The article of record as published may be found at https://doi.org/10.1287/ opre.10.6.799A method is given for determining optimum ship routes on a digital computer. A ship is assumed to be in a fallout field whose intensity is a known function f(x, y, t) of position and time. A typical problem is that of choosing a route to a point where f ≦ fm, the maximum intensity that can be tolerated indefinitely. The route is to be such as to minimize the dose z = ∫0Tfdt, the termininal time and point are not specified. The problem of sending the ship to a given point with z minimized is also discussed. These are equivalent to corresponding problems of choosing a route to minimize the probability of detection while going through a region where the probability of detection is a known function of position and time.Office Naval Researc

Faulkner, Frank D.

English

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveDSpace RepositoryFaculty and Researchers Faculty and Researchers' Publications1962-06Determining Optimum Ship RoutesFaulkner, Frank D.INFORMSFrank D. Faulkner, (1962) Determining Optimum Ship Routes. Operations Research 10(6):799-807.http://hdl.handle.net/10945/70153This publication is a work of the U.S. Government as defined in Title 17, UnitedStates Code, Section 101. Copyright protection is not available for this work in theUnited States.Downloaded from NPS Archive: CalhounDownloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.DETERMINING OPTIMUM SHIP ROUTES Frank D. Faulkner United St,a,tes Naval Postgraduate School, Monterey, Calif (Received June, 1962) A method 1s given for determmmg optimum ship routes on a digital com-puter A ship 1s assumed to be m a fallout field whose mtens1ty 1s a known funct1onf(x,y,t) of pos1t1on and time A typical problem 1s that of choos-mg a route to a pomt wheref;;i,f,,., the maximum mtens1ty that can be toler-ated mdefimtely The route 1s to be such as to m1mmize the dose z = f '[ f dt, the termimnal time and pomt are not specified The problem of sending the ship to a given pomt with z mimmized 1s also discussed These are eqmvalent to corresponding problems of choosmg a route to m1mm1ze the probab1hty of detection while gomg through a region where the probab1hty of detection 1s a known function of position and time THE METHOD of solut10n is based on differential formulas which G A BLISS mtroduced m Ballistics, based on the adJomt system of differential equations These can be combmed with the Euler equations and mverted to determme constants associated with the solution, m a Newton-Raphson iteration The method IS qmte general and can be apphed to, say, related problems of rendezvous, 1f the rendezvous pomt either follows a known course or cooperates The correct10n routme for the constants vanes greatly with the end conditions, the method can best be demonstrated by some examples and a statement of the general conditions To Bllllphfy the programmmg 1t was assumed that distances were small enough that negligible error was mtroduced by assummg the earth to be flat and that the speed of the ship was constant In most 'practical' prob-lems 1t can be shown that if f 1s mdependentof the speed as above, maXJ.-mum speed IS optimal BASIC FORMULAS FOR VARIATIONS IN THIS sect10n formulas are denved for vanations of the end values of the vanables The method seems somewhat deVIous at first, but numencal methods for calculatmg differentials and vanational theory generally rest on the equations of the type denved here, employmg Lagrange multipliers and the adJomt system of d1fferent1al equations When approached through these, mtroductory calculus of vanat1ons 1s qmte straightforward The governmg equations for the course and the dose are x=vcosp, y=vsmp, z=f, 799 (1) Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.800 Frank D. Faulkner where x,y are coordmates, z IS the dose accumulated, and p, a vanable, is the course headmg, wlu.ch IS to be found, a dot ( ) over a vanable mdicates its time denvative We shall be concerned with neighbonng courses and 1t will be assumed that the varmt10ns ox,oy etc, satisfy the equations ox+vsmp op = 0, oy-vcosp op =0, oz-fJ,x-f 11 oy=0 (2) We shall be mterested m formulas for the change ox(T), etc, m x at some future trme T as a functional of the varmt1on op of the control van-able p Followmg Bliss's method, let us multiply each of the three equat10ns (2) through by new unspecified vanables )l.,µ,v, respectively, Lagrange multiphers, add, and mtegrate to get £\x(ox+vsmp op)+µ(oy-vcosp op)+v(oz-fJ,x-f 11oy)] dt=0 (3) Now mtegrate this by parts to ehmmate ox,oy,oz from the mtegrand, (Mx+µoy+voz)~ = iT[ox()l.+vf,,) +oy(µ+vfy) +ozv+vop(-)1.smp+µcosp)] dt (4) To simplify tlu.s, let us choose )l.,µ,v as solutions to the differential equations )l.+vf,,=0, µ+vfu=0, v=0 (5) These operations of mtegrat10n by parts in (3) and settmg to zero the coefficient of the vanat10ns ox,oy,oz of the dependent vanables m ( 4) defines the system ( 5) of equat10ns adJotnt to the system ( 2) If also the termmal value T changes, the total differential changes m the termmal values of x,y,z are AT=ox(T)+x(T) oT, Ay=oy(T)+y(T) oT, Az=oz(T)+z(T) oT (6) In the problems of mterest here the m1tml values are assumed given so that ox( 0) = 0, etc Equation ( 4) then reduces to Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.Optimum Slnp Routes 801 [XAx+µ.:1y+vAz-(xX+yµ+vz)aT]t=T = 1T (->.smp+µcosp)op dt (7) = 1T A 't\,ap, where A =Xi+µj+vk, V=v(7cosp+jsmp), and the subscript p mdi-cates the partial derivative In the followmg, no d1stmction will be made between a vector A and a column matrix with the same elements ;\ µ Three solutions to ( 5) are of particular mterest, JI where X3+f,,=0, µ3+f11=0, Xa( T) =0, µa(T)=0, these constitute a fundamental set For them we get Ax=x(T) oT+ 1T A1 Vpopdt, .:1y=y(T) oT+ 1T A2 Vpapdt, Az=z(T) oT+ 1T A 3 VP ap dt (Sa) (Sb) (9) Fmally, with every optrmum 1t can be shown that there 1s a solution A to the adJomt c1 "1 +c2 I\ 2+ca A a such that or or A Vp=O, ->.smp+µcosp=0, A V = extremum, (10) (10') ( 10") as a function of p for each value of t, and a fourth eqmvalent form T l A Vdt=extremum, (10"') the last extremum bemg achieved by choosmg p properly The second of these, with the adJomt system, constitute the classw Euler, or Euler-Lagrange, equations m calculus of variations The last form (10"') has Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.802 Franlc D. FaullnNr attracted considerable attention recently due to Pontriagm as a maximum principle As a philosophy of approach, 1t 18 somewhat more general, but for our purposes all are eqmvalent Smee equation (7) IS homogeneous m the set 'A.,µ,11, we may choose one relation among the e's 1t will be con-vement to choose the coefficient c3 of /lz m (7) to be one (1) This 1s taken to be the case henceforth No philosophy ehmmates the basic problem of deternunmg the con-stants c1,~, T Curves on which ( 10) 18 satisfied for some choice of A , a solution to the adJomt system, will be called extremals Curves which sat1Sfy the con-stramts on the end values will be called admissible Minimum Dose in Attaining a Specified Curve Let us consider the followmg problem A ship ism a fallout area We W1Sh to route it to the curve whereon f = f m = ma:xunum allowed contmual mtens1ty, so that the resultmg dose 18 a mm1mum (11) Suppose we have two neighboring admissible paths, the drlierential of the end values must sat1Sfy the relation (12) smce both end on the hypersurface f = f m The transversal condition ( see Bl1Ss,r11 p 196) that the end values must sat1Sfy may be expressed fx f11 0 ft rankO O 1 0 =2, (13) 'A µ 1 -(Xx+µy+z) and the 2X4 submatnces must all have rank 2 The terms m the first row are from the constramt (12) The terms m the second row are from the quantity to be maxmuzed ( 11) The terms m the last row are the coefficients of the vanations on the left side of (7) From the transversal condition, we see that if we regard fxJ 11J, and 'A.,µ,-(>..x+µy+f) as com-ponents of two vectors, those two must be parallel Hence at t = T whence ci/f,=c-iff11= -(c1x+c1y+f)/f,, c1 = -f,f m/(df /dt), C%= -f,J,,./(df /dt) (14) (15) Smee we want to m1mm1ze z( T), the choice of p must be to mmimize A V Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.Optunum Ship Rautea 803 Computational Routine Guess startmg values for A(O),µ(O) Integrate systems (1) and (5), usmg (10'). Contmue unttlf~fm AdJust the end time step If n denotes an 1teration mdex, note that An+1( Tn)-An+1(0) =A3 n+i( Tn)-A3 n+1(0) = -A3,,,+1(0) Now assume Aan+1(0) =Aa ,,(0) =X,.(T,.)-X,.(O) A slffillar relation holds for µ and we get An+1(0) =c1,.+X,.(O)-"A,.( T,.), µ,.+1(0) = 1:2,.+µ .. (0)-µ..( Tn) (16) Smee every term on the left IS known, we are ready to start the n+ 1st round Convergence Criterion The above routme IS contmued until some convergence cntenon IS satisfied In thIS case, let E= 1 _ [f,,X+f11µ-(vAcosp+vµsmp+J)ft]2 (f,.2+fil+fi2)(">.2 +µ 2+[v">.cosp+vµamp+JJ2) evaluated at t= T The routme IS assumed to have converged when EIS below some specified number The transversal condition specified that two vectors be parallel, E IS the square of the Sign of the angle between them Ship Sent to Prescribed Point Let us now consider the problem of sendmg the ship to a specified pomt x1,y1 with a ffilillfilum final value of z The transversal condtt10n becomes 1 0 0 0 0 1 0 0 rank =3, 0 0 1 0 (18) A µ 1 -(AX+µy+z) the first two rows are from the constraints Hence (AX+µy+z)T=O (19) To get a corrective routme for the e's, note that on any extremal i\ V,.=0, Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.804 Frank D. Faulk1rer hence 1f we change c1,C2,P with the other vanables fixed, A1 t\6c1+i:2 V,,~+i\ V,,,,op=O (20) Smee X v,,,,=vl °A I, (20) reduces to op= (smp oc1-cosp oD.i)/1 ""J. I, (21) and t1x=x(T) oT- [r (vsm2 p// Al) dt 6c1+ £\vsmpcosp/l A I) dt oc2, T T (22) '1y=y(T) oT+ 1 (vsmpcosp/j .i\ I) dt oc1- [ (vcos2 p// ""J. I) dt oc2 Numerical Routine Let us guess X(O),µ(O) Integrate to get x,y,Xx+µy+f,X,µ, at T and the mtegrals m (22) In general x,y wtll not assume the desrred terminal values, nor wtll f=Xx+µ,y+Jbe zero To correct 1/1 use t:.i/1=!1 oT+f,, t:.x+fy t:.y+xoc1+yoc2, everythmg evaluated at t= T Now set 111/1= -if;, 11x=x1-x(T), '1y=y 1-y(T), (23) (24) substitutmg from (22), (23) mto (24), these equations determme 6c1,oc2,0T and hence c1,C2,T Equations (16) then deterrrune X(O),µ(O) to start the next iteration The routme IS contmued unttl some convergence cr1ter10n IS satisfied, one might take (x,-x )2 + (y1-Y )2 +l < E, where E IS a preassigned number COMMENTS THE PROBLEM has been programmed and run No particular computa-tional problems were encountered In some srmple cases mvolvmg about 100 time steps, the route was determmed m from four to twenty seconds on the CDC-1604, dependmg on the rmtial cond1t1ons assumed The fallout problems above are equivalent to correspondmg problems of findmg routes that mmnmze the probability of detection m gomg from one pomt to another Assume that the probabihty of bemg detected at a pomt is given by t:.p=p(t+t:.t)-p(t)=f(x,y,t) t:.t, If undetected pnor to trme t On the path the probability of bemg first detected at tune t satisfies the relation Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.Optimum Ship Routu llp=(I-p)f flt, set z=ln(l-p) 805 In general we should keep maxunum speed l~or if we find a path that furrushes a mmunum with v a maxunum for each value of t and then con-S1der vanations, ov~O, we have, from (7), Ty • [>.11x+,,.11y+11z-(>.x+,,.y+J) oT]T=[ -" ov dt o V = - 1\;. 1 ov dt, all other terms m the mtegral varush by virtue of ( 5), ( 10) The last nunus sign occurs because we seek a mmunum z and the mtegrand must be muunuzed If the speed v IS maxunal on the ongmal path, then ov<O leads to a larger value of the mtegral and hence to a larger value of z Tlus 1s mtmtively obVIous, the proof IS constructive, and IS eqmvalent to the proof of the fundamental lemma of the calculus of vanat10ns ( see CouRANT, 121 p 200) The problem 1s more mvolved If f mvolves v and IS an mcreasmg func-t10n of v ThIS may be the case for a submanne, where the addit10nal n01se nnght mcrease the hkehhood of detection, or where the speed and probab1hty of detection both change with depth The adJunct1on of another vanable such as v leads to another Euler equation that must be adJomed to those above If more mvolved formulas are taken for the speed, the solut10ns to the adJomt system and the resultmg correct10n formulas for the constants are more mvolved In the differential formulas above 18 the unphc1t assumption that the path has no corners m x,y,t space, if it has, a term must be added to each differential correct10n, formula ( 22), for each corner In a sense thIS method 18 a simple apphcation of methods that Bhss mtroduced for calculatmg differentials m artillery dunng World War I, summarized m his book (BLiss,£31 Chap V) The advent of large-scale digital computers has made 1t possible to use these formulas and mvert them to determme the correspondmg courses If equat10ns ( 1) govern the system, we may have from none to three side conditions of the form Gm(x,y,z,t)i=T=O, m=l, ,M<4, when all lllltial conditions are given, and a function F(x,y,z,l)t=T=extremum The transversal condition may then be expressed G,,.,., Gm,11 G,,.,. rank F., F11 F. X µ. V -(>.x+,,.y+PZ) l=T } (M rows) =M+l and the ( M + 1) X 4 matnces obtamed by elunmatmg either of the last two rows must also both be of rank M + 1 The evolution of a routme to Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.806 Frank D. Faullcner correct the e's and T, from the transversal condJ.tions, seems to be the place where mgenmty IS reqmred m the routme, the rmtml values must also be guessed m a proper range or the Newton iteration may not con-verge It IS not possible to dIScuss m a short paper other condJ.tions assoc1ated With the solution which ensure that the solution yields a nummum, the above routmes yield only stationary values The conditions are dIScussed completely m Bhss 111 (Chaps VII, VIII) The treatment there IS neat and precise but the problem IS expressed m a symmetnc form that may obscure for the control engmeer the essential difference between the so-called state vanables, x,y,z, which are properly dependent, and the control vanable p, which must be determined to effect the extremum It should be stressed that the multipliers ">..,µ,11 are mtroduced by the mathematician and he later chooses them to satISfy (5) to srmplify Green's formula (4) to the form (7) One problem that we may expect is that convergence will be poor when the lllltial pomt 1s near an envelope or a focal pomt of the extremals which satISfy the transversal cond1t1ons A tnVIal example of such a pomt 1s the center of a region which has circular symmetry with respect to a fixed pomt This may not be a senous problem SIDce ne1ghbormg traJec-tones then generally lead to termmal values of z that differ 'httle' How-ever, problems such as these generally reqmre an element of skill and Judg-ment on the part of the operator It seems to the author that there are two practical methods of detenmn-mg optrmum traJectones numencally One 1s the differential method presented here, always usmg extremals and approxrmatmg the transversal cond1t10ns The other 1s the method of 'steepest descent' of BRYSON AND DENHAM14  and KELLEY {see KELLEY, KoPP, AND MoYERl6l for summary and comments) DiscUSSions suggest we are all havmg typical small problems assomated with computations ACKNOWLEDGMENT I WOULD hke to thank LCDR K F CooK and MARY lIA YNES for assistance m programmmg and runnmg the problem A d1scuss1on of the problem of determinmg the f unct1on f IS given m Cook's thesis 161 This work was supported by ONR A more complete d1scuss1on of some SIIDpler related problems 1s available ma report1 71 on nummum-trme routes REFERENCES 1 G A BL1ss, Lectures on the Calculus of V an.atwns, Untver81,ty of Chicago, Clu-cago, 1946 2 R CoURANT AND D HILBERT, Methods of Math£matical Phy81,C8, Intersc1ence, New York, 1953 Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. This article has not been copyedited or formatted. The final version may differ from this version.Optunum Slnp Routea 807 3 G A BLISS, MaJJiematicsfor Exi,erwr Balliatics, Wiley, New York, 1944 4 A E BRYSON AND W F DENHAM, "A Steepest-Ascent Method for Solvmg Optimum Programmmg Problems," Report BR-1303, Raytheon MIS8lle and Space D1V1S1on, August 10, 1961 5 H J KELLEY, R E KOPP, AND H G MOYER, "Succe.ss1ve Approximation Techmques for TraJectory Optnmzat1on," presented at IAS Vehicle Systems Optim1zation, Nov 28-29, 1961, Garden City, N Y 6 K F CooK, "A Radiological Fallout Prediction System for the Navy," ThesIS, U S Naval Postgraduate School, Monterey, Cahf 1962 7 F D FAULKNER, "Optimum Ship Routing," Research Paper No 32, U S Naval Postgraduate School, May, 1962 Part I, forthcom1ng m Navigation Downloaded from informs.org by [205.155.65.56] on 27 July 2022, at 12:49 . For personal use only, all rights reserved. Published in Operations Research on December 01, 1962 as DOI: 10.1287/opre.10.6.799. 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Determining Optimum Ship Routes

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Determining Optimum Ship Routes

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