksmumkan: Z c c L c n (.8) n Eq. (.8) s subjected to several constrants,.e. a a L a b (.9) I the varable j represents a postve phscal parameter, thus n n j (.) The smplest approach s va a graphcal method. Eample. Use the graphcal method to mamse the ollowng objectve uncton: Z 5 75 where the condtons or constrants are: oluton () 7 77, () 8 8, () 9, () 6, (5), (6). From the gure, the optmum pont s ( 8 8, ) mamum value o Z 8 9 whch produces the 9. Noted that condton () s redundant. 9

11 Chapter Optmsaton / FIGURE.5 Lnear programmng graph or Eample. One o the numercal approach s the smple method, where the searchng or the optmum pont s guded b the slag varable, as ollowed: Z c c L c n (.) n a a L a b (.) n n (.) j j I ths sstem contans k equatons and l varables ncludng the slag varables, where usuall k < l, hence there are (l k) varables whch has to be made zeros (non-bass a non-zero varable s known as bass). The Gauss-Jourdan elmnaton can be perormed to mnmse the objectve uncton. The elmnaton can be stopped when all the bass varables become zeros. Eample.5 Repeat Eample. usng the smple method.

12 Chapter Optmsaton / oluton The sstem can be rewrtten as ollowed: Mamse: Z 5 75, Wth condtons: () 7 77, () 8 8, () 9, () 6, (5),,,,., Begn wth Z. Then orm the ollowng table: Bass Z Z oluton At column, the element at row can be a pvot, hence s selected to be the nbound varable replacng. Then, perorm the Gauss elmnaton: Bass Z Z oluton 8 6 The coecent o at row Z s stll negatve, thus Z s stll not mamum. Hence, s selected to replace : Bass Z Z oluton

13 Chapter Optmsaton / Thereore the mamum o Z s.889 whch s produced at.889 and.889. In lnear and non-lnear programmng, there are our possble outcomes:. Unque soluton,. Multple solutons,. No possble soluton,. Unbounded problem. For cases -, the smple method cannot be used. (a) Multple solutons (b) No possble soluton FIGURE.6 (c) Unbounded problem Cases where the smple method s not applcable

14 Chapter Optmsaton / Eercses. Obtan the mnmum value o the ollowng uncton at usng the quadratc nterpolaton uncton usng the ntal values o.,.5 and 5., and the Newton method usng the ntal value o.5:. Obtan the mamum value o the ollowng uncton va the steepest slope wth the ntal value o (, ) (, ):.5. A compan produces two tpes o products, A and B. These products are produced durng normal workng das o hours per week and are marketed on the same weekends. The compan needs kg and 5 kg o raw materals or products A and B, respectvel. However, the compan warehouse can onl stores, kg o raw materals per week. Onl one product s produced at one tme, where product A requres.5 hour, whle product B requres.5 hour. Nevertheless, the temporar storage secton can onl keep 55 products per week. Product A s sold at RM5 per unt whle product B s sold at RM per unt. B usng the lnear programmng usng the smple method: a. Mamse the compan prot. b. Whch actor where ts ncrease leads to the astest ncrease n prot: raw materals, capact o temporar storage secton or producton tme?