Issue 8. Non-linear programmingIntroduction to Management ScienceIntroduction to Management ScienceIssue 8. Non-linear programmingIntroduction to Management ScienceTable of ContentsChapter 10 (Nonlinear Programming)The Challenges of Nonlinear Programming NLP with Decreasing Marginal Retrns! "#n$orNLP with Decreasing Marginal Retrns! Portfolio %election%eparable Programming Di&clt Nonlinear Programming Problems '(oltionar# %ol(er an$ )enetic *lgorithmsNonlinear an$ %eparable Programming'(oltionar# %ol(erIssue 8. Non-linear programmingIntroduction to Management ScienceRecall + ,ptimi-ation problems.n an optimization problem/ we see0 to minimi-e or ma1imi-e a speci2c 3antit# (the ob4ecti(e)/ which $epen$s on a 2nite nmber of inpt (ariables5These variables may be independent of one another, orThey may be related through one or more constraints. '1ample! minimize: z = x12 + x22su!ect to:x1 x2 = "x2 # "Issue 8. Non-linear programmingIntroduction to Management Science* mathematical program* mathematical program is an optimi-ation problem in which the ob4ecti(e an$ constraints are gi(en as mathematical fnctions an$ fnctional relationships5 optimize: z = $%x1& x2& x"& '& xn(su!ect to:g1%x1& x2& x"& '& xn(g2%x1& x2& x"& '& xn(g"%x1& x2& x"& '& xn('.gm%x1& x2& x"& '& xn(

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m=Issue 8. Non-linear programmingIntroduction to Management ScienceLinear programs * mathematical program is linear if f(x1, x2, x3, , xn) an$ each gi(x1, x2, x3, , xn), i = (1, 2, , m) are linear in each of their argments!$%x1& x2& '& xn( = c1x1 + c2x2 + ' + cnxnandgi%x1& x2& '& xn( = ai1x1 + ai2x2 + ' + ainxn"here cj an$ aij (i = 1, 2, , m; j = 1, 2, , n) are 0nown constants5*n# other mathematical program is non-linear5Issue 8. Non-linear programmingIntroduction to Management Science.nteger programs*n integer program is a linear program with the a$$itional restriction that the inpt (ariables be integers5.t is not necessar# that the coe&cients in the argments f(x) an$ gi(x) an$ the constants bi also be integers/ bt the# fre3entl# ma# be5Issue 8. Non-linear programmingIntroduction to Management Science6a$ratic (non+linear) programs* quaratic program is an e1ample of a non-linear program in which each constraint is linear/ bt the ob4ecti(e fnction has the form!$%x1& x2& '& xn( = i)n!)nci!x1x! + i)ndixi"here cij an$ i are 0nown constants5 '1ample!minimize: z = x12 + x22su!ect to: x1 x2 = "x2 # "Quadratic program with linear constraints, quadratic objective function, n = 2 variables, c11 = 1, c12 = c21 = , c22 = 1, and d1 = d2 = !Issue 8. Non-linear programmingIntroduction to Management Science'1amples of Linear an$ Nonlinear 7ormlas"inear #ormulas $onlinear #ormulas%8MPR,D8CT(D9!D:/ C9!C:);(D1 < D=) > D?@ A C9.7(D= BC =/ =AC?/ ?AC9)%8M.7(D1!D:/ 9/ C1!C:)%8M(D9!D:)=AC1 < ?AC9 < C:C1 < C= < C?%8MPR,D8CT(C9!C:/ C1!C?);(C1 < C=) > C?@ A D9.7(C= BC =/ =AC?/ ?AC9)%8M.7(C1!C:/ 9/ D1!D:)R,8ND(C1)M*D(C1/ 0)M.N(C1/ C=)*E%(C1)%6RT(C1)C1 A C=C1 > C=C1 F=*ata cells are located in *1:*+ and c,anging cells are in -1:-+.Issue 8. Non-linear programmingIntroduction to Management ScienceThe Challenges of Nonlinear ProgrammingNonlinear programming is se$ to mo$el nonproportional relation!hip! between acti(it# le(els an$ the o(erall measre of performance/ whereas linear programming assmes a proportional relationship5Constrcting the nonlinear formla(s) nee$e$ for a nonlinear programming mo$el is consi$erabl# more $i&clt than $e(eloping the linear formlas se$ in linear programming5%ol(ing a nonlinear programming mo$el is often mch more $i&clt (if it is possible at all) than sol(ing a linear programming mo$el5Issue 8. Non-linear programmingIntroduction to Management ScienceThe Challenge of Nonproportional RelationshipsProportionalit# *ssmption of "inear #rogramming!The contribution of each activity to the value of the objective function is proportional to the level of the activity. In other words, the term in the objective function involving this activity consists of a coefficient times the decision variable.Nonlinear programming problems arise when an# acti(it# has a nonproportional relation!hipwhere the contribtion of the acti(it# to the measre of performance is not proportional to the le(el of the acti(it#5Issue 8. Non-linear programmingIntroduction to Management SciencePro2t )raphs for "#n$or )lass Co5(Proportional Relationship) Production rate for doorsProduction rate for windowsWeekly Profit ($)Weekly Profit ($)300600900120050010001500200025003000002 4 624DWIssue 8. Non-linear programmingIntroduction to Management SciencePro2t )raphs with Nonproportional RelationshipsDecreasing Marginal ReturnsPiecewise Linear withDecreasing Marginal ReturnsIssue 8. Non-linear programmingIntroduction to Management SciencePro2t )raphs with Nonproportional RelationshipsDecreasing Marginal ReturnsExcept for DiscontinuitiesIncreasing Marginal ReturnsIssue 8. Non-linear programmingIntroduction to Management ScienceConstrcting a Nonlinear 7ormla1=?9H:IJ* E CConstructing a Nonlinear Formula.e/el o$ 0cti/it1 2ro$it2 31+4 3245 3286 3"717 3""2ro$it /s. .e/el o$ 0cti/it137353173153273253"73"57 2 4 + 8 17.e/el o$ 0cti/it1 %x(2ro$itIssue 8. Non-linear programmingIntroduction to Management Science*$$ Tren$line Dialoge Eo1Issue 8. Non-linear programmingIntroduction to Management Science*$$ Tren$line ,ptionsIssue 8. Non-linear programmingIntroduction to Management ScienceThe Tren$line (6a$ratic '3ation)Issue 8. Non-linear programmingIntroduction to Management Science%ol(ing Nonlinear Programming Mo$elsConsi$er the following mo$el in algebraic form!maximize:2ro$it = 7.5x5 +x4 + 24.5x" "8x2 + 27xsu!ect to:x 9 5x # 7Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C 01=?9H:I* E C D 'A Simple NLPMaximumx =7."61 := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 3".18Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C ?1=?9H:I* E C D 'A Simple NLPMaximumx =".12+ := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 3+.1"Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C 95I1=?9H:I* E C D 'A Simple NLPMaximumx =5.777 := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 37.77Issue 8. Non-linear programmingIntroduction to Management ScienceThe Pro2t )raphProfit ($)xIssue 8. Non-linear programmingIntroduction to Management Science,riginal "#n$or )lass Co5 %prea$sheet1=?9H:IJK10111=* E C D ' 7 )Wyndor Glass Co. Product-Mix Problem*oors ;indoours =sed 2er =nit 2roducedIssue 8. Non-linear programmingIntroduction to Management Science"#n$or )lass with Mar0eting Costs Mar0et research in$icates that "#n$or col$ sell small nmbers of $oors an$ win$ows with no a$(ertising5 Lowe(er/ e1tensi(e a$(ertising wol$ be re3ire$ to sell all that col$ be pro$ce$5 * cr(e+2tting proce$re was se$ to estimate the wee0l# mar0eting costs re3ire$ to sstain a pro$ction rate of $ $oors an$ % win$ows! Marketing cost for doors = !"*! Marketing costs for windows = #$$!%&';! The gross pro2t per $oor sol$ is abot M?IH/ an$ the gross pro2t per win$ow is abot MI005 Therefore/ the net pro2ts are as follows! (et profit for doors = &)"* * !"*! (et profit for windows = )++; * #$$!%&';! Ths/ the re(ise$ ob4ecti(e fnction is maximize: 2ro$it = 3"65, 25,2 + 3677- %3++2@"(-2 Question% Consi$ering the nonlinear mar0eting costs/ how man# $oors an$ win$ows shol$ "#n$or pro$ceNIssue 8. Non-linear programmingIntroduction to Management SciencePro2t )raphs for Doors an$ "in$ows Weekly profit ($)Weekly profit ($)0 2 4D200400600 8001,0001,2000 2 4 6W2004006008001,0001,200 1,4001,6001,800 Production rate for doorsProduction rate for windowsIssue 8. Non-linear programmingIntroduction to Management Science%prea$sheet 7ormlationIssue 8. Non-linear programmingIntroduction to Management Science)raphical Displa# of Nonlinear 7ormlation 0 1 2 3 4 5 D123465WFeasible region 3 14 5 28(3 , 4 ) = optimal solution Profit = $2,800Profit = $2,708Profit = $2,600Profit = $2,500Production rate for doorsProduction rate for windowsIssue 8. Non-linear programmingIntroduction to Management SciencePortfolio %election.t is now common practice for professional managers of large stoc0 portfolios to se compter mo$els base$ on nonlinear programming to gi$e them5.n(estors are concerne$ abot both the expecte return an$ the ri!&5,ne wa# of formlating their approach is as a nonlinear (ersion of a co!t-bene't trae-o( problem)Minimi.e/ 0isksubject to/ 1xpected return 2 Minimum acceptable levelConsi$er a portfolio with ? stoc0s5Question% "hat is the portfolio that will minimi-e the ris0 sb4ect to achie(ing at least an 1JO e1pecte$ retrnNIssue 8. Non-linear programmingIntroduction to Management ScienceData for %toc0sStoc&'(pected)eturn)is&*Standard+eviation,-airofStoc&s.oint )is&per Stoc&*/ovariance,1 =1O =HO 1 an$ = 05090= ?0 9H 1 an$ ? P0500H? J H = an$ ? P05010Issue 8. Non-linear programmingIntroduction to Management Science*lgebraic 7ormlationminimize:AisB = %7.2531(2+%7.4532(2+%7.753"(2+2%7.74(3132+2%7.775(313"+2%7.71(323"su!ect to:%21C(31 + %"7C(32 + %8C(3" # 18C31 + 32 + 3" = 177Cand31 # 7&32 # 7&3" # 7.Issue 8. Non-linear programmingIntroduction to Management SciencePortfolio selection sprea$sheet mo$elIssue 8. Non-linear programmingIntroduction to Management Science8sing %ol(er Table to e1amine tra$e+oQsbetween e1pecte$ retrn an$ ris0Issue 8. Non-linear programmingIntroduction to Management Science"#n$or )lass when o(ertime is nee$e$ "#n$or )lass has accepte$ a special or$er for han$+crafte$ goo$s to be ma$e in plants 1 an$ = throghot the ne1t for months5 7illing this or$er will re3ire borrowing certain emplo#ees from the wor0 crews of reglar pro$cts5 The remaining wor0ers will nee$ to wor0 o(ertime to tili-e the fll pro$ction capacit# of each plantRs machiner# for the reglar pro$cts5 The original constraints of Lors 8se$ S Lors *(ailable are still (ali$5 Lowe(er/ the ob4ecti(e fnction will nee$ to be mo$i2e$ becase of the a$$itional cost of sing o(ertime wor05 .n particlar/ becase of the a$$itional cost/ the pro2t per nit will be re$ce$ for those nits that re3ire o(ertime5Question% Consi$ering o(ertime costs/ how man# $oors an$ win$ows shol$ "#n$or pro$ceNIssue 8. Non-linear programmingIntroduction to Management ScienceData for "#n$or "hen ,(ertime is Nee$e$Ma(imum 1ee&l2 -roduction-ro3t per 4nit -roduced-roduct)egular5ime6vertime 5otal)egular5ime6vertimeDoors ? 1 9 M?00 M=00"in$ows? ? : H00 100(an$ ?$ < =% S 1J)Issue 8. Non-linear programmingIntroduction to Management SciencePro2t )raphs for Doors an$ "in$ows 9001,100Weekly profit ($)3 40Production rate for doors03 6Production rate for windows1,5001,800Weekly profit ($)DWIssue 8. Non-linear programmingIntroduction to Management ScienceThe %eparable Programming Techni3e7or each acti(it# that (iolates the proportionalit# assmption/ separate its pro2t graph into parts/ with a line segment in each part5Then/ instea$ of sing a single $ecision (ariable to represent the le(el of each sch acti(it#/ intro$ce a separate new $ecision (ariable for each line segment on that acti(it#Rs pro2t graph5%ince the proportionalit# assmption hol$s for these new $ecision (ariables/ formlate a linear programming mo$el in terms of these (ariables57or the "#n$or problem/ these new $ecision (ariables are *A = (umber of doors produced per week on regular time *D = (umber of doors produced per week on overtime ;A = (umber of windows produced per week on regular time ;D = (umber of windows produced per week on overtimeIssue 8. Non-linear programmingIntroduction to Management Science%eparable programming sprea$sheet mo$elIssue 8. Non-linear programmingIntroduction to Management Science%eparable Programming with smooth pro2t graphs Level of activityProfitProfit graph ApproximationIssue 8. Non-linear programmingIntroduction to Management Science*$(antages of %eparable ProgrammingThe '1cel %ol(er can rea$il# sol(e nonlinear problems that ha(e $ecreasing marginal retrns/ with the a$(antage that no appro1imation is nee$e$5Lowe(er/ the separable programming approach also has certain a$(antages! 4onverting the problem into a linear programming problem tends to make it 5uicker to solve, which can be very helpful for large problems.6 linear programming formulation makes available 3olver7s 3ensitivity 0eport.3eparable programming only re5uires estimating the profit from each activity at a few points. Therefore, it is not necessary to use a curve fitting method to estimate the formula for the profit graph.Issue 8. Non-linear programmingIntroduction to Management Science"#n$or Problem with Eoth ,(ertime Costs an$Nonlinear Mar0eting CostsThe pre(ios sprea$sheet mo$el $oes not incl$e nonlinear mar0eting costs5Recall that the cr(e+2tting proce$re was se$ to estimate the wee0l# mar0eting costs re3ire$ to sstain a pro$ction rate of $ $oors an$ % win$ows!Marketing cost for doors = !"*! Marketing costs for windows = #$$!%&';!Question% Consi$ering both o(ertime costs an$ nonlinear mar0eting costs/ how man# $oors an$ win$ows shol$ "#n$or pro$ceNIssue 8. Non-linear programmingIntroduction to Management ScienceData for "#n$or with ,(ertime Costs an$Nonlinear Mar0eting CostsMa(imum 1ee&l2 -roduction7ross 4nit -ro3t-roduct)egular5ime6vertime 5otal)egular5ime6vertimeMar&eting/ostsDoors ? 1 9 M?IH M=IH M=H$="in$ows? ? : I00 ?00 ::=>?%=Issue 8. Non-linear programmingIntroduction to Management Science"ee0l# Pro2t from Pro$cing DoorsD7ross-ro3tMar&eting/osts -ro3tIncremental-ro3t0 M0 M0 M0 T1 ?IH =H ?H0 ?H0= IH0 100 :H0 ?00? 1/1=H ==H K00 =H09 1/900 900 1/000 100Issue 8. Non-linear programmingIntroduction to Management Science"ee0l# Pro2t from Pro$cing "in$owsW7ross-ro3tMar&eting/osts -ro3tIncremental-ro3t0 M0 M0 M0 T1 I00 ::=>?:??1>?:??1>?= 1/900 =::=>?1/1??1>?H00? =/100 :00 1/H00 ?::=>?9 =/900 1/0::=>?1/???1>?P1::=>?H =/I00 1/:::=>?1/0??1>?P?00: ?/000 =/900 :00 P9??1>?Issue 8. Non-linear programmingIntroduction to Management Science%eparable Programming %prea$sheet Mo$elIssue 8. Non-linear programmingIntroduction to Management ScienceNonlinear Programming %prea$sheet Mo$elIssue 8. Non-linear programmingIntroduction to Management ScienceDi&clt Nonlinear Programming Problems '(en if a mo$el has a nonlinear ob4ecti(e fnction/ so long as the mo$el has certain properties (e5g5/ linear constraints/ $ecreasing marginal retrns)/ the %ol(er can easil# 2n$ an optimal soltion5 .n some cases separable programming can be se$ to mo$el a nonlinear problem in sch a wa# that linear programming can be se$5 Lowe(er/ if a problem has increa!ing marginal retrns/ or nonlinear fnctions in the constraints/ or $isconnecte$ pro2t graphs/ 2n$ing a soltion is often mch more $i&clt5 3uch problems may have many local optima 3olver can get stuck at local optima, rather than finding the global optimum ,ne approach with sch problems is to sol(e the problem man# times/ each time starting with a $iQerent initial soltion5 3olver Table can be used to do this process more systematically when there are only one or two variables.Issue 8. Non-linear programmingIntroduction to Management Science8sing %ol(er Table to tr# $iQerent starting points1=?9H:IJK10111=* E C D ' 7 ) L .sing Sol!er "able to "ry #i$$erent Starting PointsMaximum Startingx =7."61 := 5 2oint Solutionxx E2ro$it2ro$it = 7.5x5-+x4+24.5x"-"8x2+27x7."61 3".18 = 3".18 7 7."61 3".181 7."61 3".182 ".12+ 3+.1"" ".12+ 3+.1"4 ".12+ 3+.1"5 5.777 37.77Issue 8. Non-linear programmingIntroduction to Management Science'(oltionar# %ol(er an$ )enetic *lgorithms'volutionar2 Solver ses an entirel# $iQerent approach than the stan$ar$ %ol(er to search for an optimal soltion for a mo$el5The philosoph# of '(oltionar# %ol(er is base$ on genetics/ e(oltionan$ the sr(i(al of the 2ttest5 Lence/ this t#pe of algorithm is sometimes calle$ a genetic algorithm5The stan$ar$ %ol(er starts with a single soltion/ an$ then mo(es in $irections that will impro(e this soltion5 '(oltionar# %ol(er begins b# ran$oml# generating a whole population of soltions5*fter generating the poplation/ '(oltionar# %ol(er creates a new generation b# pairing oQ soltions in the poplation to create UoQspringV/ combining some elements from each parent5Issue 8. Non-linear programmingIntroduction to Management Science'(oltionar# %ol(er an$ )enetic *lgorithms*mong soltions in the poplation/ some will be goo$ (or U2tV) an$ some will be ba$ (or Un2tV)/ as measre$ b# e(alating the ob4ecti(e fnction5 Eorrowing from the principles of e(oltion an$ sr(i(al of the 2ttest/ the U2tV members are allowe$ to repro$ce more fre3entl# than the n2t members5*nother 0e# featre is mutation5 Li0e gene mtation in biolog#/ '(oltionar# %ol(er will occasionall# ma0e a ran$om change in a member of the poplation5 This helps the algorithm get nstc0 if it is getting trappe$ near a local optimm5'(oltionar# %ol(er 0eeps creating new generations of soltions ntil there ha(e been no impro(ements for se(eral consecti(e generations5Issue 8. Non-linear programmingIntroduction to Management Science%electing a portfolio to beat the mar0et * common goal of portfolio managers is to beat the mar0et5.f we assme that past performance is somewhat of an in$icator of the ftre/ then pic0ing a portfolio that beat the mar0et most often in the past might #iel$ a portfolio that will more than li0el# beat the mar0et in the ftre5 Consi$er a portfolio of 2(e large stoc0s tra$e$ on the New Wor0 %toc0 '1change (NW%')! 6merica 8nline #689' :oeing #:6' ;ord #;' amble #' Mc,onald7s #M4,'Question% "hat mi1 of these 2(e stoc0s will #iel$ a portfolio that is li0el# to beat the mar0et in the ftreNIssue 8. Non-linear programmingIntroduction to Management Science%prea$sheet Mo$el1=?9H:IJK10111=1?191H1:1I1J1K=0=1===?=9=H=:=I=J=K?0?1?=???9?H?:* E C D ' 7 ) L . X Y%eating t&e Mar'et ()!olutionary Sol!er*Feat MarBetGuarter Hear 0D. F0 I 2J M-* Aeturn MarBetK %NHSL(G4 2771 -".72C 1+."5C -8.54C 8.66C -1.+4C 2."8C No 8.45CG" 2771 -"6.55C -"8.5+C -28.48C 14.61C 7."7C -18.12C No -12.5"CG2 2771 "2.77C 7.76C -11.87C 2.54C 1.82C 4.85C Hes 4."8CG1 2771 15."6C -15."4C 21.28C -18.87C -21.81C -4.78C Hes -8."2CG4 2777 -"5.14C 2.55C -6.77C 16.76C 1"."6C -1.8"C No -7.8"CG" 2777 1.4+C 54.61C ".++C 18.7+C -8."5C 1".81C Hes "."1CG2 2777 -21."6C 17.88C -+."8C 7.77C -11.86C -5.6"C No -7.81CG1 2777 -11."+C -8.44C -1".8"C -48.27C -6.28C -16.82C No -7.47CG4 1888 45.82C -2.48C +.17C 1+.86C -+.68C 11.87C Hes 8.67CG" 1888 -5.47C -2.8"C -17.8+C +."8C 5.16C -1.5"C Hes -8.54CG2 1888 -25.16C 28.8"C -7.44C -17.72C -8.24C -".71C No 6."8CG1 1888 88.52C 4.21C -".41C 6.2+C 16.88C 2".11C Hes 1."1CG4 1888 166.8+C -4.82C 24.86C 28."8C 28.+8C 51.77C Hes 18.11CG" 1888 +.18C -2".77C -27."4C -21.88C -1".57C -14.51C No -12.8"CG2 1888 5".88C -14.51C -8.84C 6.8"C 15.77C 17.+6C Hes 1.74CG1 1888 57.8+C +.51C "".4+C 5.62C 25.+5C 24.4+C Hes 12.75CG4 1886 18.8+C -17.17C 6.+2C 15.56C 7.2+C +.++C Hes 2.81CG" 1886 "5.+1C 2.58C 18.65C -2.21C -1.42C 17.++C Hes 6.42CG2 1886 "7.87C 6.+7C 21.12C 2".78C 2.25C 1+.88C Hes 1+.14CG1 1886 26.82C -6."8C -2.61C +.+2C 4.1"C 5.+8C Hes 1.58CG4 188+ -+."4C 12.67C ".27C 17."8C -4.22C ".15C No +.87CG" 188+ -18.8+C 8.4+C -".46C 6.58C 1."4C -7.88C No 2.2+CG2 188+ -21.86C 7.58C -5.82C +.8"C -2.+7C -4.5+C No ".54CG1 188+ 48.""C 17.5"C 18.75C 2.11C +."6C 16.48C Hes 5.28C7C 7C 7C 7C 7C:= := := := := Sum2ort$olio 27.7C 27.7C 27.7C 27.7C 27.7C 177C = 177C:= := := := :=177C 177C 177C 177C 177CNumer o$ GuartersFeating t,e MarBet14Issue 8. Non-linear programmingIntroduction to Management SciencePremim %ol(er Dialoge Eo1Issue 8. Non-linear programmingIntroduction to Management Science%ol(er ,ptions Dialoge Eo1Issue 8. Non-linear programmingIntroduction to Management ScienceLimit ,ptions Dialoge Eo1Issue 8. Non-linear programmingIntroduction to Management Science'(oltionar# %ol(er %prea$sheet %oltion1=?9H:IJK10111=1?191H1:1I1J1K=0=1===?=9=H=:=I=J=K?0?1?=???9?H?:* E C D ' 7 ) L . X Y%eating t&e Mar'et ()!olutionary Sol!er*Feat MarBetGuarter Hear 0D. F0 I 2J M-* Aeturn MarBetK %NHSL(G4 2771 -".72C 1+."5C -8.54C 8.66C -1.+4C 17.27C Hes 8.45CG" 2771 -"6.55C -"8.5+C -28.48C 14.61C 7."7C -24.12C No -12.5"CG2 2771 "2.77C 7.76C -11.87C 2.54C 1.82C 6.72C Hes 4."8CG1 2771 15."6C -15."4C 21.28C -18.87C -21.81C -17.5"C No -8."2CG4 2777 -"5.14C 2.55C -6.77C 16.76C 1"."6C -7.8+C Hes -7.8"CG" 2777 1.4+C 54.61C ".++C 18.7+C -8."5C "".4"C Hes "."1CG2 2777 -21."6C 17.88C -+."8C 7.77C -11.86C 1."1C Hes -7.81CG1 2777 -11."+C -8.44C -1".8"C -48.27C -6.28C -18.5+C No -7.47CG4 1888 45.82C -2.48C +.17C 1+.86C -+.68C 12.11C Hes 8.67CG" 1888 -5.47C -2.8"C -17.8+C +."8C 5.16C -7.68C Hes -8.54CG2 1888 -25.16C 28.8"C -7.44C -17.72C -8.24C 6.6+C Hes 6."8CG1 1888 88.52C 4.21C -".41C 6.2+C 16.88C 22.7"C Hes 1."1CG4 1888 166.8+C -4.82C 24.86C 28."8C 28.+8C 47.51C Hes 18.11CG" 1888 +.18C -2".77C -27."4C -21.88C -1".57C -1+.81C No -12.8"CG2 1888 5".88C -14.51C -8.84C 6.8"C 15.77C 5."8C Hes 1.74CG1 1888 57.8+C +.51C "".4+C 5.62C 25.+5C 15.47C Hes 12.75CG4 1886 18.8+C -17.17C 6.+2C 15.56C 7.2+C 2.82C Hes 2.81CG" 1886 "5.+1C 2.58C 18.65C -2.21C -1.42C 6.68C Hes 6.42CG2 1886 "7.87C 6.+7C 21.12C 2".78C 2.25C 1+.24C Hes 1+.14CG1 1886 26.82C -6."8C -2.61C +.+2C 4.1"C ".45C Hes 1.58CG4 188+ -+."4C 12.67C ".27C 17."8C -4.22C 8.7+C Hes +.87CG" 188+ -18.8+C 8.4+C -".46C 6.58C 1."4C 2.62C Hes 2.2+CG2 188+ -21.86C 7.58C -5.82C +.8"C -2.+7C -2.22C No ".54CG1 188+ 48.""C 17.5"C 18.75C 2.11C +."6C 15.88C Hes 5.28C7C 7C 7C 7C 7C:= := := := := Sum2ort$olio 18.6C 52.7C 7.2C 2+.+C 1.+C 177C = 177C:= := := := :=177C 177C 177C 177C 177CNumer o$ GuartersFeating t,e MarBet18Issue 8. Non-linear programmingIntroduction to Management Science*$(antages an$ Disa$(antages of '(oltionar# %ol(er '(oltionar# %ol(er has two signi2cant a$(antages o(er the stan$ar$ %ol(er for sol(ing $i&clt nonlinear programming problems!?. The complexity of the objective function does not matter. 6s long as the function can be evaluated for a given candidate solution #to determine the level of fitness', it does not matter if the function has kinks, discontinuities, or many local optima.!. :y evaluating whole populations of candidate solutions, 1volutionary 3olver keeps from getting trapped at a local optimum. 1ven if the whole population evolves toward a locally optimal solution, mutation allows the possibility of getting unstuck. Lowe(er/ '(oltionar# %ol(er is not a panaceaZ It can take muc, longer that standard 3olver to find a final solution. 1volutionary 3olver does not perform well on models that have many constraints. 1volutionary 3olver is a random process. 0unning it again on the same model usually will yield a different solution. The best solution found is typically not optimal #although it may be very close'.Issue 8. Non-linear programmingIntroduction to Management ScienceNonlinear ProgrammingConsi$er the following mo$el for a nonlinear programming problem!maximize:2ro$it = 7.5x5 +x4 + 24.5x" "8x2 + 27xsu!ect to:7 9 x 9 5Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C 01=?9H:I* E C D 'A Simple NLPMaximumx =7."61 := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 3".18Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C ?1=?9H:I* E C D 'A Simple NLPMaximumx =".12+ := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 3+.1"Issue 8. Non-linear programmingIntroduction to Management Science%ol(er %oltion %tarting with x C 95I1=?9H:I* E C D 'A Simple NLPMaximumx =5.777 := 52ro$it = 7.5x5-+x4+24.5x"-"8x2+27x = 37.77Issue 8. Non-linear programmingIntroduction to Management ScienceThe Pro2t )raphProfit ($)xIssue 8. Non-linear programmingIntroduction to Management ScienceProblems that %ol(er will sol(e correctl#* maximization problem with linear constraints an$ a conca*e ob4ecti(e fnction5A Concave FunctionLine joining any two pointsis on or below the curveLine 4oining an# two points is on or belo+ the cr(eIssue 8. Non-linear programmingIntroduction to Management ScienceProblems that %ol(er will sol(e correctl# * minimization problem with linear constraints an$ a con*ex ob4ecti(e fnction5A Convex FunctionLine joining any two pointsis on or above the curveLine 4oining an# two points is on or abo*e the cr(eIssue 8. Non-linear programmingIntroduction to Management Science6alit# 7rnitre Corporation The 6alit# 7rnitre Corporation manfactres two pro$cts! benches an$ tables5 The# emplo# three carpenters5 Dring the ne1t wee0/ 1=0 hors of labor are a(ailable at reglar wages (MJ per hor)5 8p to ?0 hors of o(ertime can be se$ at a wage rate of M1= per hor58p to ?0 hors of wee0en$ time can be tili-e$ at a wage rate of M1: per hor5 H90 pon$s of woo$ is a(ailable at a cost of M= per pon$5 'ach bench re3ires ? labor hors an$ 1= pon$s of woo$5 'ach table re3ires : labor hors an$ ?J pon$s of woo$5Complete$ benches sell for MJ0 each/ an$ tables sell for M=00 each5Question% Low man# benches an$ how man# tables shol$ be pro$ce$NIssue 8. Non-linear programmingIntroduction to Management Science,t$oor 7rnitre Labor CostsLabor CostLabor Hours! "! #! $%! $&! $'!(#"!($%!($"!!(')hr Regular($%)hr *verti+e($")hr ,ee-en.Issue 8. Non-linear programmingIntroduction to Management ScienceNonlinear Programming %prea$sheet1=?9H:IJK10111=1?191H1:1I1J1K=0=1===?=9* E C D ' 7 )+uality Furniture Corporation (Nonlinear*Fenc,es ?alesAe/enue@=nit 385 3277?otal =sed 0/ailale.aor " + 1"5 := 187;ood 12 "8 547 := 547;ood -ost@l. 32.aor -ost >ours%per ,our( 0/ailaleAegular 38 127D/ertime 312 "7Sunda1 31+ "7Fenc,es ?ales2roduction 45 7Ae/enue 3"&825.77;ood -ost 31&787.77.aor -ost 31&147.772ro$it 31&+75.77=sage per =nit 2roducedIssue 8. Non-linear programmingIntroduction to Management Science,t$oor 7rnitre Labor CostsLabor CostLabor Hours! "! #! $%! $&! $'!(#"!($%!($"!!(')hr Regular($%)hr *verti+e($")hr ,ee-en.Issue 8. Non-linear programmingIntroduction to Management Science%eparable Programming %prea$sheet1=?9H:IJK10111=1?191H1:1I1J1K=0=1===?=9=H=:=I=J=K* E C D ' 7 )+uality Furniture Corporation (Separable*Fenc,es ?alesAe/enue@=nit 385 3277?otal =sed 0/ailale.aor " + 1"5 := 1"5;ood 12 "8 547 := 547;ood -ost@l. 32.aor -ost >ours%per ,our( 0/ailaleAegular 38 127D/ertime 312 "7Sunda1 31+ "7Fenc,es ?ales2roduction 45 7.aor =sedAegular >ours 127 := 127D/ertime >ours 15 := "7;eeBend >ours 7 := "7Ae/enue 3"&825.77;ood -ost 31&787.77.aor -ost 31&147.772ro$it 31&+75.77=sage per =nit 2roducedIssue 8. Non-linear programmingIntroduction to Management Science*$(ertising '1ample1=?9H:IJK10111=1?191H1:1I1J1K=0=1==* E CAd!ertising )xample (Nonlinear*Parameters,=nit Mariale -ost 348=nit 2rice 3+5Sales$orce Salar1 38&777Iixed D/er,ead 32"&777Seasonalit1 1.2#ecision -ariable,0d/ertising 3122&848+uarter +.=nits Sold 14884Sales Ae/enue 3864&+17-ost o$ Sales 3618&612Jross Margin 3254&888?otal Iixed -osts 3154&848Pro$it 388&848

Sales(35)(Seasonality Factor) Advertising+Sales Force2Issue 8. Non-linear programmingIntroduction to Management ScienceThe %ales 7nction

Sales(35)(Seasonality Factor) Advertising+Sales Force2727774777+77787771777712777147771+77718777277777 57777 177777 157777 2777770d/ertisingSales .e/elIssue 8. Non-linear programmingIntroduction to Management Science*ppro1imating a Nonlinear 7nction1=?9H:IJK1011* E C DApproximating t&e Nonlinear Sales FunctionSeasonalit1 = 1.2Sales Iorce = 87770d/ertising .e/el Sales .e/el Slope37 2&816357&777 8&875 7.1"883177&777 1"&566 7.76543157&777 1+&578 7.758+3277&777 18&88" 7.7486Issue 8. Non-linear programmingIntroduction to Management Science*$(ertising '1ample 8sing %eparable Programming1=?9H:IJK10111=1?191H1:1I1J1K=0=1===?=9=H=:* E C D ' 7 )Ad!ertising )xample (Separable*Parameters,=nit Mariale -ost 348=nit 2rice 3+5Sales$orce Salar1 38&777Iixed D/er,ead 32"&777Seasonalit1 1.2=nits Sold per0d/ertising *ollar0d/ertising %37-357&777( 7.1"88 357&777 := 357&7770d/ertising %357&777-3177&777( 7.7654 357&777 := 357&7770d/ertising %3177&777-3157&777( 7.758+ 37 := 357&7770d/ertising %3157&777-( 7.7486 373177&777 ?otal 0d/ertising+uarter +.=nits Sold 1"566Sales Ae/enue 3882&575-ost o$ Sales 3+51&+8+Jross Margin 32"7&878?otal Iixed -osts 31"2&777Pro$it 388&878Issue 8. Non-linear programmingIntroduction to Management Science'(oltionar# %ol(erThe stan$ar$ %ol(er has $i&clt# with problems that are@ighly nonlinear6re not smooth #have AkinksB in the objective'@ave discontinuities #the objective jumps in value'@ave many local optima #many hills and valleys''1cel fnctions li0e .7/ M*D/ *E%/ R,8ND/ etc5/ ten$ to case one or more of these problems5Issue 8. Non-linear programmingIntroduction to Management SciencePremim %ol(er.ncl$e$ on the te1tboo0 CD is the UPremim %ol(erV5 *fter installing/ a new btton (UPremimV) is a$$e$ to %ol(er5Issue 8. Non-linear programmingIntroduction to Management SciencePremim %ol(erClic0ing on the UPremimV btton switches to Premim %ol(er/ which gi(es the option of three $iQerent sol(ers5 3tandard >0> (onlinear is e5uivalent to the regular 3olver olding -ost 34Issue 8. Non-linear programmingIntroduction to Management Science*ttempts with %tan$ar$ %ol(er[arios UsoltionsV pro(i$e$ b# the stan$ar$ %ol(er/ $epen$ing on the starting point!Starting -oint *Q, Solution *Q8, /ost1 1/000 M1K9/900=00 H00 1KH/J00900 99I 1KI/IJK:00 H00 1KH/J001/=00 1/000 1K9/90011/000 10/000 =10/0901=/000 1/000 1K9/900Issue 8. Non-linear programmingIntroduction to Management ScienceCost 7nction with 6antit# Disconts($#!/!!!($#&/!!!(%!!/!!!(%!&/!!!(%$!/!!!(%$&/!!!(%%!/!!!(%%&/!!!(%!/!!!Cost*r.er 0uantity 1logarith+ic scale2$! $!! $!!! $!/!!!334&!!Issue 8. Non-linear programmingIntroduction to Management Science%ol(ing with the '(oltionar# %ol(er1=?9H:IJK10111=1?* E C D ' 7 ) L ./rdering Policy 0it& +uantity #iscountsMin DrderGuantit1 2rice1 317.77 1 Annual Cost177 38.87 := 2urc,asing 3182&777577 38.67 Drder Guantit1 1777 Drdering 34771&777 38.+7 := >olding 32&77717&777 38.57 27&777 "otal 3184&4770nnual *emand 27&777 2rice 38.+7Drdering -ost 3270nnual >olding -ost 34Issue 8. Non-linear programmingIntroduction to Management ScienceTips on 8sing '(oltionar# %ol(erEon$ing all of the (ariables greatl# ai$s the '(oltionar# %ol(er b# $ecreasing the search space5The limit options shol$ be increase$ (Ma1 Time/ Ma1 %bproblems/ an$ Ma1 7easible %ols) for challenging problems5 %etting Tolerance to 05000H an$ Ma1 Time "ithot .mpro(ements to ?0 will ensre the algorithm will stop if the Target Cell (ale has impro(e$ less than 050HO in the last ?0 secon$s5'1periment with $iQerent poplations si-es an$ mtation rates to see what wor0s well5 . ha(e fon$ that higher than $efalt mtation rates can be helpfl in problems with man# local optima5The '(oltionar# %ol(er can ta0e a (er# long time/ bt it will sall# 2n$ a goo$ soltion5Issue 8. Non-linear programmingIntroduction to Management ScienceTips on 8sing '(oltionar# %ol(erThere is no garantee that '(oltionar# %ol(er will 2n$ the be!t soltion5The '(oltionar# %ol(er performs well e(en with nast# ob4ecti(e fnctions/ bt is not (er# e&cient at han$ling constraints5Mch of the soltion process is $ri(en b# ran$om nmbers that $irect the search5 Ths/ two people rnning '(oltionar# %ol(er on the same mo$el ma# get $iQerent reslts5,nce '(oltionar# %ol(er has fon$ a goo$ soltion/ #o can se )R) Nonlinear %ol(er (the nonlinear algorithm that is incl$e$ with the Premim %ol(er software) to tr# to 2n$ a slightl# better soltion5