A Quadratic Assignment Linear Programming Approach to Ship

Document Sample
A Quadratic Assignment Linear Programming Approach to Ship Powered By Docstoc
					This material may be protected by Copyright Law (Title 17,
US Code).
        NAVAL POSTGRADUATE SCHOOL
r




                        M onterey , California




                           THESIS
         A QUADRATIC ASSIGNMENT            /   LINEAR PROGRAMMING APPROACH
              TO SHIP SCHEDULING FOR THE U.S.                  COAST GUARD




                                  Charles Edwin S i b r e

                                        J u n e , 1977



         Thesis A d v i s o r :                                Gerald G. Brown

    c
             Approved f o r p u b l i c release; d i s t r i b u t i o n u n l i m i t e d .
                                                                                                                          READ INSTRUCTIONS
                                 REPORTWCUMENTATIONPAGE                                                               BEFORE COMPLETING FORM
         1. REPORT NUMmER                                                   2. GOVT ACCESSION NO. 3.                RECIPIENT'S CATALOG NUMBER



         4.    T I T L E ( a d Subtlllo)                                                                      5.    TYPE OF REPORT & PERIOD COVERED
              A Q u a d r a t i c Assignment / L i n e a r Programming                                             Master's T h e s i s (June,1977)
c:            Approach t o S h i p S c h e d u l i n g f o r the U. S .
              Coast Guard,

         7 . AUTHORIo)                                                                                        1 CONTRACT OR GRANT HbM8ER(*)
                                                                                                               .

              Charles Edwin S i b r e

         9. PLRCORMING ORGANIZATION NAME AND AODRESS                                                          10. PROGRAM ELEMENT, PROJECT. T A S K
                                                                                                                  A R E A 4 WORK UNIT NUMBERS
              Naval P o s t g r a d u a t e S c h o o l
              Monterey, C a l i f o r n i a 93940
         11.    CONTROLLING OFCICE NAME AND ADORESS                                                           12. REPORT D A T E

              Naval P o s t g r a d u a t e School                                                                 J u n e , 1977
              Monterey , C a l i f o r n i a 93940                                                            13. NUMOER OF PAGES
                                                                                                                   86 '-
         14. MONITORING           AGENCY NAME & ADDRESS(lf dllformt from Controllln# OffleoJ                  IS.    SECURITY CLASS.   (Of   thlo t0-e)
              Naval P o s t g r a d u a t e School                                                                  Unclassified
              Monterey , C a l i f o r n i a 93940
                                                                                                                     scnEouLE

         16.    OlSTRlBUTlON STATEMENT (of thla RoporfJ

              Approved f o r p u b l i c release; d i s t r i b u t i o n u n l i m i t e d .




         17. D l S T R l l U T l O N STATEMENT (ol tho d o t r u t mloredln Block 90. If d l l f w o n l from R.pOrt)




         18. SUPPLEMENTARY NOTES




         IS.    K E Y WOROS (Continuo M rovoroe d d o If n s s o * o w a d ldontlfy by block nurbor)


                S h i p S c h e d u l i n g ; Q u a d r a t i c Assignment; L i n e a r Programming;
                Optimization ( l a r g e s c a l e ) .


         20.    ADSTRACT (Cmtlnuo an rovorao d d o If n e c o o a w m d Id.ntl*        by block m r b o r )

                         As p a r t of h i s management p l a n n i n g and c o n t r o l f u n c t i o n , t h e U . S .
                Coast Guard's P a c i f i c Area Commander s c h e d u l e s t h e o p e r a t i o n a l m i s s i o n s
                f o r a l l High Endurance C u t t e r s i n t h e P a c i f i c Area. To p r o v i d e a
 c              p o w e r f u l management t o o l t o assist t h i s s c h e d u l i n g p r o c e s s , an a n a l y t i c
                model f o r t h i s l a r g e scale problem h a s been developed and implemented.
                It c o n t a i n s m i s s i o n r e q u i r e m e n t s , r e s t r i c t e d s e q u e n c i n g of m i s s i o n s ,
                s h i p s ' p h y s i c a l l i m i t a t i o n s and crews' m o r a l e - r e l a t e d c o n s i d e r a t i o n s ,
     b
          Approved f o r p u b l i c release; d i s t r i b u t i o n u n l i m i t e d ,


      A QUADRATIC ASSIGNEEfYT / L I N E A R P R O G R A M M I N G APPROACH
           TO SHIP SCHEDGLING FOX THE U.S.                           COAST G U A R D

                                                   by
                                                         & !.
                                                        f L/-
                                  C h a r l e s E. S'bre
                BS,
                 ..
                                                         k
                    L i e u t e n a n t , U. S. o a s t Guard
                        U. S. C o a s t G u a r d A c a d e m y , 1971



              S u b m i t t e d ir partial f u l f i l l m e n t of t h e
                     r e q u i r e m e n t s f o r t h e d e g r e e s of

                  MASTER OF SCIENCE I N COlYPUTER S C I E J C E
              HASTER OF S C I E N C E I N O P E 2 A T I O i i S RESEARCH


                                            from t h e
                           NAVAL POSTGRADUATE SCHOOL
                                          June, 1977




Author:




                                               3
                                             ABSTRACT




              As p a r t o f h i s m a n a g e m e n t p l a n n i n g a n d c c n t r o l
f u n c t i o n , t h e U, S, C o a s t G u a r d ' s P a c i f i c Area Commander
s c h e d u l e s t h e o p e r a t i o n a l m i s s i o n s f o r a l l High E n d u r a n c e
C u t t e r s i n t h e P a c i f i c Area.             To provide          a   powerful
m a n a g e m e n t t o o l t o a s s i s t this s c h e d u l i n g p r o c e s s , a n
a n a l y t i c m o d e l for t h i s l a r g e s c a l e p r o b l e m h a s b e a n
d e v e l o p e d a n d implemented. It c o n t a i n s m i s s i o n r e q u i r e m e n c s ,
restricted             sequencing             of      missions,             ships'           physical
l i m i t a t i o n s a n d crews' m o r a l e - r e l a t e d c o n s i d e r a t i o n s . T h e
modeling approach i s based on t h e Geoffzion-Gravep modd f o r
p a r a l l e l p r o d u c t i o n l i n e s w i t h s i g n i f i c a n t c h a n g e o v e r costs.
The         implementation                 solves           a        large            (860        row)
Koopmans-Eieckmann               f i x e d c h a r g e Q u a d r a t i c A s s i g n m e n t node1
u s i n g a new method w i t h a n a d v a n c e d ,                     feasible starting
s o l u t i o n p r o v i d e d by a n i m b e d d e d n e t w o r k ( w i t h 1 , 7 2 0 n o d e s
a n d 7 3 9 , 6 0 0 a r c s ) . Many l i n e a r p r o g r a m m i n g p r o b l e m s ( 2 0 0
row, 4 5 0 v a r i a b l e ) a r e t h e n s o l v e d w i t h a l i n e a r p r o g r a m m i n g
s u b r o u t i n e of a d v a n c e d d e s i g n . T h e r z s u l t i n g model a n d t h e s e
implementation t e c h n i q u e s produce e x c e l l e n t q u a l i t y working
s c h e d u l e s x i t h v e r y r e a s o n a b l e e x e c u t i o n time a n d memory
requirements. Alternative s o l u t i o n s are e a s i l y generated.




                                                  4
                                   TABLE OF CONTENTS



 I    .   INTRODUCTION        .....................................                          11
          A . EACKGROUND...................................                                  11
          E . CURRENT SCHEDULING METHOD ....................                                 12
          C . GCAL O F T H I S STUDY ...........................                             13
I1    .   ANALYSIS O F SCHEDULE A N D SCHEDULING G U I D E L I N E S ... 15
          A . SCHEDULE'S PlEASURE OF E F F E C T I V E N E S S .......... 15
          B . EASIC ELEMENTS ............................... 15
          C . G U I D E L I N E S ...................................                        17
I11   .   CHARACTERIZATION O F PROBLEM A N D MODEL S E L E C T I O N .. 19
          A . CHARACTERIZATION ............................. 19
          E . S E L E C T I O N OF APPROACH ........................ 21
 IV   .   GEOFPEION-GRAVES MODEL ........................... 2 4
          A . OVERALL PHILOSOPHY ........................... 2 4
          B.  QUADRATIC ASSIGNMENT POXMULATION ............. 2 6
              1 . G e n e r a l K o o p m a n s - B e c k m a n n F o r m u l a t i o n .... 26
              2 . G r a v e s - W h i n s t o n G e n e r a l i z a t i o n ........... 27
              3 . G e o f f r i o n - G r a v e s F o r m u l a t i o n ............. 2 8
          C . LINEAR PROGRAM FORMULATION I N G-G NODEL ...... 32
          D . I EPLEME NTAT I O N ...............................                            33
  V   .   ANALYTIC HODEL FOE COAST G U A i i D PROBLZM ...........                           36
          A . OVERALL C O Z S I D E a A T I O N S .......................                    36
          B . C;UADRATIC B S S I G 8 J E N ' i ' MODEL ................... 37
              1 . C - H a t r i x ............................... 37
              2 . Q - N a t r i x ...... ........................                            39
              3 . D - H a t r i x ............................... 40
              4 . G u i d e l i n e s Not I n c o r p o r a t e d i n QA ........ 4 3




                                              5
                                    .




          C   .   LINEAR PROGRAfiMING MODEL                ..................... 4 3
                   1 . O b j e c t i v e F u n c t i o n a n d C o n s t r a i n t s ....... 43
             D . GUIDELINES            Nor ~ O C E L E D.......................              45
   VI . IHPLEHENTATION ................................... 4 7
             A . OIERALL CONSIDERATIONS .......................                              47
             B . QUADRATIC ASSIGNMENT ItlPLZMENTATION .......... 4 9
             c . A N E W QUADRATIC A S S X G N H E N T         METHOD............ 53
             D . POST-PROCESSING O F THE             QA       SOLUTION ........... 56
             E . IEPLEBENTATION OF LINZAR PROGRAHS ............ 57
             F . IMPLEMENTATION ALTERNATIVE FOR L P ............ 6 2
             G . CALLING STRATEGY FOR T H E LINEAR PROGRAMS ..... 6 5
  V I I . RESULTS ..........................................                                 69
             A . USE OF THE HODEL .............................                              69
             B . QUALITY OF SCHEDULES PRODUCED ................ 7 3
             C . QA / L P RESULTS .............................. 75
V I I I . ENHANCEMENTS ..................................... 77
             A . AFPROACH .....................................                              77
             B . IMPLEHENTATION ...............................                              78
A p G e n d i x A:      S A l E L E PROBLEY A N D RESULTANT SCHEDULE ...... 80
L I S T OF 3EFERXNCES ......................................                                 83
I N I T I A L D I S T B I B U T I O N L I S T ............................... 85




                                               6
                                        LIST OF FIGURES



 1   .   Mission t o Hission T r a n s i t i o n s             ......................            19

 2   .   Q - M a t r i x ............................................                            28

 3.      C.Matrix ............................................                                   31

 4.      From-To T r a n s i t i o n D i f f i c u l t y ....................... 38

 5.      S i n g l e M i s s i c n S t r u c t u r e ............................ 42

 6.      P a t r o l S t r u c t u r e ....................................                      43

 7.      I m p l e m e n t a t i o n F l o w c h a r t ............................ 4 8

 8.      Vectors fcr Rows a n d C o l u m n s of Q - M a t r i x ............ 50

 9.,     P e r f o r m a n c e o f Q u a d r a t i c A s s i g n m e n t d e t h o d s ......... 5 5

10 .     P r e s e n t P e n a l t y C o s t S t r u c t u r e s .....................           61

11 .     Network F c r m u l a t i o n ................................. 64

12 .     LP C a l l i n g S t r a t e g y P e r f o r m a n c e ..................... 6 8

13 .     I n f l u e n c e of More B r e a k E o i n t s i n P e n a l t y C o s t
           S t r u c t u r e for T o t a l Away H c m e p o r t ( A H P ) T i m e    ....... 77
14   .   B a l a n c i n g o f C o s t s B e t u e e n Two   C o m p o n e n t s ........... 7 2




                                                  7
                                           GLOSSARY



                                                                                         Page


1    .   M e a s u r e o f E f f e c t i v e n e s s (MOE)   ..................              15

2    .   High E n d u r a n c e C u t t e r s   (HEC)       ....................             15

3    .   Medium E n d u r a n c e C u t t e r s (MEC) ..................                     16

4.       S H I P . S F E C I P I C ...................................                       16

5.       G E N E R A L .........................................                             17

6.       M a i n t e n a n c e ( M a i n t ) .............................                   17

7    .   R e f r e s h e r T r a i n i n g ( R e f t r a ) .....................             17

 8   .   A l a s k a F i s h e r y P a t r o l ( A l p a t ) ...................             37

 9   .   O c e a n o g r a p h i c Data C o l l e c t i o n ( O c e a n ) ...........        17

10 .     Navy ASW E x e r c i s e ( N a v e x ) .......................                      17

11 .     PATROL r e q u i r e m e n t ..............................                         17

12 .     Away From H o n e p o r t (AHP) ........................                            18

13 .     T a t a l B B P time ..................................                             18

14 .     T o t a l A l p a t t i n e ................................                        18

35 .     C r u i s e ..........................................                              18

16 .     I n p o r t f o r S e a r c h a n d R e s c u e S t a n d b y ( I n p o r t ) ...   19

17 .     " L i n i t i n g " and Morale-related c o n s t r a i n t s .......                21

18 .     H y b r i d QA/LP A p p r o a c h ...........................                       24

19.       E a r l y s t a r t / L a t a f i n i s h times ...................                24


                                                 a
                                                                           .
20.      Geoffrion-Graves             (G-G)         ..........................       24
21.      Campaign ........................................                           25

22   .   Q u a d r a t i c A s s i g n m e n t ( Q A ) .......................       25
23   .   L i n e a r Program (LP) .............................                      25

24 .     Q.Matrix.         I n t e r a c t i o n s . .........................       28

25 .     Horizon (H) .....................................                           29
26 .     T i m e R e s o l u t i o n (S).............................                29

27 .     Requirement Fixed             Demand (A ) ...................               29
                                                k

28 .     Standard T r a n s i t i o n Cost (T             )      .................   31
                                                            PP'
29 .     C.Hatrix.          T r a n s i t i o n C o s t s ......................     31

30 .     D.Hatrix.          F i x e d C o s t s ...........................          31

31   .   G r a v e s - W h i n s t o n (G-W) ...........................             33

32 .     S w i t c h A l g o r i t h m for Q A ..........................            34

33 .     S l i d e A l g o r i t h m f o r LP ..........................             34

34 .     S w i t c h A l g o r i t h m for L P .........................             35

35 .     T i m e - O n i t .......................................                   36

36 .     Mission.Unit ....................................                           36

37 .     ? l i s s i o n d u r a t i o n v a r i a b l e .......................     44

38   .   P e n a l t y v a r i a b l e ................................              44

39 .     Work Factor .....................................                           49

40 .     CGNET   ...........................................                         54

41   .   XS   ..............................................                         58

42   .   P e n a l t y Cost S t r u c t u r e s   .........................          61



                                                   9
                                        ACKNOWLEDGEMENT




           I       wish t oexpress my s i n c e r e       appreciation             to
P r o f e s s o r s Gerald G. Brown a n d G o r d o n H . B r a d l e y f o r t h e i r
guidance,         constant support, complete a v a i l a b i l i t y ,           and
willingness to participate.                           I an a l s o d e e p l y grateful t o
P r o f e s s o r G l e n n W . G r a v e s for      providing          the     POBTRAN       code
from      his     s u c c e s s f u l i m p l e m e n t a t i o n of t h e Geoffr i o n - G r a v e s
Model.         Y r . D a v i d Norinan, M r . Ed D o n n e l l a n , a n d Mr.  Mannas
A n d e r s o n of C h u r c h C o m p u t e r C e n t e r are t h a n k g d for t h e i r
a s s i s t a n c e d u r i n g t h e inany h o u r s s p e n t    at E   the     center.
thank   my   wife,                 Ellen,   for  her enduring patience,
understanding,  and                assistance throughout  these  degree
prcgrams.




                                                10
                                                    ,




                                         1.     INTRODUCTION
                                                --I---




A.    BACKGROUND



      T h e U n i t e d S t a t e s Coast G u a r d i s          responsible             for many
a r e a s of n a t i o n a l c o n c e r n i n t h e maritime r e g i o n s o f t h i s
c o u n t r y . These r e s F o n s i b i l i t i e s , c a l l e d l f m i s s i o n s t f b y t h e
Coast Guard,                  i n c l u d e s e a r c h a n d rescue, law e n f o r c e m e n t ,
f i s h e r y and c u s t o m r e g u l a t i o n , s m a l l c r a f t a n d commercial
vessel safety,                    icebreaking,          I n t e r n a t i o n a l Ice P a t r o l and
c o l l e c t i o n of o c e a n o g r a p h i c d a t a .              To coordinate these
r e s p o n s i b i l i t i e s , two major commands h a v e b e e n e s t a b l i s h e d         -
o n e f o r t h e A t l a n t i c , G u l f o f M e x i c o a r e a , a n d o n e for t h e
P a c i f i c a r e a . E a c h a r e a i s f u r t h e r s u b d i v i d e d i n t o ssveral
d i s t r i c t s . The District Comnanders h a n d l e a l l o p e r a t i o n a l
a n d a d m i n i s t r a t i v e matters a r i s i n g i n t h e i r d i s t r i c t s . The
Area Comlnandor, h o w e v e r , assumes o p e r a t i o n a l c o n t r o l f o r a l l
missions t h a t are ctmulti-districttt i n geographical region,
sccpe, or resources,                        Some of t h e s e m i s s i o n s a r e :             law
e n f o r c e m e n t a n d f i s h e r y p a t r o l s i n t h e Alaska r e g i o n ,
oceanographic data c o l l e c t i o n , and readiness t r a i n i n g


      To f u l f i l l h i s r e s p o n s i b i l i t i e s ,  t h e Area Commander m u s t
s u p e r v i s e t h e a l l o c a t i o n of r e s o u r c e s f o r t h e s e m i s s i o n s .
Periodically,            the      Pacific Area C o m m a n d e r ' s s t a f f p r e p a r o s
a n d formally i s s u e s t h e o p e r a t i o n a l s c h e d u l e s f o r t h e High
and     Medium          Cutters i n t h e P a c i f i c Area. T h i s
                      Endurance
published schedule also s a t i s f i e s t h e maintenance, military
readiness,            and t r a i n i n g requirements.           It s e r v e s a s t h e
o p e r a t i o n a l o r d e r s t o t h e Commanding O f f i c e r s of t h e s h i p s ,


                                                   11
balancing,          maxinum c r u i s e l e n g t h s ,     m i n i m a l times t o b e s p e n t
in     homegort,          and     other      items        that       affect     the          crew's
 m ra1
  o   ell     .
       These morale-related               limitations result               in     goals        that
the s c h e d u l e r a t t e m p t s t o s a t i s f y .      The evaluation and
c o m p a r i s o n o f two d i f f e r e n t s c h e d u l e s t h a t s a t i s f y t h e
mission requirements and morale-related g o a l s i n d i f f e r e n t
b a l a n c e s can not b e performed using s t r i c t l y o b j e c t i v e
criteria. Judgement a n d i n d i v i d u a l p r e f e r e n c e s are involved.

       U s i n g t h e g u i d e l i n e s , two    schedulers          spend         about     one
week         manipulating                a      large magnetic Gantt c h a r t and
c a l c u l a t i n g summary t o t a l s f o r e a c h c a n d i d a t e s c h e d u l e (e. g.
t o t a l t i m e away from h o m a p o r t , t o t a l t r a v e l time f o r e a c h
ship).          T h e n , a d d i t i o n a l time i s s p e n t w i t h t h e r e s u l t a n t
s c h e d u l e i n n e g o t i a t i o n s between t h e d e c i s i o n makers, t h e
s c h e d u l e r s , a n d t h e Commanding O f f i c e r s of t h e s h i p s .           The
d e s i r e d f i n a l s c h e d u l e is o n e t h a t s a t i s f i e s t h e n i s s i o n
requirements with consideration and                               respect          for      the
morale-r e l a t e d g o a l s .

       E a c h J a n u a r y a n d J u l y a f o r m a l s c h e d u l e is 2 u b l i s h e d
c o v e r i n g t h e n e x t 18 month p e r i o d .     A t e n t a t i v a s c h e d u l a for
t h e s u b s e q u e n t s i x month p e r i o d ( m o n t h s 19 t o          24)     also
                                                                                        is
prspared,            but distributed inforaally                  .
                                                                The f i n a l p u b l i s h e d
schedule           p r o v i d e s one-day         tiine   iesolution,       specifying
required           on-scene a r r i v a l and d e p a r t u r e d a t e s and e s t i m a t e d
hcmeport          d e p a r t u r e and r e t u r n dates.



C.     GOAL O F THIS S T U D Y



       The        immediate       g o a l o f t h i s e x p l o r a t o r y r e s e a r c h is t h e
development            of    an     analytic            scheduling      model         based      an


                                                   13
mathematical           programming             techniques.         T h i s a n a l y t i c inodel,
i n c o r p o r a t i n g t h e most i m p o r t a n t of t h e m i s s i o n r e q u i r e m e n t s
and     morale-related               goals,      w i l l g e n e r a t e p o t e n t i a l working
s c h e d u l e s u s i n g a r e a s o n a b l e a m o u n t of   computer           resources.
These       potential         s c h e d u l e s w i l l p r e s e n t t h e d e c i s i o n makers
and     schedulers           with      more      choices           for      objective               and
subjective          e v a l u a t i o n t h a n t h e c u r r e n t manual method.                This
managerial tool u i l l a i d           strategic and tactical d e c i s i o n
prccesses,          and p o s s i b l y improve them by r e l i e v i n g a heavy
c l e r i c a l uor k l c a d .

        Historically,            scheduling problems (as an a p p l i c a r i o n of
m a t h m a t i c a l programming) h a v e e i t h e r b e e n o v e r s i m p l i f i e d
w i t h a r e s u l t a n t loss o f e f f e c t i v e n e s s , o r s o d e t a i l e d t h a t
computational d i f f i c u l t i e s prevent economical                         solution.
Certain            recent        advances i n optimization c a p a b i l i t i e s ,
h o w e v e r , h a v e made p o s s i b l e t h i s e x p l o r a t o r y r e s e a r c h t o
a p p l y m a t h e m a t i c a l programming models t o t h e Coast Guard
s c h e d u l i n g Froblein.


       The     chapters         of     t h i s t h e s i s follow a s y s t e m s a n a l y s i s
apFroach :

  1.    F o r a u l a t i o n ( i d e n t i f y problem and o b j e c t i v e s )     ,
  2.    Analysis   (examine    elements,                             i n t e r r e l a t i o ns h i p s ,
        variables, and constraints) ,

  3.    S y n t h e s i s ( c l a s s i f y t h e problem and            study      alternative
        m e t h c d s of s o l u t i o n ) ,

  4.    S e l e c t i o n of Method a n d D e t a i l e d M o d e l i n g ,

  5.    I m p l e m e n t a t i o n ( t h e t e s t of a l l t h a t p r e c e d e s ) , a n d

  6.     E v a l u a t i o n a n d aeform u l a t i o n .




                                                  14
          11       ANALYSIS 02 SCHEDJLE
                   -_I--
                                                                SCHEDULILNG G U I D Z L I N E S




A.     SCHEDULE'S MEASUBE OF E F F E C T I V E N Z S S



       Unlike        typical         production           or job shop scheduling, t h e
Coast Guard problem d o e s n o t p o s s e s s                       a s i n g l e measure of
e f f e c t i v e n e s s on w h i c h o b j e c t i v e e v a l u a t i o n of c a n d i d a t e
schedules can e a s i l y be based.                          T h e r e a r e some o b j s c t i v e
evaluation                 criteria.          However,           t h e subjective criteria
d e r i v e d from t h e m o r a l e - r e l a t e d             considerations play t h e
dominant role i n e v a l u a t i o n . Calendar-related e v e n t s are
a l s o c o n s i d e r a t i o n s (e.g.        f i s h i n g s e a s o n s , major h c l i d a y
periods,            a n d weekends)        .     Individually, each requirement or
m o r a l e - r e l a t e d g o a l is r e a s o n a b l e .       U n f o r t u n a t e l y , t h e r e is
n o s c h e d u l e t h a t c a n s i m u l t a n e o u s l y meet all t h e d e s i r e d
g o a l s . S i n c e any s c h e d u l e w i l l have v i o l a t i o n s of t h e
guidelines,                i t is n e c e s s a r y t o a s s e s s t h e t r a d e - o f f s among
t h e c o n f l i c t i n g requirements, and t o b a l a n c e t h e v z r i o u s
goals.            M a n a g e r i a l j u d g e m e n t a n d p a r s o n a l p r 2 f e r e n c e s (of
t h e Area a n d D i s t r i c t Commanders,                     their         staffs,         and each
ship's            Commanding            Officer)             ultimately                determine t h e
                                                               /
p r o p e r t i a s a n d s t r u c t u r e of t h e f i n a l s c h e d u l e .



B.     B A S I C ELXMENTS




       There       are t h r e e basic elements i n t h e schodule                          -   ships,
time,       and      missions.            There        are      presently            seven         High
Endurance          Cutters          (HEC)       a n d f i v e Medium E n d u r a n c e C u t t e r s


                                                    15
(MEC) i n t h e P a c i l i c Area t h a t a r e s c h e d u l e d by t h e                    Area
Commander.       An e i g h t h HEC a r r i v e s 1 J u n e 1 9 7 7 .                           The
IfHamilton           c l a s s t 1 of H E C I S ,      b e c a u s e of t h e i r d i f f a r e n t
c a p a b i l i t i e s , are u s e d f o r d i f f e r e n t t y p e s of m i s s i o n s t h a n
t h e s i n g l e o l d e r HEC i n t h e Area.                 Scheduliag i n t e r a c t i o n s
between t h e s e s h i p t y p e s and a l l o t h e r s h i p s are minimal.
Thus,        t h e s c h e d u l i n g p r o c e s s can b e d i v i d e d i n t o two
s e p a r a t e cases      -      o n e f o r t h e I1Hamiltonli c l a s s H E C I S a n d o n e
fOK       the o t h e r ,         s m a l l e r cuttP-rs.           To l i n i t s c o p e ,    and
e m p h a s i z e basic issues,                 t h i s s t u d y c o n c a t r a t a s on the
s c h e d u l i n g p r o c e s s f o r just t h z q l H a a i l t o n l l i l i g h Z n d a r a n c e
Cutters           ( H E C I S ) . T w o of t h e s e H E C ' s a r e b a s e d ( h o m e p o r t e d )
i n ~ o n o l u l u , Hawaii; two i n S a n ~ r a a c i s c o , C a l i f o r n i a ; a n d
tho, r e s t i n S e a t t l e , H a s h i n g t o n .


        As m e n t i o n e d a b o v e , each p u b l i s h a d s c h e d u l e c o v e r s a n
e i g h t e e n rnoilth p e r i o d .      M o n t h s 19 t o 2 4 a r e t e n t a t i v e l y
s c h e d u l e d b u t not f o r m a l l y d i s t r i b u t e d . Betueen s u c c e s s i v a
s c h e d u l e s , t h e r e i s a d e s i r e t o m i n i m i z e t h e number of major
changes.           U n l e s s major c h a n g e s i n m i s s i o n r e q u i r e m e n t s h a v e
o c c u r r e d o r unforeseen e v e n t s have t a k e n place, t h e first
s i x s o n t h s o f a new s c h e d u l e w i l l b e i n close a g r e e m e n t w i t h
t h e same p e r i o d f o r t h e p r e v i o u s s c h e d u l e .               As t h e time
frame of t h e s c h e d u l e         progresses,            more c h a n g e s a n d t h u s
g r e a t e r d e v i a t i o n s occur.         The i n f o r m a l     19 t o 2 4 inonth
segment. i s t h e i n i t i a l p r o j e c t i o n f o r t h i s c a l e n d a r p e r i o d ;
i t is, t h e r e f o r e , h i g h l y s p e c u l a t i v e .


       The m i s s i o n s performed by t h e tlHamilton" class High
Z n d u r a n c e c u t t e r s c o v e r most o f t h e C o a s t G u a r d * s a r e a s o f
responsibility             -    fishery        and       custom          regulation,          law
enforcement,             oceanography,        search a n d r e s c u e ,               military
readiness, and personnel training.                      T w o b a s i c c a t e g o r i e s of
requiremants o r i g i n a t e from these r e s p o n s i b i l i t i e s .                   The
first t y p e w i l l be termed SHIP-SPECIFIC requirements.                                 These
a s s i g n m e n t s arise froa r e q u i s i t e s p h y s i c a l l y a s s o c i a t e d with


                                                 16
a p a r t i c u l a r s h i p a n d h e r crew. T h e r e is a n e e d t o m a i n t a i n
her e q u i g m e n t a n d m a c h i n e r y , a n d t r a i n h e r crew t o f u n c t i o n
as a u n i t so t h a t t h e s h i p c a n p e r f o r m g e n e r a l o p e r a t i o n a l
missions.              For e x a m p l e ,   R e f r e s h e r t r a i n i n g ( R e f t r a ) , Navy
ASW e x e r c i s e s (Navex) , a n d N a i n t e n a n c e p e r i o d s         (YAINT)         are
SHIP-SPECIPIC               requirements.          Each s h i p m u s t b e s c h e d u l e d t o
i n d i v i d u a l l y satisfy these needs.                       T h e s e c o n d t y p e of
requirements,               termea GENERAL, c o v e r t h e d i r e c t o p e r a t i o n a l
a r e a s o f r e s p o n s i b i l i t y . S u c c e s s i o n s of s h i p s a r e u s e d to
c o l l e c t i v e l y s a t i s f y t h e s e requirements.                  Alaska f i s h e r y
p a t r o l s ( A l p a t ) , Academy C a d e t t r a i n i n g ,        and oceanographic
data            collection             (Ocean)       are         examples         of        GENERAL
requirements.

      The         g e o g r a p h i c a l area w h e r e t h e m i s s i o n i s p e r f o r m e d is
a l s o a n i m p o r t a n t c o n s i d e r a t i o n . T r a v e l time t o a n d f r o m t h e
mission            area          is       n e c e s s a r i l y p r e s e n t i n t h e one-day
r e s o l u t i o n of t h e p u b l i s h e d s c h e d u l e .        (For instance, t r a v e l
time f r o m Hauaii t o A l a s k a is 7 d a y s . )                       T h e time s p e n t by a
s h i p away from i t s h o m e p o r t i s a f f e c t e d b.y t h i s t r a v e l t h e .
Also,         i n o r d e r t o m i n i m i z e t r a v e l time t o t h e m i s s i o n a r e a ,
d i f f e r e n t m i s s i o n s are p e r f o r m e d b y t h e Hawaii-based                 ships
t h a n by t h e c o n t i n e n t a l - b a s e d (CONUS) s h i p s .



C.     GUIDELINES



      T h e GENERAL a n d SHIP-SPECIFIC                  requirem2nt.s are s p e c i f i e d
i n t h r e e ways by t h e s c h e d u l i n g g u i d e l i n e s .         T h e f i r s t is
t h e s e t t i n g of t h e n u m b e r o f s h i p s t h a t a r s t o b e o n - s c e n e
i n a g i v e n area a t a g i v e n time (e.g. D u r i n g May, two s h i p s
s h o u l d be n e a r t h e A l e u t i a n I s l a n d s ) .             This typc             of
s p e c i f i c a t i o n is c a l l e d a PBTaOL r e q u i r e m e n t . T h e s e c o n d is
t h e s e t t i n g o f a r e p e t i t i v e f r e q u e n c y t o b e met     (e.g.        once
per quarter).                T h e f u l f i l l m e n t of t h i s t y p e of r e q u i r e m e n t



                                                 17
u s u a l l y t a k e s a s t a n d a r d a m o u n t of time a n d i s t o b e
accomplishsd             without         interruption.     The t h i r d i s t h e
s p e c i f i c a t i o n of a go&        anm&           gg Ligs t o        be    spent       in   a
given p e r i o d , T h i s time q u o t a w i l l u s u a l l y be d i v i d e d i n t o
several segments (e.g.         T h e m a i n t e n a n c e r s q u i r e m e n t of   13
w e e k s p e r yaar may b e d i v i d e d i n t o s e g m e n t s of 4 , 4 , a n d 5
weeks e a c h . )


      Three       principal          morale-related                       affect t h e
                                                                  guidelines
a m o u n t o f ti!ue e a c h s h i p c a n be s c h e d u l e d t o be a w a y from
its h o m e p o r t , F i r s t , a maximum l i m i t i s s e t f o r t h e t o t a l
llAway H o m e p o r t " (AHP) time p e r y e a r f o r e a c h s h i p .     Second,
i t is d e s i r a b l e t o b a l a n c e AHP            time b e t w e e n s h i p s s i n c e it
f u n c t i o n s a s a p s e u d o - m e a s u r e of    each s h i p ' s s h a r e of t h e
workload.          T h i r d , a maximun l i m i         t is s e t f o r t h e d u r a t i o n of
a n y s i n g l e c r u i s e ( t h e time from           d e p a r t u r e from h o n e p o r t   to
return).


      O t h e r g u i d e l i n e s c o n c e r n t h e l o n g e s t a n d inost       difficult
GENERAL        requirement           -    Alaska          Fishery       Patrol        (Alpat), A
maximum l i m i t o n t h e t o t a l y e a r l y            Alaska       Patrol   time per
s h i p i s set.            If      a s h i p i s a s s i g n s d back-to-back Alpat
m i s s i o n s , t h e minimun i i n t e r v e n u n g i n - h o m e p o r t    p e r i c d is
e i g h t weeks.          B e f o r e a n y A l p a t , a minimum f o u r week i n p o r t
period is desired for adequate preparation.                                   It i s a l s o
d e s i r e d t o a l t e r n a t e s h i p s t h a t are a s s i g n e d A l p a t m i s s i o n s
i n t h e r o u g h weather m o n t h s of w i n t e r .




                                                 18
A.    CHARACTERIZATION



      Ex3mination            of        the    p u b l i s h e d s c h e d u l e s f o r t h e period
J a n u a r y 1 9 7 5 t o December 1 9 7 6 snows t h e f o l l o w i n g b a s i c
elements:             6    ships,        2 y e a r time s p a n , 6 a r e a s o f
responsibility generating requirements,                                a n d 129 s e p a r a t e
m i s s i o n s used t o s a t i s f y t h e r e q u i r e m e n t s .

      T h e f r e q u e n c y of m i s s i o n - t o - m i s s i o n   t r a n s i t i o n s made       by
t h e s h i p s i s shown i n            F i g u r e 1.




 ----
 From:
        Inport                                  3          12             7          5         27
        Alpat                     11                           4                                    1
        Ocean                      4          14
        Ref tra                    7                                                                1
        Nav ex                     6                                      1                         1
        naint                     15            2              2                     4              1


                                             Figure        1



      Forinstance,                      ships   nade   the                  t r a n s i ti o n from
Oceanographic data                     collection  (Ocean)                t o A l a s k a Patrols


                                                    19
( A l p a t ) f o u r t e e n times i n t h e t u o y e a r s .

      The       s c h e m a u s e d b y Conway, M a x w e l l , a n d Miller [ 3 ] t o
d e s c r i b e s c h e d u l i n g problems is a u s e f u l c l a s s i f i c a t i o n and
h e l p s t o s u c c i n c t l y d e f i n e w h i c h f a c t o r s a r e known a n d
unknown by t h e s c h e d u l e r :

 1.    -- q-----m e n t -----v a l ----c e s s -
       Re uire          Arri       Pro                        A t   the         s t a r t of t h e
       scheduling process, t h e t o t a l                r e q u i r e m e n t s f o r each
       area of r e s p o n s i b i l i t y a r e        'specified within a given
       expected range.      T h e n u m b e r of         separate inissions t h a t
       w i l l be necessary t o f u l f i l             l each r e q u i r e m e n t i s n o t
       specified    prior         to       the              scheduling                 process.
       Bequirements        are not simultaneously a v a i l a b l e but
       become a v a i l a b l e a t i n d i v i d u a l times a c c o r d i n g t o
       frequencies and timings given i n t h e guidelines.

 2.    Resourcgs         AvW&lable         -    The     number        and        individual
       c a p a b i l i t i e s of a l l s h i p s a r s known b y t h e s c h e d u l e r .

 3.    g&oz g a t t e r n - As s h o w n            i n Z'igure 1, t h e i a i s s i o n
       t r a n s i t i o n s t h a t occur have d e f i n i t e p a t t e r n s . The
       s e q u e n c i n g of m i s s i o n s i s c r i t i c a l , m o s t l y b e c a u s e of
       t h e l i m i t a t i o n on c r u i s e l e n g t h .          Also,      t h e PATROL
       requirement             introduces           precedence                and         i
                                                                                        1 nking
       r e l a t i o n s h i p s between t h e i n d i v i d u a l m i s s i o n s used t o
       f u l f i l l t h e r e q u i r e m e n t : t h e e n d of a p a t r o l f o r o n e
       s h i p s h o u l d i m p l i c i t l y s p e c i f y t h e s t a r t of a p a t r o l
       for a n o t h e r s h i ? .

 4.    -------
       Measure         Ql      Effectiveness             LMOEl_       -    As     e x ? l a i ned
       p r e v i o u s l y , t h e r a is n o s i n g l e YOE b u t rather a complex
       c o m b i n a t i o n of r e q u i r e m e n t s and goal's.



      T h i s problem also p o s s e s s e s c o n s t r a i n t s beyond t h e s c o p e
of t h o s e g i v e n b y Conway, Max wel l , a n d H i l l e r .         These
constraints a r e caused b y t h e p h y s i c a l l i m i t a t i o n s of t h e


                                               20
    s h i p s a n d t h e n e e d s of t h e i r crews. T h e t o t a l AHP time
    s c h e d u l e d f o r e a c h s h i p s h o u l d be e q u a l l y b a l a n c e d w i t h
    respect        to t h e o t h e r s h i p s . T h e l e n g t h of s i n g l e c r u i s e s
    s h o u l d b e l e s s t h a n t h e maximum l i m i t of     t h e e n d u r a n c e of
    the     ship  o r crew.          For e a c h s h i p , t h e t o t a l A l a s k a P a t r o l
.   time p e r y e a r s h o u l d n o t exceed t h e s p e c i f i e d l i m i t .     These
    three constraints,                ref erred t o a s t h e l l l i m i t i n g ' l c o n s t r a i n t s
    i n f u r t h e r d i s c u s s i o n s , are n o t t y p i c a l, p r o d u c t i o n o r j o b
    shop constraints.


          To summarize t h e Coast G u a r d p r o b l e m ,                   t h e schaduler has
    definite         r e s o u r c e i n f o r m a t i a n a n d d e f i n i t e k n o w l a d g z ci t h e
    t y p e of t r a n s i t i o n s t h a t     can        occur.       He    knows        the     total
    m i s s i o n area r e q u i r e m e n t s a n d m u s t d e t e r m i n e t h e number a n d
    d u r a t i o n s of t h e s e p a r a t e m i s s i o n s t h a t c o l l e c t i v e l y s a t i s f y
    t h e s e r e q u i r e m e n t s . The r e q u i r e m e n t s are n o t s i m u l t a n s o u s l y
    a v a i l a b l e a t t h e b e g i n n i n g of t h e p e r i o d b u t                      arrive
    dynamically.            The t * l i m i t i n g f l c o n s t r a i n t s i m p o s e a d d i t i o n a l
    restictions           on     the      sequencg           and     durations         of      separate
    missions.


           T h i s problem is v e r y            large.            The    final       schedule          has
    one-day   resolution              for a b o u t 5 , 0 0 0 s h i p - d a y s a n d a b o u t 6
    mission-types.   Any              e c o n o m i c a l l y f e a s i b l e a n a l y t i c modal
    m u s t a g g r e g a t e and d i v i d a t h e s c h e d u l e p e r i o d i n t o weeks or
    e v e n m o n t h s a n d t h e n s c h e d u l e i n terms of t h e s e u n i t s .



    B.     SELECTION OF APPROACH



           An        Integer         Linear          Programming a p p r o a c h h a s been
    considered.             This approach t o scheduling problems                                  (as
    d e m o n s t r a t e d by P r a b h a k e r [ l o ] ) i s n o t f e a s i b l e b e c a u s e of
    t h e l a r g e s c a l e of t h i s C o a s t G u a r d p r o b l e m . A s c h e d u l e f o r
    6 ships,             6 m i s s i o n - t y p e s a n d 2 y e a r time s p a n x i t h 1 week



                                                       21
r e s o l u t i c n w o u l d y i e l d a b o u t 3 , 6 0 0 0/1 v a r i a b l e s . T h i s s i z e
i s t o o l a r g e f o r commercially a v a i l a b l e c o m p u t e r c o d e s . A
s p e c i a l c o d e f o r t h i s p a r t i c u l a r a p p l i c a t i o n would h a v e t o
be d e v e l o p e d ,       (There is n o a p r i o r i g u a r a n t e e t h a t even t h e
b e s t p o s s i b l e a p p r o a c h would s o l v e t h e problem w i t h a
r e a s o n a b l e a m o u n t of c c m p u t e r r e s o u r c e s . )

        S i m u l a t i o n h a s been d i s c a r d e d a s a s o l u t i o n method f o r
s e v e r a l reasons.               b o n g d e v e l o p m e n t time is r e q u i r e d t o
i n p l e m e n t t h e p r o p e r s e q u e n c e of r a n d o m s t a r t i n g s o l u t i o n s
a n d p r i o r i t y o r h e u r i s t i c - g u i d e d s e a r c h methods. ( P a n w a l k a r
and I s k a n d e r [ 9 ] list o v e r 100 s t a t i c and dynamic s c h e d u l i n g
or d i s p a t c h i n g r u l e s . )       T h i s a p p r o a c h is t y p i c a l l y e x p e n s i v e
to run. D i f f i c u l t i e s e x i s t i n i n t e r p r e t i n g t h e r e s u l t s : data
s e n s i t i v i t y and e x p e r i m e n t a l v a r i a t i o n are d i f f i c u l t t o
d i f f e r e n t i a t e , E f f e c t i v e e x t e r n a l c o n t r o l s for g u i d a n c e o r
model c o e r s i o n a r e u s u a l l y l a c k i n g .            T h e r e are a l s o serioils
reservations about t h e potential                                   flexibility            of       the
r s s u l t a n t c o m p u t e r p r o g r a m t o meet f i l t u r e p r o b l s m c h a n g e s
 ( i n the g u i d e l i n e s , f o r example).

       A     f o u n d a t i o n of o p t i m i z a t i o n t h e o r y a n d t e c h n i q u e s i s
desired.           T h e g s n e r a l a p p r o a c h i n i t i a l l y selected t o a d d r e s s
t h i s Coast- G u a r d s c h e d u l i n g p r o b l e m i s t h e h y b r i d Q u a d r a t i c
A s s i g n m e n t / L i n e a r P r o g r a m m i n g m o d e l of G e o f f r i o n a n d G r a v e s
[ 4 1.      T h e d e t a i l e d d e s c r i p t i o n o f t h i s n o d e 1 is p r e s e n t e d i n
t h e n e x t C h a p t e r b u t a b r i e f summary i s a p p r o p r i a t e h e r e .


       A      Q u a d r a t i c A s s i g n m e n t model d e t e r m i n e s a c a n d i d a t e
s e q u e n c e of s e p a r a t e m i s s i o n s f o r each s h i p .         A Linear
P r c g r a m m i n g modei t h e n d e t e r m i n e s t h e d u r a t i o n o f each
m i s s i o n . T h e n , two h e u r i s t i c s e a r c h e s v a r y t h e c a n d i d a t e
s e q u e n c e a n d new d u r a t i o n s a r e o b t a i n e d i n a n a t t e m p t t o
imFrove tho schedule.




                                                   22
      This     model       has      been      selected            for the reasons listed
below:

 1.    The     flexibility           in     t h e p r o b l e m s t a t e m e n t accomodatas
       nicely t h e i n d e f i n i t e information                 on    the        number     and
       d u r a t i o n of i n d i v i d u a l m i s s i o n s .

 2.    N simplifying assumptions
        o                                             o r d e l a t i o n of g u i d P l i n e
       restrictions                appear t o b e necessary, a l l o w i n g an
       u n c o m p r o m i s i n g v i e w of t h e e n t i r e s c h e d u l i n g p r o b l e m .

 3.    The       Q u a d r a t i c A s s i g n m e n t model n a t u r a l l y h a n d l e s t h e
       s e q u e n c i n g r e s t r i c t i o n s (which n o r m a l l y c a u s e g r s a t
       d i f f i c u l t y and com2utational complexity f o r analytic
       models)     .
 4.    The         c o n t i n u o u s L i n e a r P r o g r a m m i n g model n a t u r a l l y
       h a n d l e s t h e f f l i m i t i n g ( t c o n s t r a i n t s of t h i s problem and
       allows t h e l e n g t h s of t h e missions t o be optimally
       a d j u s t e d t o b e s t meet t h e g u i d e l i n e s .

 5.    The       L i n e a r P r o g r a m m i n g model    (     using f l e x i b l e peilalty
       f u n c t i o n s ) i s a n a t u r a l way                to   incorporate          the
       g u i d e l i n e s a n d morale-related                   goals into a linear
       objective function.

 6.    The     model h a n d l e s d y n a m i c , n o n - s i m u l t a n e o u s   a r r i v a l of
       requirements.

 7.    The     model       pcssesses          special           structure         amenable        to
       implementation techniques                     appropriate            for      the      large
       s c a l e of t h i s p r o b l e m .

 a.    T h i s h y b r i d model h a s b e o n s u c c e s s f u l l y i m p l e m e n t e d for
       a n i n d u s t r i a l p r o d u c t i o n a p p l i c a t i o n of smaller scale.




                                               23
A.    O V E R A L L PtiILOSOPiIY



      T h e model p r o p o s e d b y G e o f f r i o n a n d G r a v e s inakes s e v e r a l
s i g n i f i c a n t c o n t r i b u t i o n s t o s o l v i n g an i n t e r e s t i n g a class
of        SchGduling            problems:           the          s epa r a t i o n      of     the
combinatorially                   difficult             s e q u e n c i n g p r o b l e a from t h e
c o n t i n u o u s a l l c c a t i o n and t i m i n g problem; a f l e x i b l e problem
statsment reflecting true aanagerial discretion;                                             and a
method t o e x p r e s s t h i s c l a s s of s c h e d u l i n g p r o b l e m s i n t h e
rigid,           finite           f o r m u l a t i o n reqilired f o r the Q u a d r a t i c
Assignment Froblea.

      The      class      of     scheduling          problems addressed by t h i s
hybrid       method       is     stated i n          production terminology    as
follows:

        There are          s e v e r a l ffsimilarfl c o n t i n u o u s 2rocess
        f a c i l i t i e s ( l i n e s ) o p e r a t i n g i n p a r a l l e l . Each i s
        a b l e t o m a n u f a c t u r e soma s u b s e t of p r o d u c t s w i t h
        knoun p r o d u c t i o n r a t o s a n d costs.                    Significant
        c o s t s a r e i n c u r r e d by e a c h c h a n g e o v e r of a l i n e
        from p r o d u c i n g o n e p r o d u c t t o a n o t h e r .        Production
        o r d e r s are r e c e i v e d d y n a m i c a l l y a n d are               not
        simultaneously                available for scheduling.                      Each
        o r d e r i s f o r a s p e c i f i c t o t a l a r n o u n t of p r o d u c t i o n
        t o o c c u r b e t w a e n a n e a r l y s t a r t time a n d a l a t e
        f i n i s h time. An o r d e r c a n b e s p l i t , a m o n g l i n e s o r
        produced non-contiguously                         on t h e same l i n e . T h e


                                                24
        d e s i r e d s c h e d u l e h a s t h e minimum t o t a l           production
        and t r a n s i t i o n costs o v e r a s p e c i f i e d horizon.


        T h e s o l u t i o n a p p r o a c h f o l l o w e d b y t h e i r mcdel allows t h e
b r o a d e n i n g of t h i s p r o b l e m s t a t e m e n t t o a l l o u lower a n d
u p p e r l i B i t s o n t h e o r d e r demands r a t h e r t h a n a n e x a c t
amount.            Also, v i o l a t i o n o f t h e s t a r t a n d f i n i s h times of
o r d e r s , and e a r l y or late c o m p l e t i o n b y i n d i v i d u a l l i n e s is
allowed b u t p e n a l t y c o s t s a r e i n c u r r e d ,


       I n t h e h y b r i d model, a Q u a d r a t i c A s s i g n m e n t ( Q S ) p r o b l e m
is     f o r m u l a t e d a n d s o l v e d u s i n g t h e e x a c t demands, t h e
specified horizon,                and    the      earlystart and l a t e f i n i s h
times. V i o l a t i o n s of t h e h o r i z o n o r s t a r t / f i n i s h times i n c u r
penalty costs.          T h e Q A s o l u t i o n is n o t a n o p t i m a l ,    minimal
cost s o l u t i o n .      The problem i s s o l v e d by h e u r i s t i c m s t h o d s
t h a t o b t a i n very good,      low cost f i n a l r e s u l t s t h a t a r e
l*locally optimal"            (i.e.      c e r t a i n a a s y c h a n g e s made t o t h e
solution cause increased cost).                          F o r t h e remainder         of t h i s
thesis,                            m
              ~ l m i n i m a l ~i ~ p l i e s t h i s l o c a l l y o p t i m a l c o n d i t i o n .
The r e s u l t a n t a s s i g n m e n t ( l i n e s t o o r d e r s o v e r t h e f i x e d
time s p a n )       i s t h e n u s e d o n l y f o r t h e s e q u e n c e of c r d e r s
assigned t o each                   production           line.         Each        continuous
                                    \
production          run      on     a    line, called a                                i s noted.
T h e q u a n t i t y a n d t i m i n g i n f o r m a t i o n is d i s c a r d e d .


       G i v e n t h e p r o d u c t s e q u e n c e from t h e Q A s o l u t i o n , a
L i n e a r P r o g r a m (LP) is g e n e r a t e d a n d s o l v e d to f i n d t h e
d u r a t i o n of e a c h c a m p a i g n o n e a c h l i n e , u s i n g t h e b o u n d e d
d e m a n d s , t h e d e s i r e d e a r l y s t a r t a n d l a t e f i c i s h times,         and
a       flexikle schedule horizon,                            T h e LP m i n i m i z e s t o t a l
p c o d u c t i o n a n d p s n a l t y c o s t s ( i n c u r r e d by v i o l a t i o n s of t h e
d e s i r e d o r d e r times a n d / o r h o r i z o n ) . T h e s e q u e n c e c o s t s a r e
d e t e r m i n s d by t h e QA s e q u e n c e a n d are n o t c h a n g e a b l e by t h e
LP.         Next,       two l o c a l s e a r c h e s a r e e m p l o y e d t o e x a m i n e t h e
s e q u e n c e of c a m p a i g n s a n d l o c a t e p o t e n t i a l l y b e t t e r p r o d u c t


                                                  25
sequences.              For        each favorable candidate sequence, another
LP      solution       is o b t a i n e d .     The f i n a l s o l u t i o n i s t h a t
c o m b i n a t i o n of s e q u e n c e a n d p r o d u c t d u r a t i o n s w i t h t h e
minimum t o t a l p r o d u c t i o n , s e q u e n c e a n d p e n a l t y c o s t s .



B.      Q U A D R A T I C A S S I G N B E N T FORHULATION



        Although            many     variants        of        the     Quadratic       Assignment
problem            have      been      presented          in     the      literature,          this
discussion             c o v e r s t h e m o d e l s i n f l u e n c i n g t h e Coast Guard QA
model.         Discussed a r e t h e o r i g i n a l             Koopmans-Beckaann            modal
C7 1,        the     generalization             addressed by Graves-Whinston [6 1,
the            specialization                05        G raves-Whinston          used       by
G e o f f r i o n - G r a v e s i n t h e i r s c h e d u l i n g model, and ( i n t h e n e x t
C h a p t e r ) t h e l a r g e s c a l a Q A model a r i s i n g i n t h e C o a s t Guard
prcblem.            T h e r e a d e r w i l l n o t e t h a t t h e s p e c i a l i z a t i o n s made
in t h e C o a s t Guard model a r e a p p r o p r i a t e t o t h i s p s r t i c u l a r
problem.          T h e t e c h n i q u e s t o h a n d l e t h e l a r g e s c a l e c a n be
a p p l i e d t o g e n e r a l i z a t i o n s of t h e Coast Guard model with
some a d d i t i o n a l effort.


        1.     G e n e r a l Koopmans-Beckmann                 Formulatio;


               T h e o r i g i n a l s t a t e m e n t of a Q u a d r a t i c A s s i g n m e n t
 ( Q A ) p r o b l e m nas made      by Roopmans a n d Beckmann f o r t h e
a s s i g n m e n t cf n " i n d i v i s i b l e 8 1 m a n u f a c t u r i n g p l a n t s t o n
fixed           geographical              locations     so          that        inter-plant
t r a n s p o r t a t i o n costs         of o n e c o m m o d i t y a r e m i n i m i z e d . As
s t a t e d b y Lawler [ 81,            let

                   c        = t r a n s p o r t a t i o n c o s t p e r u n i t from
                       jk
                               location j t o location k;




                                                    26
                q         = q u a n t i t y s h i p p e d from
                    iP
                             plant i to plant p (interaction quantity) ;
                x         = 1 i f p l a n t i is a s s i g n e d t o l o c a t i o n j,
                    i j
                          = 0 otherwise.


               The         object         is         to        minimize          the        interplant
t r a n s p o r t a t i o n costs,




suk ject t o


  ,
                             2
                           i = l
                                       xij       =        1        for     j = 1 rn




                             2
                           j = l
                                      X
                                          i j
                                                          1        for     i = 1,n      .

That  is,           the      optimal            pairs         of    ( p l a n t i, l o c a t i o n j) a r e
desired.




        Graves   and       Hhiaston       have          generalized           the
Koopmans-Beckmann m o d e l by t h e a d d i t i o n o f a f i x e d c o s t term
t o t h e objective function.                The a s s i g n m e n t of p l a n t i t o
l o c a t i o n j c a n i n c u r known,              fixed costs              ( f o r example,
p u r c h a s e of l a n d ,   i n s t a l l a t i o n of h i g h w a y s a n d             other
services). The p r o b l e m i s t o m i n i m i z e t o t a l t r a n s i t i o n p l u s
f i x e d costs i n d e t e r m i n i n g t h e ( p l a n t , l o c a t i o n ) p a i r i n g s .




                                                      27
              The G e o f f r i o n - G r a v e s    formulation is a specialization
of     the          Graves-Uhinston                 QA problem.    R e s t r i c t i o n s are
p l a c e d o n t h e l l p l a n t l t - t o - l l p l a n t l l i n t e r a c t i o n s t h a t occur.

                I n t h e G-G           QA f o r m u l a t i o n , t h e s c h e d u l i n g h o r i z o n
 (0,H) on e a c h p r o d u c t i o n l i n e i s d i v i d e d i n t o                          equal
i n d i v i s i b l a time-units.             Each time-unit i s a l*plantll i n t h e
Koopmans-Beckmann s e n s e . T h e i n t e r a c t i o n b e t w e e n t i m e - u n i t s
 (plants)            affects          only        those          time-units           immediately
p r e c e e d i n g a n d f o l l o w i n g t h e m o n t h e same p r o d u c t i o n l i n e .
T h e i n t e r a c t i o n Q matrix               ( c o m p o s e d of a l l 4 ) h a s t h e
                                                                                      iP
s t z u c t u r e shown i n F i g u r e 2 .

                                                    4- 3a t r ix
                                                       TIME-UNITS
                                                                   m2            m3
                                              ml          I                I




      TIME-UN I T S          m2    {                                     Ol
                                                                         -t----



                                                          I                I
                                                          I                I
                                             Figure           .2

  (Note: T h e G e o f f r i o n - G r a v e s       model is g e n e r a l i z e d t o            handle


                                                     28
flsimilarf1             production          lines.      I d e n t i c a l production and
transition                costs       between        lines       are      not    required.
P r o p o r t i o n a l i t y b e t w e e n l i n e s f o r t h e s e c o s t s may e x i s t .
T h e + 1 e n t r i e s i n t h e Q-matrix a r e r e p l a c e d b y t h e l i n e ' s
t r a n s i t i c n cost p r o p o r t i o n a l i t y c o n s t a n t .         )


             The q u a n t i t y A of e a c h p r o d u c t i o n o r d e r i s d i v i d e d
i n t o e q u a l i n d i v i s i b l e I1product-units.        E a c h p r o d u c t - u n i t is
a Koopmans-Beckmann f l l o c a t i o n .

              The  f i n i t e q u a n t i z a t i o n is d e t e r m i n e d by t h e choice
of t h e b a s i c time r e s o l u t i o n S.         T h e n u s b e r of      time-units
p e r l i n e 1 is:

                          m        = H / S ,
                               1
         where            H        = t h e schedule horizon.



            The n u m b e r of p r o d u c t - u n i t s per p r o d u c t i o n               order      k
f o r a p r o d u c t p is:

                          n        = A       / (R              *S),:                               (3)
                               k         k             P

         nhere            A        = t h e Drder's s p e c i f i e d d e m a n d
                               k
                          R        = the p r o d u c t I s s t a n d a r d p r o d u c t i o n rate.
                               P

             I i e q u i r i n g t h a t t h e t o t a l n u m b e r of t i m e - u n i t s       equal
t h e t o t a l number of p r o d u c t - u n i t s y i e l d s :

                           n       =   C
                                       1
                                              m
                                                  1
                                                           =        C
                                                                    k
                                                                        nk.


         The p r o b l e m is                         infeasible              if t o t a l product-units
exceed t o t a l t i m e - u n i t s .                A lgslackll p r o d u c t can be added             to
enforce (4).




                                                               29
                The q u a d r a t i c a s s i g n m e n t problem c a n now             be        stated
as    the        set       of p a i r i n g s ( t i m e - u n i t i, p r o d u c t - u n i t j) s u c h
that total             t r a n s i t i o n c o s t s p l u s t o t a l f i x e d c o s t s is
minimized,



     2
     p = l

subject to
                i =1
                          'ip   '   52
                                    j = l k = l
                                                            jk   i j   pk                              i j      i j


                                                                                                                (5)




                             2
                            j = l
                                        X
                                            ij
                                                 -      1        for



where:

      C         = t r a n s i t i o n cost between p r o d u c t - u n i t s         j a n d k;
          jk

      9         = 7 if time-units i and p i n t e r a c t ,
          i P
                = 0 otherwise;

      d         = fixed         cost        of   assigning             product-unit               j        to
          i j
                   time-unit         i;


      X         = 1 if product-unit                  j is a s s i g n e d t o t i m e - u n i t       i,
          i j
                = 0 otherwise.



                The C    matrix, shoun i n                  Figure 3, contains t h e
product-unit           t o p r o d u c t - u n i t t r a n s i t i o n c o s t , d e r i v e d fr0.m
t h e s t a n d a r d product-to-product costs.                      E a c h p r o d u c t - u n i t is
a s s o c i a t e d with a n o r d e r which is f o r a p a r t i c u l a r product.
 (The s t a n d a r d t r a n s i t i o n c o s t between p r o d u c t p a n d p r o d u c t


                                                      30
 p' i s d e n o t e d by T               .)
                                   FP'
                                                     C 1 Hatrix
                                                            MISSION-UNITS
                                              nl
                                                                               ...              ...
                                                                 *2
                                                                         1                I
                                                        I                                  I




MISSION-UNITS                  \

                                                        I
                                                                          I
                                                                                           I
                          .              TP P
                                           31           '
                                                        I T P3 2
                                                             P            !
                                                                                  0        I
                                                                                           I
                                                                                                . ' *




                          #




                                                   Figure         3



              The     d             form       the          D    m a t r i x of f i x e d c o s t s .       This
                          ij
  matrix c o n t a i p s           costs           (incurred          when      product-unit            j     is

 assigned        to       time-unit                i)       which a c c o u n t f o r t h e f o l l o w i n g
  effects:

   1.    The p r o d u c t i o n c o s t s ;

   2.    The t r a n s i t i o n c o s t ,           for         each        line,    from       the        last
         product          on        the p r e v i o u s s c h e d u l e t o t h e first i n t h i s
         scheduling period;

   3.    P r o h i b i t i v e p e n a l t i e s for p r o d u c i n g a n o r d e r b e f o r o t h e


                                                            31
        e a r l y s t a r t time o r a f t e r t h e l a t e f i n i s h time;

 4.     Infeasibility            of c e r t a i n p r o d u c t s w h i c h c a n n o t be
        p r o d u c & d by c e r t a i n l i n e s (a h i g h c o s t o f i n f e a s i b i l i t y
        is used).

                (Note: T h e r e i   s an          explanation           of      how      this      QA
                                             f o r s h i p s a n d m i s s i o n s of t h e
f o r m u l a t i c n is i n t e r p r e t e d
Coast G u a r d p r o b l e m i n t h e n e x t C h a p t e r . Also,     Appendix A
c o n t a i n s an example a n d p i c t u r e . )



C.     L I N E A R PROGaAM F O R M U L A T I O N I N G-G         MODEL



       A c h a n g e of v a r i a b l e s     occurs       between         the      (time-unit,
product-unit)            p a i r s o f t h e Q A a n d t h e time d u r a t i o n s o f t h e
LP.         Each c o n t i n u o u s         production           ran       of   a        product
 ( p r o d u c t - u n i t s for t h e same order) i s n o t e d a n d c a l l e d a
                           The p r i m a r y LP v a r i a b l e s a r e t h e d u r a t i o n s o f
e a c h campaign.


       Without changing t h i s               sequence       of c a m p a i g n s on e a c h
l i n e , t h e LP d e t e r m i n e s t h e d u r a t i o n ( i n c o n t i n u o u s time) of
e a c h c a m p a i g n so t h a t t h e m i n i a a l t o t a l c o s t i s o b t a i n e d .
The     three       cost      components           i n the LP*s objective function
are:

  1.    Production costs,

  2.    P e n a l t i e s for violating orders'                  early        start       or     late
        f i n i s h time, a n d

 3.     P e n a l t i e s for m o d i f y i n g t h e s p e c i f i e d h o r i z o n ,   H,   on    a
        line.

T h e t o t a l q u a n t i t i e s for a n o r d e r a r e        constrained            to     fall
b e t w e e n t h e s p e c i f i e d lower a n d u p p e r l i m i t s .



                                                 32
a s s u m p t i o n i s made t h a t t h e a s s i g n m e n t p r o b a b i l i t y of e a c h
p o s s i b l e p e r m u t a t i o n of       assignments             is      equal.           The
r e p l a c e m e n t of e n u m e r a t i o n for u n s p e c i f i e d s s s i g n m e n t c o s t s
by t h e c o n d i t i o n a l e x p e c t e d v a l u e           is      a       significant
c o n t r i b u t i o n cf t h e G-W method.

       T h e p a i r , d e s i g n a t e d by     (i', j'),       w i t h t h e minimum
e x p e c t e d t o t a l c o s t is s e l e c t e d by t h e a s s i g n m e n t c r i t e r i a .
N e x t , t h e f i x e d c o s t s , d , of t h o s e t i m e - u n i t s                  that
                                                 i j
immediately           preceed         or fcllow i' a r e u p d a t e d . For example,

let time-unit           i p r e c e e d i t . Any p r o d u c t - u n i t    j     assignsd to
time-unit           i w i l l be         f o l l o w e d by p r o d u c t - u n i t      j c . The
t r a n s i t i c n c c s t of j t o     j * i s now known a n d is t h u s a f i x e d
cost. The f i x e d cost                    elements for time-unit i a r e t h u s
u p d a t e d by t h e a d d i t i o n   of t h i s known t r a n s i t i o n c o s t . A
s i m i l a r u p d a t e f o r the      t i m e - u n i t f o l i o w i n g i l i s a l s o done.
                                                         th
The methcd t h e n p r o c e e d s t o t h e k                stage.


        S i n c e n o c o n d i t i o n 3 1 c o m p u t a t i o n s o c c u r a t each stage,
t h e c o m p u t a t i o n a l e f f o r t a n d memory r e q u i r e m e n t s o f     this
p o r t i o n o f t h e G-id              method may b e d e t e r m i n e d a s a s i m p l e
f u n c t i o n cf n.

        A l c c a l search ( c a l l e d S w i t c h ) is n e x t Ferformed on t h s
 (time-unit, Froduct-unit) pairings.                         All p a i r u i s e s x c h a n g 2 s
of        product-units         are       c y c l i c a l l y tested for p o s s i b l e
i m p r o v e m e n t i n total c o s t . F o r each new p a i r i n g , t h e c h a n g e
i n f i x e d costs and t r a n s i t i o n costs is d e t s r m i n i s t i c .

    At t h e termination                   or'    this        Switch       algorithm,         a QA
solution is obtained.                       N e x t , t h e s e c ; u e n c e of c a m p a i g n s i s
e x t r a c t e d a n d made t h e i n c u m b a n t s e q u e n c e f o r t h e L P stage
of t h e G-G model. A L i n e a r P r o g r a m m i n g s o l u t i o n i s o S t a i n e d
for t h i s s e q u e n c e .  T h e n , two d i f f e r e n t l o c a l s e a r c h e s a r e


                                                  34
D.        IHPLEHENTATION.



          Geoffrion and G r a v e s h a v e             implemented          their      model      to
scheOule        s i x c h e m i c a l reactors f o r a o n e month p l a n n i n g
h o r i z o n for Dart I n d u s t r i e s , Inc. V a r i o u s k i n d s of plastics
are t h e products.                  The Q A p r o b l e m i s s o l v e d by t h e G r a v e s
a n d W h i n s t o n m e t h o d [ 6 3.


          The    Graves       a n d Whinston method s o l v e s a g e n e r a l i z a t i o n
of t h e Koopmans-Beckmann                     QA problem.              To s i m p l i f y t h i s
discussicn            ( a n d w i t h o u t l o s s of g e n e r a l i t y ) , t h e s p e c i a l
s t r u c t u r e of t h e Q - m a t r i x o f t h e G-G m o d e l i s a s s u m e d .       The
G-W      method is a n n - s t a g e d e c i s i o n p r o c e s s where n is t h e
                                                                                     th
t o t a l n u m b e r of t i m e - u n i t s f o r a s s i g n m e n t . A t t h e k      stage,

k-1  p a i r i n g s of    (time-unit           i, p r o d u c t - u n i t j) h a v E b e e n
made. I n t h e k a s i c m e t h o d , n o b a c k t r a c k i n g o c c u r s so t h e s e
                                                             th
k-1 a s s i g n m e n t s are p e r m a n e n t .  The k           stage examines a l l

p o s s i b l e p a i r i n g s of t h e n- ( k - 1 ) u n a s s i g n e d u n i t s . For e a c h
pairing,          t h e e x p e c t e d f i n a l t o t a l c o s t , c o m p r i s e d of t h r e e
c o m p o n e n t s , is c a l c u l a t e d .     T h e s e three c o m p c n e n t s a r e :

     1.    T h e c o s t s i n c u r r e d b y t h e p r e v i o u s k-1 p a i r i z g s ,

  2.       The     immediate          cost        of        assigning    the       pair          being
                                             th
           considered a t t h i s k                stage, a n d

  3.       The     expected         value         of        future   costs      that     will b e
           i n c u r r e d by t h e r e m a i n i n g n-k u n s p e c i f i e d a s s i g n m e n t s ,
           g i v e n t h a t t h e c u r r e n t p a i r i s made p e r m a n e n t .


          The      e x p e c t e d v a l u e of f u t u r e          costs is c a l c u l a t e d
e x p l i c i t l y DY a formula r a t h e r t h a n                 by e n u m e r a t i o n . The



                                                       33
p e r f o r m e d on t h t c a n d i d a t e s e q u e n c e .           The f i r s t s e a r c h
 ( c a l l e d S l i d $ ) moves e a c h campaign i n t o a l l o t h e r p o s s i b l e
positions.               The s e c o n d s e a r c h      ( c a l l e d S w i t c h ) makes a
p a i r v i s e i n t e r c h a n g e of a l l c a m p a i g n s .          F o r e a c h new,
f a v o r a b l e s e q u e n c e o f c a m p a i g n s s e l e c t e d by a s c r e e n i n g
c r i t e r i a , a n LP s o l u t i o n i s o b t a i n e d .        If  t h e t o t a l cost
 ( s e q u e n c e p l u s LP c o s t s ) i s i m p r o v e d , t h e c a n d i d a t e s e q u e n c e
r e p l a c e s t h e i n c u m b e n t a n d t h e two l o c a l s e a r c h e s a r e
restarted.               When both s e a r c h e s t e r m i n a t e ,          t h e incumbent
s e q u e n c e and d u r a t i o n s c o n s t i t u t e t h e " l o c a l l y "       optimal,
f i n a l solution.

       T h e LP p r o b l e m s a r e s o l v e d b y G-G         w i t h a n LP    subroutine
described          in     Graves       [ 51. T h i s a l g o r i t h m i n c l u d e s a d v a n c e d
f e a t u r e s for d e a l i n g with degeneracy, employs a complete
mutual primal-dual                s i m p l e x mxhanism, and u s e s a basis
r6presentation with a                      small, s e g r e g a t e d i n v e r s e .      The
p r c g r a m uses c o m p l e t e         in-core  s t o r a g e of t h e c o e f f i c i e n t
m a t r i x , a n d h a n d l e s l o g i c a l v a r i a b l e s i m p l i c i t l y . R a n g i n g of
e q u a t i o n s and bounded v a r i a b l e s a r e n o t a v a i l a b l e .




                                                  35
A.    OVERALL CONSIDERATIONS




      The      Coast          Guard s c h e d u l i n g model i s based o n t h e
Geoffrion              and        Graves         approach.           T h e Area C o m m a n d e r ' s
s c h e d u l i n g g u i d e l i n e s , e x p r e s s i n g m i s s i o n r e q u i r e m e n t s and
t h e morale-related g o a l s a n d c o n s t r a i n t s , a r e c o n v e r t e d i n t o
t h e o b j e c t i v e f u n c t i o n s a n d c o n s t r a i n t s or' t h e Q u a d r a t i c
A s s i g n m e n t a n d L i n e a r P r o g r a m m i n g models. The o b j e c t i v e
f u n c t i o n e x p r e s s e s p s e u d o - c o s t s t h a t are              incurred           by
d e v i a t i o n s from t h e r e q u i r e m e n t s a n d g o a l s . The o p t i m i z a t i o n
models, b y m i n i m i z i n g t h e i r o b j e c t i v e f u n c t i o n s ,            seek the
closest             co E p l i a n c e    of t h e r e s u l t i n g        schedule t o the
g u i d e l i n e s . The o b j e c t i v e f u n c t i o u s are summations,                       each
term e x p r e s s i n g w i t h p e n a l t y c o s t s t h e d e g r e e of c o m p l i a n c e
w i t h o n e of t h e r e q u i r s m e n t s or g o a l s .


          I n t h e C o a s t G u a r d problem, a s h i p c o r r e s p o n d s t o a G-G
l l l i n e . l l T h e time a v a i l a b l e o n a s h i p i s d i v i d e d i n t o
discrete t i m e - u n i t s ,         Each m i s s i o n r e q u i r e m e n t is a G-G
" p r o d u c t i o n order"          and     is    similarily           divided      into
mission- units.

       The g u i d e l i n e ' s   requirements and              g o a l s have n o t been
c o m p l e t e l y c a p t u r e d b y t h e QS a n d LP        models.       Some f e a t u r e s
are i n c l u d e d i n t h e Q A t h a t a r e n o t            i n t h e iP a n d v i c e
versa. T h e g u i d e l i n e s n o t modeled a t               a l l a r e l e f t f o r human
a c t i o n upon t h e r e s u l t a n t s c h e d u l e .




                                                  36
B.     Q U A D R A T I C ASSIGNPlENT YODEL



       The       m o d e l i n g f o r t h e Q u a d r a t i c A s s i g n m e n t a l g o r i t h m is
p e r f o r m e d by s e t t i n g t h e s t r u c t u r e s a n d c o m p o n e n t s of the
t h r e e matrices:

                 C   -   mission-unit t o mission-unit t r a n s i t i o n
                         costs,
                 Q   -   i n t e r a c t i o n between time-units, and
                 D   -   f i x e d costs.


Each of t h e s e matrices a r e                 s p e c i a l l y s t r u c t u r e d for t h i s
p a r t i c u l a r Coast Guard p r o b l e m . T h e C - m a t r i x w i l l coerce t h e
m i s s i o n s E g u e n c e s t o f o l l o w t h e a c t u a l p a t t e r n s shown i n
F i g u r e 1.            The Q-matrix is t h e same a s t h e G-G m a t r i x . T h e
D-matrix             w i l l express          in      fixed        costs           the   dssiiad
s t a r t / f i n i s h times, t h e two t y p e s of m i s s i o n r e q u i r e m e n t s           -
PATEOL a n d s i n g l e c o n t i n u o u s d u r a t i o n , a n d o t h e r i t e m s .



       1.       - Mat
              c-------r i x

              The mission-unit            t o mission-unit             t r a n s i t i o n costs
a r e d i r e c t l y d e t e r m i n e d from t h e r e q u i r e m e n t - t o - r e q u i r e m e n t
t r a n s i t i c n costs.      Frequently occurring mission t r a n s i t i o n s
are        strcngly         encouraged         b y low c o s t s a n d u n d e s i r e d
t r a n s i t i o n s a r e d i s c o u r a g e d by h i g h c o s t s .        (A   transition
betueen mission-units                       of t h e same r e q u i r e m e n t i n c u r s n o
c o s t . ) F i g u r e 7 i n C h a p t e r I11 s h o w s t h e f r e q u e n c y o f t h e
v a r i o u s t r a n s i t i o n s i n a c t u a l s c h e d u l e s . The m i s s i o n I n p o r t
 ( s h i p i n h o m e p o r t o n s e a r c h a n d r e s c u e s t a n d b y ) plays t h e
dominant role.                 Most of           t h e time,        I n p o r t is t h e m i s s i o n
o c c u r r i n g b e t w e e n a l l o t h e r m i s s i o n s . The e x c e p t i o n ,   Ocean
 ( o c e a n o g r a p h i c d a t a c o l l e c t i o n ) , i s p e r f o r m e d e n r o u t e or
r e t u r n i n g frcm A l a s k a P a t r o l . T h e number of o c c u r r e n c E s o f


                                                  37
each e x t e n d e d s e q u e n c s i s s h o w n b e l o w .

                  I n p o r t/Ocea n/Al p a t                  14 ( a p p r o x )
              ,   Alpat/Ocean/Inport                            4


                 w i t h o n l y a "from I l i s s i o n A t o H i s s i o n Btt t r a n s i t i o n
cost,           a       difficulty       arises          in    modeling          t h e s e two
a l t e r n a t i v e s . T h e g o a l of t h e model i s t o match t h e
s c h e d u l e r u s l o g i c a l r a n k i n g with t h e r a n k i n g d e t e r m i n e d by
t h e pseudo-costs structure.                    The e x a m p l e b e l o w s h o w s t h e
problem.

                                -----
                                From-To       Traqsizion Difficulty

   Sample T r a n s i t i o n Costs:
                                                                                    -
                                                                                    To:
                                                                --E---t
                                                                In or               -
                                                                                    Alpat            --
                                                                                                     Ocean
                                            IApOrt                   0                30                  lo
                          Fromy             Alp at                  70                 0                  30
                                            Ocean                   90                10                   0


                                             Log i c a 1        P seudo-Cost                T r a n s it i o n
                                              Rank                   Rank                         cost
---
Sam@e   Sequence:
                                             -----              -------                     ------
Inport/Ocean/Alpat/Inport                         1                         1                        30
Inport/Alpat/Ocean/Inport                         2                         3                       150
InFort/O cean/Inpor t                             3                         2                      100


                                             Figure        4



                The s e q u e n c e I n p o r t / O c e a n / I n p o r t        would n e v e r be
s c h e d u l e d a n d thus i s h i g h l y u n f a v o r a b l e .               The p s e u d o - c o s t
mechanism c a n n o t convoy t h i s a s t h e d i f f e r e n c e i n t h e
r a n k i n g s shows.          T h e ltfrom A t o 8" s t r u c t u r e c a n n o t h a n d l e
ltfrom A t o B t o C" s e q u e n c e s .            A p o s s i b l e way t o p r e v e n t t h e
u n d e s i r e d a n d u n r e a l i s t i c I n p o r t / O c e a n / I n p o r t s e q u e n c e from



                                                      38
h a v i n g a n i n a p p r o p r i a t e l y low c o s t      is      to        eliminate
OCean/AlFat            o r Alpat/Ocean.         E a s e d on t h e f r e q u e n c y T a b l e ,
O c c a n / A l p a t is r e t a i n e d .

               T h e same d i f f i c u l t y o c c u r s i f T r a v e l time i s m o d e l e d
a s a m i s s i o n t o p r e c e e d or follow t h o s e m i s s i o n s r e q u i r i n g
it. T h e i r r a t i o n a l s e q u e n c e I n p o r t / T r a v e l / I n p o r t w o u l d h a v e
a lower c o s t t h a n a l o g i c a l l y p r e f e r e d s e q u e n c e .                  (e.g.
InFort/Travel/Reftra/TraveL/Inport)                          .    Thus, T r a v e l time as a
m i s s i o n is n o t e x p l i c i t l y m o d e l e d i n e i t h e r t h e Q A or LP
models.          T h i s e x c l u s i o n is n o t c o n s i d e r e d s e r i o u s because of
t h e times ( 2 t o 7 d a y s ) a r e c o n s t a n t a n d d e p e n d e n t u p o n t h e
s p e c i f i c s h i p . T h u s , a l l o w a n c e r'or t h i s T r a v e l t i m e c a n b e
made i n t h e d u r a t i o n of t h e I n p o r t m i s s i o n s .

              T h e t r a n s i t i o n s t h a t a r e a s s i g n e d low ( e n c o u r a g i n g )
c o s t s a r e l i s t e d . A l l o t h e r t r a n s i t i o n s a r e d i s c o u r a g e d by
h i g h costs.


                  InpOrt/All missions
                  All m i s s i o n s / I n p o r t
                  Oc E a n / A l p a t



       2.    Q    =   Matrix        '



              T h e Coast G u a r d Q-Matrix              i s t h e same a s t h a t o f G-G
with        the           specialization               that     the    transition        cost
p r o p o r t i o n a l i t y c o n s t a n t s z q u a l one f o r a l l s h i p s .       This          I




m a t r i x s e r v e s t o a s s o c i a t e each t i m e - u n i t w i t h only t h e
i m m e d i a t e l y a d j a c e n t t i m e - u n i t s o n t h e same s h i p ( f i r s t a n d
l a s t t i m e - u n i t s on a s h i p a r e s p e c i a l c a s e s ) . This s t r u c t u r e
is t h e same a s t h a t of t h e M u l t i p l e T r a v e l i n g S a l e s m a n
problem i n t h e open l i t e r a t u r e .




                                                      39
             The       D-Hatrix         c o n t a i n s t h e f i x e d c o s t s of p a i r i n g
t i m e - u n i t i t o m i s s i o n - u n i t j. F i v e d i f f e r e n t f i x e d cost
c o m p o n e n t s a r e u s e d t o m o d e l f i v e g u i d e l i n e s (The r e a d e r c a n
r e f e r to A p p e n d i x A where t h e i n i t i a l D-Matrix               for t h e
s a m p l e p r o b l e m c o n t a i n s l a b e l e d e x a m p l e s of each of t h e s e
c o m p o n e n t s ) . T h e f i r s t t h r e e a r e t h e same a s i n t h e G-G
model.

             The f i r s t component models t h e r e q u i r e m e n t s t h a t c a n
b e f u l f i l l e d by a s p e c i f i c s h i ? ,    The g e o g r a p h i c a l l o c a t i o n s
of h o m e p o r t a n d m i s s i o n a r e a p r i m a r i l y d e t e r m i n e p o s s i b l e
a s s i g n m e n t s for t h e GENERAL r e q u i r e m e n t s . T h i s component
a l s o models t h e S H I P - S P E C I F I C r e q u i r e m e n t s b y a l l o w i n g o n l y
t h e i n t e n d e d s h i p t o be a s s i g n e d its SHIP-SPECIFIC m i s s i o n s .
If a s h i p c a n n o t s a t i s f y a r e q u i r e m e n t , t h e i n t e r s e c t i c n s of
a l l associated time-units                     and     mission-units           are assigned
v e r y h i g h costs,

             The    s,ecog4 c o m p o n e n t p r o v i d e s   t h e mission-to-mission
t r a n s i t i c n cost between t h e m i s s i o n a s s i g n e d t o a s h i p
immediately prior to t h e s t a r t of t h e c u r r e n t schedule
p e r i o d a n d t h e f i r s t m i s s i o n i t w i l l be a s s i g n e d i n t h i s
period.

              T h e t h i r 4 c o m p o n e n t coerces t h e e n f o r c e m e n t of e a r l y
s t a r t and l a t e f i n i s h d a t e s t h a t a r e associated w i t h each
r e q u i r e m e n t , Each t i m e - u n i t t h a t f o l l o w s t h e l a t e f i n i s h
time o f a r e q u i r e m e n t i s assessed a p e n a l t y c o s t f o r
l a t e n e s s . This c c s t i n c r e a s e s l i n e a r l y a s t h e l a t e n e s s
increases.             S i m i l a r l y , a p e n a l t y c o s t for p r e c e e d i n g t h e
s p e c i f i e d e a r l y s t a r t time i s a s s e s s e d . T h e s e a s s e s s m e n t s
are          made       for         a l l mission-units          associated w i t h t h e


                                                40
requirement.           To m o d e l t h e p r e d e t e r m i n e d n a t u r e of t h e f i r s t
three       t o s i x m o n t h s of t h e s c h e d u l e , a n d t h e freedom of t h e
l a s t s i x m o n t h s , t h e r a t e of l i n e a r i n c r e a s e of t h e p e n a l t y
can   be   controlled          over       t h e time h o r i z o n f o r e a c h
requirement.  A r e q u i r e m e n t i n t h e b e g i n n i n g of t h e s c h e d u l e
can b e assiped a higher penalty rate t h a n a requirement a t
t h e e n d . T h u s , a d e v i a t i o n of t h e same a m o u n t c a n b e
p e n a l i z e d d i f f e r e n t l y , depending upon t h e p r o x i m i t y to t h e
b e g i n n i n g of t h e s c h e d u l e a n d t h e f l e x i b i l i t y a l l o w e d for
t h e requirement.

              Ihe     ----r t h
                      fou           c o m p o n e n t o f t h e D-Matrix         doss n o t
c o r r e s p o n d t o o n e i n t h e G-G               model.      Some r e q u i r e m e n t s
s t i p u l a t e t h a t t h e d e s i r e d d u r a t i o n of t h e r e q u i r e m e n t
c a n n o t b e s p l i t a n d m u s t b e c o m p l e t e d a t o n e time.             This
t y F e is u s u a l l y a SHIP-SPECIFIC r e q u i r e m e n t .               As t h e QA
model           allows      s p l i t t i n g of r e q u i r e m e n t s i n t o m u l t i p l e
m i s s i o n s , t h i s f o u r t h component             attempts         to      prevent
splitting               by      imposing          a s t r u c t u r e on t h e p e r t i n e n t
 (time-unit, mission-unit) p a i r s .                      F i g u r e S shows w i t h a n d
uithout            states           for       a small e x a m p l e .              An- a d d i t i o n a l
i n t e r p r e t a t i c n is p l a c e d 3 n t h e m i s s i o n - u n i t s ; t h e y a r e t o
be a s s i g n e d i n a l e f t t o r i g h t o r d e r { t h e left-most first,
a n d t h e r i g h t - m o s t l a s t ) . As o n l y o n e m i s s i o n - u n i t        can be
a s s i g n e d t o each t i m e - u n i t , t h e f i r s t t i m e - u n i t a f t e r t h e
e a r l y s t a r t time o f t h e r e q u i r e m e n t i s e n c o u r a g e d t o h a v e
t h e leftmost (first) mission-unit                                 a s s i g n e d t o it. The
t i m e - u n i t p r o z e e d i n g t h e l a t e f i n i s h time i s a l s o e n c o u r a g e d
t o h a v e t h e r i g h t m o s t ( l a s t ) m i s s i o n - u n i t a s s i g n e d t o it.
Similar e n c o u r a g e m e n t s are u s e d as a p p r o p r i a t e f o r t h e
remaining               mission-units.            The e k x o u r a g e m e n t is done b y
a p p r o p r i a t e d i s p l a c e m e n t o f t h e time p e n a l t i e s of t h e t h i r d
 component.

              T h e e f f e c t o f t h i s i m p o s e d s t r u c t u r e is t o          decrease
t h e number of p a i r i n g s t h a t i n c u r n o time p e n a l t y c o s t .                  The



                                                   41
  ( t i m e - u n i t , n i s s i o n - u n i t ) p a i r i n g s are t h u s encouraged t o
 f o l l o w o n e of t h e " p a t h s "                   s h o w n by t h e arrows. T h i s
 s t r u c t u r e does n o t guarantee t h a t t h e requirement w i l l not
 be s p l i t , b u t d o e s p e n a l i z e a s p l i t .

                                  -n q l e
                                  Si          tlission Structure




         1
         2
Time-    3
Units 4
      5
         6    I   1       1        1                                3        2        1

             (Note: The n u m b e r s a r e u n i t s of T i m e P e n a l t y . )

                                           Figure        5


              The      fifth       component of t h e D-Matrix                        explicitly
 models t h e P A T R O L r e q u i r e m e n t s ( g i v e n number o f s h i p s on
 s c e n e a t same t i m e ) .               A similar left t o r i g h t                         time
 o r i e n t a t i o n i s p l a c e d on t h e mission-units.                 By t h e n a t u r e
 o f t h i s t y p e of r e q u i r e m e n t ,          a s t r o n g r e l a t i o n s h i p is
 t h e r e b y created between p a r t i c u l a r t i m e - u n i t s .         T h a t is, t h e
 first mission-unit                s h o u l d o n l y be a s s i g n e d t o t h e o n e
 time-unit             of each s h i p t h a t c o r r e s p o n d s t o t h e same a c t u a l
 c a l e n d a r time.        Figure 6 depicts t h e following situation:
 for a s c h e d u l e h o r i z o n of t h r e e w e e k s ( w i t h o n e week
 r e s o l u t i o n ) , two s h i p s m u s t s a t i s f y a P A T X O L r e q u i r e m e n t of
 1 ship            always on scene.             T h u s , t h e s e c o n d m i s s i o n - u n i t of
 t h e r e q u i r e m e n t is s t r o n g l y associated with only t h e one
 time-unit             of each s h i p t h a t c o r r e s 2 o n d s t o t h e s e c o n d w e e k
 o f t h e p e r i o d . All o t h e r t i m e - u n i t s      a r e d i s c o u r a g e d from
 possible a s s i g n m e n t by h i g h costs ( d e n o t e d b y a ) .


                                                  42
                                          Patrol Structure
                                          ---c             ------

                                                    ---------
                          Ship A            1                    €4    il
                                            2         l
                                                      l                M
      Time-                                 3         Iy         M
      Units               ShiF B            1   I                M     Iy


                                            21 :
                                            3                    M
                                                                       M



                                            Figure           6



       4.     Guidelines          Not    Incorporated                 Q4

              Those           parts         of t h e q u i d e l i n a s s p a c i f y i n g t h e
m o r a l e - r e l a t e d a n d I 1 l i m i t i n g 1 *c o n s t r a i n t s a r e n o t m o d e l s d i n
t h e ;2A model.              No m e c h a n i s m h a s b e e n f o u n d t o c o n d i t i o n t h e
                                  th
a s s i g n m e n t of t h e k           stage o n , f o r e x a i n p l e , t h e a m o u n t of

AHF   time p r e v i o u s l y a s s i g n e d each s h i p .          Tbase c o n s t r a i n t s
are i n c o r p o r a t e d i n t h e s u b s e q u e n t L i n e a r Z r o g r a a r n i n g model.



C.     L I N E A R P R O G R A P I H I N G HODEL




              T h e L i n e a r P r o g r a m m i n g rnodel          for    tha      Caast        Guard
problem c o n t a i n s t h e           sine    oasic s t r a c t u r e a s t h a         G-G    model,
A d d i t i o n a l Fenalties            are        incorporated            in     the      objective


                                                     43
function            tc         model        the         "limiting".        and    morale-rdated
guidelines.              The     Froblem            statement         of    t h e QA m o d e l i s
broadened.               The     s i n g l e t a r g e t a u r a t i o n of a r e q u i r e m e n t i s
e x t e n d e d t o h a v e l c w e r / u p p e r l i m i t s a n d t h e time h o r i z o n can
v a r y on e a c h s h i p .

              The     p r i m a r y v a r i a b l e s of t h e LP r e p z e s e n t t h e
mission durations.                 ( N o t e : F o r t h e r a m a i n d e r of t h i s t h e s i s ,
a m i s s i o n r e f e r s t o a G-G c a m p a i g n . )          A l l other        variables
are         non-negative             peoalty         variables t h a t measure t h e
d e v i a t i o n s from d e s i r i d c o n d i t i o n s . The s i x t y p e s of p e n a l t y
v a r i a b l e s i n t h e o b j e c t i v e f u n c t i o n a r e l i s t e d below.

  1.    F o r each s h i p , t h e d i f f e r e n c e b e t w e e n t h e         sum
                                                                                  all        of
        m i s s i o n s on t h e s h i p a n d t h e time h o r i z o n m i n u s t h e
        s t a r t i n g time of t h e s h i p .

 2.     For      each          ship,      t h e d i f f e r e n c e b e t w e e n t h e sum of a l l
        Away Homeport m i s s i o n s a n d t h e s p e c i f i e d t o t a l                 desired
        amount for t h a t s h i p .

  3.    For each s h i p a n d e a c h m i s s i o n - t y p e               w i t h a maximum
        limit,         t h e p o s i t i v e d i f f e r e n c e b e t w e e n t h e sum cf a l l
        m i s s i o n s on t h e s h i p for t h i s m i s s i o n - t y p e           and t h e
        specified l i m i t .

  4.    For each mission,                 the negative difference                  between         the
        starting            time of t h e m i s s i o n a n d t h e a s s o c i a t e d
        requirement's             e a r l y s t a r t time,        and t h e     positive
        d i f f e r e n c e b e t w e e n t h e a c t u a l s t a r t time a n d t h e l a t e
        s t a r t time s p e c i f i e d .

  5.    For      each          mission, t h e p o s i t i v e d i f f e r e n c e between t h e
        s t a r t time plus t h e d u r a t i o n o f t h e m i s s i o n                    and   the
        a s s o c i a t e d r e q u i r e m e n t ' s l a t e f i n i s h time.

  6.    For      each          sequence            of        Away    Homeport    missions,         the
        positive           d i f f e r n c e between t h e s u m of d u r a t i o n s i n t h e
        c r u i s e and t h e          l i m i t        on     the    duration    of     a     single
        cruise.

                The c o n s t r a i n t s of t h e model, w i t h n o p r o v i s i o n f o r
v i o l a t i o n a t any p e n a l t y cost, p e r t a i n t o t h e upper/lower
l i m i t s o n t h e t o t a l r e q u i r e m e n t s a n d t h e d u r a t i o n s of e a c h
mission.

 1.     For e a c h r e q u i r e m e n t ,     sum of d u r a t i o n s o f t h e
                                              the
        missions         associated w i t h t h e requirement is within t h e
        lower a n d u p p e r r a n g e s .

 2.     For e a c h m i s s i o n , its             d u r a t i o n i s between s p e c i f i e d
        lower a n d u p p e r bounds.

             The     PATROL r e q u i r e m e n t s a r e n o t h a n d l P d e x p l i c i t l y .
This type    of r e q u i r e m e n t is - a n d l e d         i m p l i c i t l y by t h e
priority     assigned          to      t h e s t a r t / f i n i s h timas of t h e
requiremant.     T h e p r i o r i t y w e i g h t i n g of t h e l i n e a r p e n a l t y
r a t e is t h e same a s t h a t e x p l a i n e d i n t h e Q u a d r a t i c
A s s i g n m e n t model.       G u i d e l i n e s s u c h a s t h e minimum o f e i g h t
ueeks i n p o r t between s u c c e s s i v e A l p a t n i s s i o n s and t h e
minimum cf f o u r weeks p r i o r t o a n A l p a t f o r p r e p a r a t i o n a r e
e x p l i c i t l y h a n d l e d b y m a n i p u l a t i o n of t h e m i s s i o n d u r a t i o n
bounds.



D.    G U I D E L I N E S N O T MODELED




      Some         f a c e t s of t h e C o a s t G u a r d p r o b l e m h a v e n o t b e e n
a n a l y t i c a l l y mod4ed.     T r a v e l time i s n o t e x p l i c i t l y h a n d l e d
b e c a u s e of t h e s e q u e n c i n g d i f f i c u l t y a l r e a d y d i s c u s s a d
previously i n t h i s chapter.                The t a r g e t t o t a l Away     Homeport
time o f e a c h s h i p c o u l d be a d j u s t e d t o p a r t i a l l y a c c o u n t f o r
t h e e f f e c t o f t r a v e l time. T h e s i n g l e c r u i s e l i m i t c o u l d
also be adjusted.




                                                45
        Multi-ship         c o n f l i c t s are n o t modeled.              Overlag           of
f i r s t / l a s t w e e k s . of a e f t r a , o v e r l o a d i n g of d i s t r i c t s w i t h
ships simultaneously                      in   maintenance,             staggering           ths
Hawaii-based             s h i p s , a n d a l t e r n a t i n g w i n t e r A l p a t s are
examples. These m u s t be checked and p o s s i b l y c o r r e c t e d by
t h e scheduler.

      The d e s i r e t o h a v e a t l e a s t o n e s h i p o n        Inport      (search
a n d rescue s t a n d b y ) a t a l l times h a s n o t b e e n m o d e l e d .
Presently,            t h e s h i p workload i s s u c h                that       this      is
p r a c t i c a l l y guaranteed.            T h e e f f e c t s o f t h e new 2 0 0 mile
f i s h i n g l i u i t may p r e v e n t t h i s I n p o r t            coverage       from
occurring naturally.                I n p o r t c a n t h e n b e h a n d l e d a s a PAT3OL
requirement missicn.                D i s t r i c t t r a i n i n g a n d non- s c h e d u l e d
o p e r a t i o n s are n o t modeled s i n c e t h e y are n o t m a n u a l l y
scheduled a t present.

        S i n g l e . d a y t i m i n g of h o m e p o r t , r e t u r n , d e p a r t u r s , a n d
on-scene            d a t e s must b e l e f t - t o t h e manual s c h e d u l e r .
O v e r l o a d i n g of t h e f u e l p i e r i n Kodiak, Alaska, a n d S u n d a y
a r r i v a l s a t R e f t r a a r e e x a m p l e s of c o n s i d e r a t i o n s t h a t t h e
s c h e d u l e r s h a n d l e when p r o d u c i n g t h e s i n g l e d a y r e s o l u t i o n
of t h e p u b l i s h e d s c h e d u l e .


        T h e items n o t a n a l y t i c a l l y modeled c a n b e i n c l u d e d i n a n
                                                                            .
e d i t program t h a t would examine t h e r e s u l t a n t s c h e d u l e f o r
t h e s e r e l a t i o n s h i p s and i n f o r m t h e s c h e d u l e r s of               hand
c o r r e c t i o n s t o b e made. S u c h a u t i l i t y p r o g r a m c o u l d a l s o
p r o v e v e r y u s e f u l f o r c o m p u t i n g " p s e u d o - c o s t s " of a l t e r n a t e
s c h e d u l e s manually produced.




                                                 46
                                      VI      ------.---
                                              I MPLEHENTATION




A.    OVERALL C C N S I D E R A T I O N S



       T h e G-G        m o d e l a n d c o m p u t e r program d e s c r i b e d in [ 4 ]
p r o v i d s a n o u t s t a n d i n g e x a s p l e of s u c c e s s f u l i m p l e m e n t a t i o n
of     a coniplex o p t i m i z a t i o n m o d e l for p r o d u c t i o n u s e . T h e
model c o n c i s e l y c a p t u r e s t h e p e r t i n e n t features of t h e
p r c d u c t i o n s c h e d u l i n g problem i n a n o p t i m i z a t i o n c o n t e x t .
The c o m p u t e r s y s t e m e f f i c i e n t l y p r o v i d e s g o o d s o l u t i o n s to
t h e model f o r t h e intended c l i e n t .               Unfortunately, t h e sheer
size of t h e Coast Guard p r o b l e m makes t h e G-G                                     system
h o p e l e s s l y e x p e n s i v e t o u s e . A c c o r d i n g l y , a new system h a s
b e e n d e s i g n e d for the l a r g e s c a l e Coast G u a r d p r o b l e m w h i c h
p r e s e r v e s all t h e e x c i t i n g model f e a t u r e s of G - G I w h i l e
i n c o r p o r a t i n g a d d i t i o n a l model f e a t u r e s a n d y i e l d i n g v a s t l y
iIUFrOVed computational e f f i c i e n c y              .
        T h e p r i m a r y c o n s i d e r a t i o n s f o r t h e i m p l e m e n t a t i o n of t h e
a n a l y t i c a l model are: t o overcome t h e formidabla c o m p u t e r
memory a n d e x e c u t i o n time d i f f i c u l t i e s a r i s i n g from t h e
l a r g e s c a l e of t h e p r o b l e m , a n d t o t e s t and e v a l u a t e t h e
p r c p o s e d a n a l y t i c model.            T h e G e o f f r i o n a n d G r a v e s code
 (TCBTRAN I V ) i s used a s a s t a r t i n g p o i n t .                       It i s d e s i g n e d
f o r commercial use f o r s m a l l e r scale i n d u s t r i a l p r o d u c t i o n
scheduling problems a s discussed i n Chapter I V .

        An I9M 360/67 a t t h e W. R . C h u r c h C o m p u t e r C e n t e r of t h e
U.        S.       Naval     P o s t g r a d u a t e S c h o o l h a s b e e n u s e d for
d e v e l o p m e n t work i n t h i s s t u d y .


                                                  47
                 Impleg en ta tion F lo wcha rt




            QUADRAT I C ASS I OHHENT
                               I        LINEAR
    GRAVES-WHINSTON                     ASSIGNMENT
        METIiOD
                                        TRANSITION
                   --    -.LIIoo
                                        CnST UPDATE

          S W I T C H OF M I S S I O N - U N I T S
+




                           Figure        7


                                   48
       T h e o v e r a l l structure o f t h e i m p l e m e n t a t i o n of t h e C o a s t
G u a r d a n a l y t i c a l m o d e l i s d e p i c t e d i n F i g u r e 7.



B.    Q U A D R A T I C ASSIGNMENT IYPLEi'lENTATION



      The Quadratic Assignment c o d e used f o r a n i n i t i a l                       start
i s a n i n p l e m e n t a t i o n of t h e a l g o r i t h m d e v e l o p e d b y G r a v e s
and Whinston. The c o d e u s a s e x p l i c i t i n - c o r e                   s t o r a g e for
e a c h of t h e t h r e e matrices:                        D,      C,   a n d Q.     Thus, d a t a
s t o r a g e for a 1 , 0 0 0 u n i t p r o b l e m w o u l d b e a p p r o x i m a t e l y 3
m i l l i o n w c r d s which is c l e a r l y a n u n c o m f o r t a b l e s i z e . Thus,
t h e f i r s t o b s t a c l e t o o v e r c o m e t o e n a b l e t h e " s o l v i n g of a
1 , 0 0 0 u n i t q u a d r a t i c a s s i g n m e n t p r o b l e m i s coze s i z e .


    T h e G-W a l g o r i t h m is i m p l e m e n t e d f o r t h e most g e n e r a l
case o f t h e Q A p r o b l e m .         W i t h t h e g e n e r a l Q-matrix, many
complex r e l a t i o n s h i p s between flplantsfl c a n be s p e c i f i e d ( t h e
preceeding/f cllowing                 relationship              of       time-units       is a
specialization).             T h e work f a c t o r ( a p p r o x i m a t i o n f o r n u m b e r s
of c o m p u t e r o p e r a t i o n s or e x p e n s e , a s a f u n c t i o n of p r o b l e m
s i z e ) f o r t h e g e n e r a l a l g o r i t h m is e s t i m a t e d to b e o n t h e

o r d e r of n
                    4
                      .    By s p e c i a l i z a t i o n of t h e a l g o r i t h m to t h e

d e f i n e d s t r u c t u r e s of t h e C a n d Q m a t r i c e s of t h e C o a s t
Guard         a n a l y t i c model,       r e d u c t i o n of core a n d i m p r o v e d
performance u i l l r e s u l t .


      The    ( m u l t i p l e t r a v e l i n g salesiaan) s p e c i a l i z a t i o n used
f o r t h e Q m a t r i x i s t o c o l l a p s e t h e n+n m a t r i x i n t o two
vectors         cf s i z e n. One v e c t o r a c t s a s a n i n d e x s e t t o g i v e
f o r e a c h row t h e c o l u m n n u m b e r o f t h e s i n g l e n o n - z e r o
element          ( o r zero i f n o n e i s p r e s e n t ) . T h e s e c o n d vector
p e r f o r m s t h e i n v e r s e m a p p i n g from each c o l u m n t o t h e row


                                                49
with the non-zero                   element, i f any.       Figure 8 depicts these
two v e c t o r s f o r             t h e Q-matrix  i l l u s t r a t e d i n Figure 2
 ( C h a p t e r IV)       .

Row i:                                        1         2           3        ...    m -2 m -1
                                                                                     1       1
                                                                                                          m
                                                                                                              1
C o l u m n # cf Non-Zero:                   2          3           4        ...    m -1
                                                                                     1
                                                                                            m
                                                                                                 1
                                                                                                      0


                  ----- --- C o l u m n s
                  Vector f o r ----                  gf Q - M a t r i x      fog   F i r s t Shig

Column i:                                     1         2           3        ...    m - 2 m -1
                                                                                     1       1
                                                                                                          m
                                                                                                              1
Row           of Non-Zero:                   0          1           2        ...    m - 3 m - 2 m -1
                                                                                     1       1            1

                                             Figure             8


      For i n s t a n c e ,        f o r row 3 , t h e n o n - z e r o
                                                                     dement appears i n
c o l u m n 4 ; f o r c o l u m n 3 , t h e n o n - z e r o e l e m e n t a p p e a r s i n row
2.        Row m       h a s no non-zero               e l e m e n t a s it i s t h c l a s t
                       1
t i m e - u n i t o n t h e s h i p ( n o t i m e - u n i t follows i t ) .                 Column                1

h a s no non-zero              element ( t h e first t i m e - u n i t on t h e s h i p has
no predecesscr).


                                                                                                 2
      This         c h a n g e r e d u c e s s t o r a g e r e q u i r e m e n t s from n            t o 2n,

and t h e uork factor from n
                                                 4
                                                     to n
                                                            3
                                                                .
      The         s p e c i a l b l o c k s t r u c t u r e of t h e C m a t r i x e n a b l e s t h e
e l i m i n a t i o n of t h i s m a t r i x r e s u l t i n g i n a s t o r a g e r e d u c t i o n

of    n
          2
              .   ay       using     several          levels            of   indexing       from              the

mission-tc-mission    transition      costs,          the     appropiate
mission-unit   t o mission-unit t r a n s i t i o n c o s t is o b t a i n a b l e


                                                      50
from d a t a a r r a y s a l r e a d y u s e d e l s e w h e r e i n t h e model.

      The          f i n a l s t o r a g e r e d u c t i o n i s o b t a i n e d from t h e
observation t h a t a t most two u p d a t e s a r e made t o e a c h
element of the D matrix.                          A f t s r a (time-unit,mission-unit)
a s s i g n m e n t is made, t h e f i x e d c o s t D m a t r i x i s u p d a t e d . The
i n c u r r e d t r a n s i t i o n c o s t i n t o o r o u t of t h e a s s i g n e d u n i t is

added t o each d               i n t h e rows p r e c e e d i n g a n d        following       the
                        ij

row     just      assigned,            T h u s , t h e maximum v a l u e a d               element
                                                                                     i j

can a t t a i n i s t h e maximum i n i t i a l f i x e d c o s t p l u s t u i c f            the

maximum t r a n s i t i o n c o s t .      By s c a l i n g t h e i n i t i a l f i x e d c o s t s

and t h e t r a n s i t i o n costs s u c h t n a t t h e l a r g e s t p o s s i b l e        d
                                                                                                   i j
                          15
is    less      than 2           (32768 is a b s o l u t e m a g n i t u d e ) .      Integer*2

(16 b i t ,    half-word)         s t o r a g e is u s e d   rather        than       integer*4

(32     bit,      full-word)       .  I n t e g e r arithmetic o p e r a t i o n s are
a b o u t 10% s l o w e r t h a n f l o a t i n g p o i n t a r i t h m e t i c on t h e
m a c h i n e u s e d i n t h i s s t u d y , b u t the d a t a s t o r a g e r e d u c t i o n
h a s t o b e made.


      By t h e s e c h a n g e s , t h e t o t a l d a t a s t o r a g e r e q u i r e m e n t f o r
                                                                               2
the     three      matrices          is      reduced         from         3n         words         to

                         2
approximately           n /2       words.       T h u s , a p r o b l e m w i t h 1 , 0 0 0 time


u n i t s would r e q u i r e 500K w o r d s r a t h e r t h a n 3,000K w o r d s              for

data storage.


      From            trial       runs,      three        additional                algorithmic
s p e c i a l i z a t i o n s h a v e b e e n n o t i c e d f o r the              Coast Guard



                                                51
problem.         Mission-units for Inport comprise                               one     fourth    to
one t h i r d of t h e t o t a l number of m i s s i o n - u n i t s .                          All
m i s s i o n - u n i t s of I n p o r t h a v e i d e n t i c a l i n i t i a l f i x e d c o s t s .
All d y n a m i c t r a n s i t i o n c o s t u p d a t e s m a i n t a i n t h i s i d e n t i t y .
Thus, d u r i n g t h e e x p e c t e d v a l u e c a l c u l a t i o n m a t c h i n g           each
unassigned time-unit with a l l unassigned mission-units,                                         only
o n e of t h e u n a s s i g n e d I n p o r t m i s s i o n - u n i t s    need b e t r i e d      as
a c a n d i d a t e r a t h e r t h a n a l l n/3.

       The s e c o n d s p e c i a l i z a t i o n a r i s e s      from        the    observation
that        alincst a l l of t h e I n p o r t m i s s i o n - u n i t s a r e always t h e
last mission-units                   assigned t o time-units                       and t h a t t h e
e x p e c t e d v a l u e of t h e f u t u r e c o s t of t h e s e new a s s i g n m e n t s
is zero.           O n c e t h e p o i n t i s r e a c h e d w h e r e only I n p o r t
m i s s i o n - u n i t s r e m a i n u n a s s i g n e d , t h e i d e n t i c a l n a t u r e of a l l
                                                                                                           \
or' t h e s e u n i t s m e a n s t h a t t h e e x p e c t e d f u t u r e c o s t o f          their
a s s i g n m e n t becomes p e r m u t a t i o n i n d e p e n d e n t . These u n i t s c a n
be a s s i g n e d t o any u n a s s i g n e d t i m e 7 u n i t s and i n a n y order.

       The      t h i r d s p e c i a l i z a t i o n d e p e n d s on t h e l i k e l i h o o d t h a t
i n f e a s i b l e (strongly undesired                  or      impossible)           assignments
occur.           Prior         to       calculating              the      expected        value     of
a s s i g n m e n t of   ( t i m e - u n i t i, m i s s i o n - u n i t   j ) , the fixed         cost

for t h i s a s s i g n m e n t , d         , can    be tested.            If t h i s f i x e d c o s t
                                       ij
h a s exceeded a predetermined i n f e a s i b i l i t y value,                          the trial
assignment can be greemptively skipped.                               A d i f f i c u l t y arises
If a n i n f e a s i b l e a s s i g n m e n t s h o u l d r e a l l y t a k e p l a c e via t h e
lowest           mean c r i t e r i o n .      With t h i s e f f i c i e n c y - m o t i v a t e d
modification, abnormal termination with                                      an      incomplete
a s s i g n m e n t map w o u l d t h e n o c c u r s i n c e t h e a l g o r i t h m would
bypass t h i s i n f e a s i b l e assignment.


       These       three       specializations                have        not     been     made for
r e a s o n s a p p a r e n t l a t e r i n t h i s C h a p t e r (a new a l g o r i t h m f o r
t h e Q A model is d e v e l o p e d ) .               I t is s p e c u l a t e d t h a t t h e i r



                                                    52
a f f e c t w i l l r e d u c e t h e work f a c t o r o f t h e s p e c i a l i z e d G-W
                      3         3
m e t h o d from n t o n /4 i f I n p o r t m i s s i o n s c o m p r i s e o n e t h i r d
of t h e s c h e d u l e ,

        F o l l o w i n g t h e c o m p l e t i o n of a (time-unit,mission-unit)
map b y minimum e x p e c t e d v a l u e s ,            an additional algorithm
 ( c a l l e d 11SwitchI~) is t r i e d t o o b t a i n q u i c k l y a n d e a s i l y
f u r t h e r improvement i n t h a o b j e c t i v e f u n c t i o n .        A l l pairwise
i n t e r c h a n g e s of t h e m i s s i o n u n i t s a r e c y c l i c a l l y t e s t e d b y
e x h a u s t i v e enumeration.             For t h i s C o a s t G u a r d p r o b l e m ,
s t r i k i n g improvements are achieved, T h i s effect and o t h s r
b a s i c p e r f o r e a n c e d a t a of      t h e G-W    method i s l i s t e d i n
F i g u r e 9.         (The 5 d a y time r e s o l u t i o n h a s n o t bzen r u n
b e c a u s e t h e e x e c u t i o n time u o u l d b e e x c e s s i v e . )



C.     A HEM Q U A D R A T I C ASSIGNYEl4T IYETHOI)



       The         Graves-Whinston              method, w i t h t h e m o d i f i c a t i o n s a n d
s p e c i a l i z a t i c n s d e s c r i b e d a b o v e , i s e s t i m a t e d t o use f o u r t o
e i g h t h c u r s of c o m p u t e r time ( I B M 360/67) t o s o l v e a 1 , 0 0 0
u n i t Q u a d r a t i c Assignment problem.                   S i n c e t h i s is e x c e s s i v e
 ( e s p e c i a l l y c o n s i d e r i n g t h a t t h e p u r p o s e of t h e o v e r a l l
a n a l y t i c a l model i s u l t i m a t e l y t o a i d manual s c h e d u l e s i n a n
i n t e r a c t i v e f a s h i o n ) , a s e c o n d method of i m p l e m e n t a t i o n h a s
been e x p l o r e d .         Other reasons for introducing                            a second
method are:

  1.    T h e t e l i m i t i n g f ea n d m o r a l e - r e l a t e d   constraints     are     not
        modeled i n t h e Q A ;

 2.     The      QA       sequences will be s u b j e c t           to        increasing
        m o d i f i c a t i o n a s t h e LP m o d e l i n c o r p o r a t e s t h e s e new
        t y p E s of c o n s t r a i n t s ;

 3.     Significant             cost        reductions             are     mads   by t h e S w i t c h



                                                     53
        transformation; and

 4.     The  f i x e d c o s t m a t r i x f o r t h e C o a s t G u a r d p r o b l e m is
        highly structured.


      F o r t h e s e r e a s o n s , t h e new method i s d e s i g n e d t o q u i c k l y
o b t a i n a f e a s i b l e a s s i g n m e n t of ( t i m e - u n i t   i, m i s s i o n - u n i t
j) p a i r s u s i n g o n l y t h e f i x e d c o s t D-matrix.                      A Linear
A s s i g n m e n t F r o b l e m is s o l v e d t o o b t a i n a s e t of p a i r i n g s
w i t h t h e minimal t o t a l f i x e d cost.                     Then, t h e f i x e d cost
m a t r i x i s u F 6 a t e d w i t h t r a n s i t i o n c o s t s i n t h e same manner
a s t h e G-W method.                  T h e same S w i t c h a l g o r i t h m i s t h e n u s e d
t o minimize t o t a l f i x e d and t r a n s i t i o n costs.                    The f i n a l
s o l u t i o n , o b t a i n e d a t t h e e n d of t h i s a l g o r i t h m , is a
s o l u t i o n to t h e o r i g i n a l Q u a d r a t i c Assignment problem.

      The         L i n e a r A s s i g n m e n t problem i s s o l v e d by a s p e c i a l
s u b r o u t i n e c a l l e d CGNET, w h i c h is a n a d o p t i o n f o r t h e C o a s t
G u a r d p r o b l e m of t h e w e l l - k n o w n p r i m a l s i m p l e x n e t w o r k
p a c k a g e SNET [ l ] .         T h e p a c k a g e is . m o d i f i e d t o U S E t h e
c o m p l e t e l y dense assignment (fixed) c o s t m a t r i x and t o
e x g l o i t where p o s s i b l e t h e s p e c i a l s t r u c t u r e of t h e s e
extremely               l a r g e problems.         Several external parametric
c o n t r o l s p e r m i t t u n i n g of t h e p a c k a g e           for    efficient
p e r f o rm nce
             a    .
      The network problem f o r t h e 5                       d a y r e s o l u t i o n h a s 860
n o d e s a n d 7 3 9 , 6 0 0 arcs. CGNET c o n s t r u c t s a n o p t i m a l L i n e a r
A s s i g n m e n t s o l u t i o n i n 11. 3               minutes      (IBH 360/67).          A
literature              search i n d i c a t e s t h a t t h i s i s t h e l a r g e s t
a s s i g n m e n t problem f o r which an o p t i m a l s o l u t i o n h a s been
constructed.                T h e p r e v i o u s r e c o r d was a 4 5 0 , 0 0 0 a r c inodel
d o n e i n a f e a s i b i l i t y s t u d y f o r t h e Navy C o m p u t e r Assisted
D i s t r i b u t i o n a n d A s s i g n m e n t ( C A D A ) model.




                                                54
                                                   2
       The memory r e q u i r e m e n t o f n / 2            w o r d s for d a t a s t o r a g e is
the      same       a s t h e G-V           method's.             Figure      9 shows t h e
c o m p a r i s o n b e t w e e n t h e G-W a n d CGNET m e t h o d s f o r c o m p u t a t i o n
time, e f f e c t i v e n e s s a n d c o m p u t a t i o n a l c o s t of t h e S w i t c h
a l g o r i t h m , a n d t o t a l p s e u d o - c o s t of t h e f i n a l  (time-unit,
mission-unit)               pairing.         Note t h a t CGNET p r o d u c e s e x c e l l e n t           ,
q u a l i t y s o l u t i o n s with s i g n i f i c a n t l y less computational
expense.




                 --------- c e
                 Performan           of Q u a d r a t i c Ass&qngnnt M e t h o d s


Resolution                20 D A Y              15 D A Y                   10 D A Y              5 DAY

Size (n)                     206                       288                    420                     860

Hethod                G-W       CGNEll       G-W         CGNET         G-W      CGNET       G-U         CGNET
Assignment
T i m e (min)         8.23         0.38     20.78            0.90     69.00         2.37    --          11.32

Switch
T i m e (min)         0.27         0.42      0.52            0.92      1.55         1.91    --              7.66

Total                                                                                      est
T i m e (min)         8.50         0.80     21.30            1.82     70.55         4.28   480.         18.98

Assignment
Cost                 81110 1 4 8 7 3 0      96120 1 7 2 7 4 0 1 1 0 0 6 0 1 8 2 0 7 0       --         168230

F i n a l cost       37470      52570       38590       36480         45960     40940       --          11100

96 S w i t c h
ImFroved              53.8         64.7      59.9           78.9       58.2         77.5    --              93.4

                                          Figure        9




                                              55
   D.     POST-PROCESSING           O F THE Q A SOLUTION



           The G r a v e s - W h i n s t o n a n d CGNET m e t h o d s f o r t h e 2A p r o b l e m
   do n o t g u a r a n t e e t h a t t h e o p t i m a l s o l u t i o n w i l l b e f o u n d .
   In        addition,            the        solution          may        contain     infaasible
   a s s i g n m e n t s , m u l t i p l e c a m p a i g n s when o n l y o n e is d e s i r e d ,
   and i l l e g a l transitions.                     To d e t e r i n i n e i f these u n d e s i r e d
   e v e n t s h a v e c c c u r r e d , q u i c k and e f f e c t i v e e d i t c h e c k s are
   conducted.

          I n t h e h y b r i d a p p r o a c h of t h e a n a l y t i c model 8 t h e
  Q u a d r a t i c A s s i g n m e n t s o l u t i o n is e x a m i n e d a n d u s e d € o r o n l y
  t h e p u r e s e q u e n c e cf m i s s i o n s c o n d u c t e d b y e a c h s h i p .         The
  consecutive time-units                      of a s h i p a r e c h e c k e d f o r e a c h
  s e q u e n c e of m i s s i o n - u n i t s         associated          with        the        same
  requirement.              C h e c k s a r e m a d e Tor a s s i g n m e n t of m i s s i o n s t o
  ships for requirements t h a t they cannot f u l f i l l .                            Any s u c h
  m i s s i o n t h a t i s f o u n d is r e l o c a t e d t o b e t h e f i r s t m i s s i o n
  o n t h e first s h i p able t o c o n d u c t t h e m i s s i o n .                             The
  s t r u c t u r e Flaced i n t h e D-matrix                       to encourage a sin gle
  m i s s i o n for a r e q u i r e m e n t when t h i s is d e s i r e d d o e s n o t
  guarantee t h i s result.                 Thus, an e d i t check t h a t r e t a i n s t h e
  f i r s t missicn e n c o u n t e r e d a n d d i s c a r d s                the       rest        is
, performed.              A n o t h e r q u a l i t y n o t g u a r a n t e e d b y t h e QA
  s o l u t i o n is t h e a b s e n c e of a l l i n f e a s i b l e o r u n d e s i r e d
  transitions,            s u c h as Alpat/Reftra.                  An o p t i o n a l e d i t c h e c k
  c a n e l i m i n a t e t h i s t y p e of t r a n s i t i o n b y t h e i n s e r t i o n of an
  InFort            mission        wherever            needed.           However,        as later
  p r o c e d u r e s may c a u s e t h e r e i n t r o d u c t i o n of s u c h t r a n s i t i o n s ,
  t h i s c h e c k is b e s t d e l a y e d u n t i l a f t e r t h e f i n a l L P s o l u t i o n
  is o b t a i n e d .




                                                    56
E.     IHPLEil!lENIATION         OF L I N E A R PROGRAMS



        I n t h i s h y b r i d model,            o n c e t h e m i s s i o n s e q u e n c e is
o b t a i n e d , a l i n e a r p r o g r a m is s o l v e d t o d e t e r m i n e t h e
mission d u r a t i o n s t h a t minimize t h e o b j e c t i v e f u n c t i o n .
Then, n e i g h b o r i n g m i s s i o n s e q u e n c e s a r e g e n e r a t e d and t e s t s d
t o see i f i m p r o v e m e n t o c c u r s .            A n e i g h b o r i n g s e q u e n c e is
d e f i n e d a s a n y s e q u e n c e t h a t is g e n e r a t e d b y a p p l y i n g o n e of
t h e following o p e r a t i o n s t o t h e p r e s e n t sequence:

  1.     HOVE one m i s s i o n t o a n o t h e r p o s i t i o n        (called S l i d e ) .

 2.     Interchange              the     position          of     two     missions          (called
        Switch).

T h i s t r i a l of n e i g h b o r i n g s e q u e n c e s i s a b s o l u t e l y c r i t i c a l
f o r t h i s C c a s t G u a r d p r o b l e m because of t h e d i f f e r e n c e s i n
t h e c o n t e n t of t h e 9A a n d LP models f o r t h i s p r o b l e m .                     The
QA s o l u t i c n ' s s e q u e n c e w i l l b e f a r from o p t i m a l b e c a u s e t h e
t r l i m i t i n g t t c c n s t r a i n t s are n o t c o n s i d e r e d . Many n e i g h b o r i n g
s e q u e n c e s w i l l b e f a v o r a b l e c a n d i d a t e s to r e d u c e t o t a l c o s t
a n d f o r each s u c h n e i g h b o r i n g s e q u e n c e , a n LP s o l u t i o n i s
necessary.                Thus, t h e computational requirements,                      core and
time, m u s t b e m i n i m i z e d f o r each LP c a l l .

       The i n i t i a l LP code ( u s e d        i n t h e G-G       implementation)
s o l v e d a F r o b l e m cf 330 rows a n d 330 v a r i a b l e s i n t h e r a n g e
of 17 t o 2 1 s e c o n d s o n t h e I3M 360/67.                   Analysis of t h e
c o n s t r a i n t e q u a t i o n s s h o w s t h a t o n l y 1 % of t h e t o t a l
c o e f f i c i e n t matrix i s n o n - z e r o a n d t h e o n l y n o n - z e r o value
i s +1.           Thus,        a n LP code u s i n g a d a t a s t r u c t u r e           for the
s p a r c e c o n s t r a i n t c o e f f i c i e n t s is d 3 s i r e d . Also,           a code
a l l o w i n g a r e d u c t i o n i n t h e number o f e q u a t i o n s b y             ranging
a n d b o u n d i n g methods r e d u c e s t h e d i m e n s i o n a l i t y              of t h e


                                                  57
p r o b l e m a n d t h u s r s d u c e s c o m p u t a t i o n a l time.

        S i n c e t h e Q A a n d LP p o r t i o n s of t h e h y b r i d m o d e l a r e
d i s t i n c t , t h e p r o g r a m a n d d a t a a r e a s of each m o d e l c a n be
o v e r l a y e d i n main memory, r e s u l t i n g i n s i g n i f i c a n t E a v i n g s
of core. W i t h a p r o j e c t e d LP p r o b l e m s i z e of 3 0 0 rows a n d
5 0 0 variables, t h e o r i g i n a l LP c o d e w o u l d h a v e s t o r a g e
r e q u i r e m e n t s l a r g e r t h a n e i t h e r s p e c i a l i z e d Q A code f o r time
r e s o l u t i c n s cf 20 d a y s or l e s s .

      A      new LP s y s t e m c a l l e d XS h a s b e e n i n s t a l l e d f o r u s e
w i t h t h e Ccast Guard problem.                            XS [ 2 ] is a             prototype
o p t i m i z a t i o n s y s t e m d e s i g n e d t o s e r v e a s a resaarch t e s t b e d
for e v a l u a t i c n of a d v a n c e d d e s i g n f e a t u r e s f o r l a r g e s c a l e
l i n e a r , n o n l i n e a r and i n t e g e r problems.              The s y s t e m i s u s e d
a s a s u b r o u t i n e , w i t h t h e LP p r o b l e m s e t u p p r o v i d e d b y t h e
calling discipline.                      S a l i e n t f e a t u r e s of t h e v e r s i o n of XS
e m p l o y e d i n c l u d e i m P l i c i t r a n g i n g ( u p p e r a n d lower v a l u e s
for each c c n s t r a i n t ) , l o g i c a l b o u n d i n g ( u p p e r a n d lower
values for each variable), and e f f e c t i v e external controls
f o r t u n i n g t h e s y s t e m t o p e r f o r m well on t h e c l a s s of
p r o b l e m a t hand.

       The     LP       c b j e c t i v e f u n c t i o n , d e s c r i b e d i n C h a p t e r V , is
c o m p o s e d cf n o n - n e g a t i v e     p e n a l t y v a r i a b l e s whose values
m e a s u r e t h e d e v i a t i o n o f t h e s c h e d u l e from t h e d s s i r e d
g o a l s . The d e v i a t i o n i s o b t a i n e d by e x p r e s s i n g t h e g o a l a s a
constraint,             w i t h t h e i n s e r t i o n of t h e a p p r o p r i a t e s l a c k or
s u r p l u s p e n a l t y v a r i a b l e i n t h e e q u a t i o n or i n e q u a l i t y .




                                                 58
       For e x a m p l e , t h e d e v i a t i o n of a s h i p from t h e d e s i r e d
t o t a l A H F time is d e t e r m i n e d by a d d i n g t h e f o l l o w i n g
c o n s t r a i n t t c t h e LP:

        O b j e c t i v e Function:           (cost   *   s l a c k ) + (cost      *   surplus)
        Constraint:

        Total
        AHP
        Limit
                  =   1:ii     Mi ssi o n s
                                                +     Slack     -     Surplus



        T h u s , t h e i a p l e m e n t a t i o n of t h e L P model h a s t h e two
t y p e s of n o n - v i o l a b l e c o c s t r a i n t s a n d s i x p e n a l t y - r e l a t e d
constraints             ( t h a t d e t e r m i n e t h e v a l u e of t h e . p e n a l t y
variables).           S e v e r a l of t h e s e t y p e s of c o n s t r a i n t s a r e
i n d e p e n d e n t of t h e o v e r a l l n u m b e r a n d s t r u c t u r e of t h e
s e q u e n c e of m i s s i o n s .  T h e s e a r e t h e t o t a l h o r i z o n time, A H P
t i m e a n d A l p a t time on e a c h s h i p , a n d t h e e a r l y / l a t e s t a r t
a n d l a t e f i n i s h time p e n a l t y c o n s t r a i n t s .     T h e n u m b e r of
time p e n a l t y e q u a t i o n s is f i x e d b y t h e number o f m i s s i o n s
for r e q u i r e w n t s p o s s e s s i n g t h e s e p e n a l t i e s . S i n c e no
m i s s i o n s a r e d e l e t e d or a d d e d b y t h e LP, no c h a n g e i n t h e
n u m b e r of m i s s i o n s o c c u r s .

      The        numker of e q u a t i o n s f o r t h e cruise d u r a t i o n p e n a l t y
and t h e q u a n t i t y bounds on t h e requirements can be reduced
for each i n d i v i d u a l LP b a s e d on i t s p a r t i c u l a r m i s s i o n
s e q u e n c e . The n u m b e r of A w a y H o m e p o r t c r u i s e s is d e p e n d e n t
on t h e i n d i v i d u a l sequence.              For e a c h s e q u e n c e of tuo o r
more A H P m i s s i o n s , a n e q u a t i o n i s i n t r o d u c e d .    Rather than
i n t r o d u c e a n e q u a t i o n for a s o l i t a r y AHP m i s s i o n , t h e
mission d u r a t i o n ' s u p p e r b o u n d i s c o m p a r e d t o t h e AHP c r u i s e
l i m i t and lowered t o t h i s value if necessary.                          T h i s method
does n o t allow a s o l i t a r y m i s s i o n t o e x c e e d t h e c r u i s e
l i m i t ,     even for a p e n a l t y assessment.              ( T h i s is a change t o



                                                59
t h e s t r i c t d e f i n i t i o n of t h e LP m o d e l g i v e n i n C h a p t e r V.)


        T h e l a s t c o n s t r a i n t t y p e is t h e q u a n t i t y r e s t r i c t i o n o n
each requirement.                   These are h a r d c o n s t r a i n t s w i t h no
p r o v i s i o n f o r p e n a l t i e s for n o n - c o m p l i a n c e . An e q u a t i o n i s
i n t r o d u c e d o n l y i f a r e q u i r e m e n t is s p l i t i n t o two or more
missions. If only one mission produces t h e requirement, a l l
p r e v i o u s b o u n d s on t h e m i s s i o n d u r a t i o n a r e s u p e r c e d c d b y
t h e minimum a n d maximum q u a n t i t i e s of t h e r e q u i r e m e n t .               Note
t h a t i f t h e s i n g l e m i s s i o n forms a s o l i t a r y AHP c r u i s e a n d
t h e maximum q u a n t i t y e x c e e d s t h e AiIP s e q u e n c e l i m i t , n o
p e n a l t y w i l l be c h a r g e d i f t h e c r u i s e l i m i t i s e x c e e d e d a s
n o e q u a t i o n for c r u i s e p e n a l t y i s g e n e r a t e d .

      The t y p i c a l LP f o r a 6 s h i p ,         y e a r s c h e d u l e h a s 171
                                                       2
rows a n d 4 3 5 v a r i a b l e s . T h e n u m b e r of rows for e a c h t y p e of
c o n s t r a i n t i s : 6 each f o r t o t a l h o r i z o n , t o t a l AHP time a n d
t o t a l ALPAT time; 86 f o r e a r l y / l a t e s t a r t times; 51 f o r l a t e
f i n i s h times; 1 2 for c r u i s e s of 2 o r more m i s s i o n s ;           and 4
requirement             duration         totals.              There    are      145     mission
v a r i a b l e s a n d 290 p e n a l t y v a r i a b l e s .

       The      n a t u r e a n d b a l a n c e of t h e o b j e c t i v e f u n c t i o n i s
c r i t i c a l t o o b t a i n i n g t h e d e s i r e d t y p e of s c h e d u l e . T h e
b a l a n c e b e t w e e n t h e s i x t y p e s of p e n a l t i e s w i l l d e t e r m i n e
t h e r e l a t i v e i m p o r t a n c e a n d p r e f e r e n c e s of   the     tradeoffs
t h a t w i l l b e made d u r i n g t h e m i n i m i z a t i o n p r o c e s s .


       T h e c o s t s t r u c t u r e for e a c h t y p e o f p e n a l t y is        also     an
important             factor.            Figure 10           depicts          the       presont
structures.




                                                60
                   ---- ------y ---s t ----- c t ur es
                   Present P e n a l t C o S tr u



                                      ICOST
 Horizon       ~




                                      U
                                      a           D E V I AT1 ON




 Total
 Alpat
 Time


                                      0.          D E V I AT I ON

Early/ L a
Start
Times

                                                  START TIME
                                           COST
Late
Finish
Time

                                      0           DEV I AT I ON

AHP
Crulse
Du ra t io n

                                                  D E V I A T 1 ON



                            Figure         10




                                 61
F.    INPLEHENTATION ALTESNATIVE FOR LP



      E x a m i n a t i o n of t h e s o l u t i o n s t o a l l t e s t LP's        shows t h a t
m i s s i o n d u r a t i o n s were a l w a y s exact i n t e g e r s .       This leads to
t h e i n v e s t i g a t i o n of a N e t w o r k f o r m u l a t i o n of t h e L P .              A
complete t r a n s f o r m a t i o n t o a network has n o t been found for
t h e CG m o d e l b e c a u s e o f a few c o m p l i c a t i n g c o n s t r a i n t t y p e s .
The horizon l i m i t ,                 time p e n a l t y a n d r e q u i r e m e n t q u a n t i t y
c o n s t r a i n t t y p e s (which c o l l e c t i v e l y comprise t h e e n t i r e
o r i g i n a l G-G         model a n d a b o u t 85% of t h e e q u a t i o n s o f t h i s
Coast G u a r d m o d e l ) c a n be r e f o r m u l a t e d a s a p u r e n e t w o r k
model.            T h i s r e f o r m u l a t i o n h a s n o t been implemented, but a
s i g n i f i c a n t r e d u c t i o n i n time f o r each L P c a l l , e s p e c i a l l y
for t h e G-G model, would o c c u r .

       The v a r i a b l e s   (Arcs)      f o r t h e network formulation are t h e
l e n g t h a n d t h e s t a r t i n g t i m e for e a c h m i s s i o n .              F i g u r e 11
p i c t o r a l l y s h o w s t h e f u n d a m e n t a l s t r u c t u r e of t h e n e t w o r k .
There           is cne e q u a t i o n         ( n o d e ) f o r e a c h n i s s i o n of a
r e q u i r e m e n t t h a t h a s a n y of t h e s e q u a l i t i e s :                   a late
completicn penalty, a f o l l o w i n g mission with an early start
or f i n i s h date. Each requirement v i t h m u l t i p l e m i s s i o n s
w i l l have an equation.                        s i n g l e m i s s i o n r e q u i r e m e n t s are
h a n d l e d hy a F p r o p r i a t e b o u n d s on t h e m i s s i o n ' s l e n g t h .         The
t o t a l t i n o l i m i t e q u a t i o n for each s h i p i s a d d e d b e t w e e n the
l a s t m i s s i o n node for each s h i p a n d t h e r o o t n o d e .

       The      L a l a n c i n g of AHP time is h a n d l e d i n t h e n e t w c r k by
i t s c o m p l e m e n t , t h e time s p e n t o n I n p o r t m i s s i o n s .   Three
t y p e s of m i s s i o n s e m e r g e from t h e r o o t n o d e . T h e f i r s t t y p e
is a l l AHP time r e q u i r e m e n t s .          The s e c o n d t y p e i s a l l
SHIP-SPECIFIC h o m e p o r t r e q u i r e m e n t s .     T h e t h i r d t y p e is a l l
GENERAL s h i p i n - h o m e p o r t       requirements,         usually only t h e


                                                 62
I n p o r t ( S A R s t a n d b y ) r e q u i r e m e n t t h a t is t h e s y s t e m ' s s l a c k .
T o o b t a i n t h e t o t a l non-AHP time o n a s h i p , t h e t y p e two a n d
three     m i s s i o n s o n a s h i p n e e d t o b e a d d e d i n some m a n n e r .
I f t h e t y F e two m i s s i o n s do n o t h a v e a f i x e d t o t a l , t h i s c a n
n o t be d o n e t y t h e n e t w o r k .  I f t h e y do h a v e a f i x e d l i m i t ,
a p p r o p r i a t e l i m i t s a n d p e n a l t i e s c a n b e p l a c e d o n t h e sum of
type         three         missions          g o i n g t o each s h i p .             Actually,
c a l c u l a t i o n s to c h e c k for t h i s s i t u a t i o n may n o t                  be
worthwhile            i n a c t u a l implementation.            Replacement by o n e
non-network             e q u a t i o n p e r s h i p is s i m p l e r .     Non-network
e q u a t i o n s a r e a l s o n e c e s s a r y f o r each AHP c r u i s e s e q u e n c e o f
2 cr more m i s s i o n s a n d o n e p e r s h i p for each m i s s i o n - t y p e
w i t h maximum l i m i t s ( i . e . A l p a t ) .


       An i m p o r t a n t p r o p e r t y o f t h e n e t u o r k    is t h e r e s u l t i n g
p i c t o r i a l d i s p l a y of t h e m o d e l . Based on t h i s p i c t u r e , a
method t o p e r f o r m d y n a m i c c h a n g e s t o t h e f o r m u l a t i o n m a y b e
developed t o e l i m i n a t e t h e coinplete problem g e n e r a t i o n f o r
e a c h new LP m i s s i o n s e q u e n c e . A l t e r n a t i v e l y , a m i s s i o n t h a t
i s ! a c a n d i d a t e f o r r e l o c a t i o n may b e p l a c e d i n s e v e r a l
p l a c e s a t once. S e n s i t i v i t y a n a l y s i s and r e o p t i m i z a t i c n c a n
d e t e r m i n e t h e one b e s t p l a c e m e n t f a s t e r t h a n a s u c c e s s i o n of
c o m p l e t e LP c a l l s . T h i s a n d a n a b i l i t y t o r e o p t i m i z e b a s e d
on t h e c u r r e n t s o l u t i o n may d r a s t i c a l l y r e d u c e t h e t i m e a n d
i n c r e a s e t h e i n f o r m a t i o n g a i n e d from each LP c a l l .




                                                 63
F i r l u r e 11
    64
G.    CALLING STRATEGY F O 2 THE LINEAR PROGRAMS



        T h e f i r s t s e q u e n c e of m i s s i o n s o b t a i n e d from                   the
Q u a d r a t i c A s s i g n m e n t s o l u t i o n a n d t h e post-QA/pre-LP                edit
p r c c e s s i n g w i l l b e f a r from o p t i m a l b e c a u s e t h e I c l i m i t i n g l '
c o n s t r a i n t s a r e f i r s t i n t r o d u c e d i n t h e L P model.           Thus t h e
a b i l i t y t o improve t h e s o l u t i o n                   by       evaluating           many
t a n e i g h b o r i n g l t m i s s i o n s e q u e n c a s is v i t a l t o t h e s u c c o s s f u l
s o l u t i o n of t h e s c h e d u l i n g p r o b l e m .

        The first algorithm ( c a l l e d S l i d e )                       p e r f o r m e d on t h e
m i s s i o n s e q u e n c e r e m o v e s e a c h n i s s i o n i n t u r n a n d i n s e r t s it
i n t o a l l other p o s s i b l a l o c a t i o n s .          T h e n u m b e r of d i f f e r e n t

m i s s i o n s e q u e n c e s t o b e e v a l u a t e d i n some m a n n e r i s         n
                                                                                               2
                                                                                                   .   An

a c t u a l c a l l for a f u l l L P s o l u t i o n ,                   t h a t is, c o m p l e t e
e x p l i c i t enumeration,            is c l e a r l y i n p o s s i b l e .        A calling
s t r a t e g y i m p l i c i t l y e v a l u a t e s each p o s s i b l e s e q u e n c e a n d a n
LP i s s o l v e d o n l y            f o r t h e most            promising             candidate
sequences.           Addit i o n a l l y ,        a f t e r a n e i g h b o r i n g s e q u e n c e is
f o u n d w i t h a n i m p r o v e d t o t a l s e q u e n c e a n d LP p e n a l t y c o s t s ,
                                                                                    2
i t becomes t h e i n c u m b e n t s o l u t i o n a n d a n o t h e r n c a s e s a r e

e v a l u a t e d u n t i l no f u r t h e r improvements c a n              be     found          with
t h i s transform.


                                      (called Switch) e v a l u a t e s p a i r w i s e
       The s e c o n d a l g o r i t h m
interchanges            cf     missions.  T h e n u m b e r of p o s s i b l e s w i t c h e s
                                                         2
f o r e a c h i n c u m b e n t s e q u e n c e is ( n - n ) / 2 .        Again,        aftEr          an

imFroving switch i s found, t h e transform i s r e s t a r t e d .

       For each t y p e of t r a n s f o r m , a c a l l i n g s t r a t e g y t o d e c i d e


                                                  65
when         t o i n v e s t i n a n LP s o l u t i o n i s c r i t i c a l . Those
t r a n s f o r m e d s e q u e n c e s t h a t i n t r o d u c e a s s i g n m e n t s on s h i p s
i n c a p a b l e o f f u l f i l l i n g t h e requirement can e a s i l y be
detected.             But for feasible assignments, t h e expected change
i n e a c h of t h e s i x t y p e s of c o s t p e n a l t i e s m u s t b e
c a l c u l a t e d or estimated.                   The change i n t h e                     sequence
t r a n s i t i . c n c o s t is d e t e r m i n i s t i c a n d e a s i l y c a l c u l a t e d . The
changes i n t h e penalties,                    however,          have a n unpredictable
n a t u r e b e c a u s e t h e y are d e p e n d e n t o n t h e m i s s i o n l a g t h s
w h i c h a r e s e t by t h e L P t o o b t a i n t h e m i n i m a l o b j e c t i v e
function,


       To s o l v e t h i s problem           for Slide, a detailed calling
s t r a t e g y is i n p l e m e n t e d f o r f i v e of t h e p e n a l t y t y p s s .    (The
horizon Fenalty is n o t included.)                       The t r a n s i t i o n c o s t i s
d e t e r m i n i s t i c a l l y f o u n d a n d e s t i m a t e s a r e made f c r t h e
c h a n g e i n s t a r t a n d f i n i s h time, t o t a l AHP, t o t a l A l p a t , a n d
AHF c r u i s e p e n a l t i e s .         Amazing a c c u r a c y h a s b e e n a c h i e v e d .
I n t h e e a r l y s t a g e s of t h e a l g o r i t h m , a h i g h p e r c e n t a g e of
t h e LP c a l l s t h a t a r e m a d e l e a d t o i m p r o v e m e n t s e x t r e m e l y
close t o t h e p r e d i c t i o n .


       Other      methods         tomaximize t h e c o n t r i b u t i o n o f e a c h LP
c a l l a r e tkeoretically similar t o p i v o t                      pricing        and
s e l e c t i o n rules.        For Slide,      a m i s s i o n i s moved t o e a c h
f e a s i b l e new p o s i t i o n a n d t h e e x p e c t e d c o s t c h a n g e i s
calculated,            The        proposed        new      position         with      the      most
b e n e f i c i a l e x p e c t e d c h a n g e is t h e o n l y o n e c o n s i d e r e d
f u r t h e r . The p r e d i c t e d c h a n g e is t h e n compared t o a d e c i s i o n
r u l e c u t o f f value.      B a s e d on t h i s t e s t , a n LP c a l l i s           either
made        or   the     Slide       algorithm          discards        t h a t mission as a
c a n d i d a t e a n d p r o c e e d s f o r t h e n e x t mission.


       In
        another strategy,      t h e d e c i s i o n cutoff   v a l u e is
changed a s t h e algorithm progresses.        Presently, t h e sliding
decision         r u l e h a s f o u r v a l u e s , e a c h h i g h e r i n v a l u e (-8000,


                                                 66
-4'000, -1000 a n d - 6 0 0 ) .         Those s e q u e n c e       changes t h a t cause
v e r y l a r g e a n d b e n e f i c i a l cost i m p r o v e m e n t s a r e a c t i v e l y
s o u g h t i n t h e o p e n i n g s t a g e s . After o n e d e c i s i o n r u l e v a l u e
l e a d s t o a c o m p l e t e c i r c u i t of S l i d e w i t h no i m p r o v e m e n t s
being found, t h e r u l e assumes t h e n e x t h i g h e r v a l u e and t h e
algorithm i s restarted. The c o m p u t a t i o n a l t h e s p e n t d o i n g
t h e s e c a l c u l a t i o n s a n d t e s t s f o r a comslete c i r c u i t o f t h e
a l g o r i t h m is less than h a l f t h e time of a s i n g l e L P c a l l .
F i g u r e 1 2 shows t h e dramatic i m p r o v e m e n t s made i n t h e
objective function,                     t h e p e r c e n t a g e of LP c a l l s b e i n g made
t h a t l e a d t o t r u e i m p r o v e m e n t s , a n d t h e o v e r a l l s u c c e s s of
t h e s e c a l l i n g s t r a t e g i e s for s l i d e .

        For t h e s e c o n d a l g o r i t h m , S w i t c h , a p r e d i c t i o n o f t h e
c o s t c h a n g e is made for o n l y t h e s t a r t a n d f i n i s h time
p e n a l t i e s a n d t h e t o t a l AHP time p e n a l t i e s .        The t r a n s i t i o n
cost c h a n g e i s d e t e r m i n i s t i c a n d , t h u s , i s c a l c u l a t e d . O n l y
o n e v a l u e of t h e d e c i s i o n r u l e c u t o f f      value is used.                   LP
solutions are obtained for a l l favorable candidate sequencss
r a t h e r t h a n f o r o n l y t h e most f a v o r a b l e .       T h e p e r f o r m a n c e of
t h i s a l g o r i t h m i s a l s o i n d i c a t e d i n F i g u r e 12.




                                                67
                                                                                                        I
                                 Lp C a l l i n q S t r a t e q y P e r f o r m a n c p



---- e --- ----- h
S l i d and Switc           Algorithms

Slide:       4 l e v e l s of d e c i s i o n criteria:            -8000, -4000, -1000,               -600
Switch:      1    l e v e l of d e c i s i o n c r i t e r i a :      +10

        I f t h e e s t i m a t e d q o l u t i o n i m p r o v e m e n t is l e s s t h a n           the
c u r r e n t d e c i s i c n c r i t e r i a , t h e LP is f o r m u l a t e d a n d s o l v e d .

   --------o n :
   Definiti

             H i t Ratio:        =           - .......................... made
                                             0 L2's w i t h i m p r o v e m e n t
                                             T o t a l # L P ' s called

Time R e s o l u t i o n                     2 0 DAY       2 0 DAY       20 DAY           20 DAY

I n i t i a l cost                           348500          319000      348000           810000

SLIDE:       C o s t End o f -8000            109000         112000      121000           174000
             H i t Ratio                       7/7            6/6         6/6             10/10
             C c s t End of -4000              95900          90000      101000           149000
             H i t Ratio                       2/2            4/8         4/8              6/20
             C o s t End of'-1000              91600          70000       81000           122000
             H i t Ratio                       4/20           9/20       10/23            11/49
             C o s t End o f -600              88630          63000        71000          121650
             Hit R a t i o                     2/70           6/13         9/5 1           1/31

SWITCH:      C o s t End o f +10               82650          62700         68416          98740
             H i t Ratio                       7/3 0          0/3           3/22          15/59


I n i t i a l Ccst                           348500        319000        348000           810000
F i n a l cost                                82650         62700         68416            98740
Composite. H i t Ratio                       22/129        25/50         32/110           45/169
A v e r a g e T i m e / LP (secs)             5.32          6.50          4.79             4.47

                                          Figure        12




                                                 68
                                          VII.        RESULTS




A.    USE OF T E E MODZL



        Before d i s c u s s i o n cf t h e q u a l i t a t i v e p r o p e r t i e s of t h e
s c h e d u l e s p r o d u c e d by t h e i m p l e m e n t a t i o n o f t h e C o a s t G u a r d
a n a l y t i c a l s c h e d u l i n g model, t h e g e n e r a l t y p e s o f i n p u t d a t a
and u s e r e x t e r n a l c o n t r o l s must be understood.                     There are
s e v e n t y p e s of i n p u t d a t a u s e d b y t h e m o d e l :

 1.     Number of s h i p s , n u m b e r o f m i s s i o n t y p e s , t h e t i m e s p a n
        of t h e s c h e d u l e , a n d t h e time r e s o l u t i o n of t h e Q A
        model.

 2.     D e s c r i p t i o n of t h e m i s s i o n t y p e s : a w a y / i n h o m e p o r t ,
        s i n g l e ( n o s p l i t of r e q u i r e m e n t ) , a n d / o r P a t r o l .

 3.     Description           of     the ships:        c a p a b i l i t y for e a c h mission
        t y p e , l a s t m i s s i o n on p r e v i o u s s c h e d u l e , i n i t i a l       fixed
        p o r t i o n of p r e s e n t s c h e d u l e .                                     I


 4.     Description           of     requirements:            mission         type,          target
        demand f o r QA f o r m u l a t i o n a n d l o n e r / u p p e r l i m i t s for LP
        f o r s u l a t i o n , e a r l y and l a t e +art t i n e s and l a t e f i n i s h
        time,         F r i o r i t y of times (for c o n t r o l of time p e n a l t y
        rates).

 5.     Pseudo-Costs:             mission-to-mission      t r a n s i t i o n costs and
        cost rates f o r t h e               penalty     variables            (horizon,
        s t a r t / f i n i s h times,   t o t a l AHP time, t o t a l B l p a t time
        and cruise duration).




                                                 69
 6.       D e s i r e d g o a l s f o r t o t a l AHP time, t o t a l A l p a t t i m e , a n d
          c r u i s e duraticn.

 7.       Implementation           controls:        LP c a l l i n g s t r a t e g y d e c i s i o n
          cut-off     v a l u e s , l e v e l of o u t p u t , l e n g t h of maximum r u n .


      By t h e s p e c i f i c a t i o n of t h e i n p u t d a t a , t h e s c h e d u l e r
h a s e f f e c t i v e c o n t r o l of t h e r e s u l t i n g s c h e d u l e . The cost
rates f o r t h e penalty v a r i a b l e s determine t h e balances and
t r a d e o r ' f s t h a t w i l l b e made b y t h e o p t i m i z a t i o n p r o c e s s t o
a r r i v e a t t h e f i n a l proposed schedule.              By a d j u s t i n g a r a t e ,
t h e i n e l u e n c e a n d i m p o r t a n c e of    that guideline's        g o a l is
changed with r e s p e c t t o a l l                    others. An e x a m p l e o f c o s t
b a l a n c i n g between t r a n s i t i o n costs and             the     cruise        psnalty
r a t e is g i v e n i n F i g u r e 13.

       A n o t h e r i m p o r t a n t c o n t r o l i s p r o v i d e d by t h e c o s t
p e n a l t y s t r u c t u r e s shown i n F i g u r e 10 ( C h a p t e r 81). T h e
influence          this      structure         has       on       the       tradtoffs       and
p r e f e r e n c e s w i t h i n e a c h g u i d e l i n e g o a l is i l l u s t r a t e d i n
F i g u r e 14.       A s t r u c t u r e with l a r g e r    penalties for large
d e v i a t i o n s forces t h e AHP b a l a n c e c l o s e r t o t h e d e s i r e d
goal.


      B       p o w e r f u l p r o p e r t y of t h e h y b r i d QA/LP a p p r o a c h is t h e
s e F a r a t i o n of t h e c o m b i n a t o r i a l s e q u e n c i n g o f . m i s s i o n s f o r
e a c h s h i p a n d t h e d e t e r m i n a t i o n of t h e d u r a t i o n o f e a c h
missioh.         This approach and               the     implementation              techniques
provide   t h e c a p a b i l i t y of b y p a s s i n g t h e QA, using a
previously generated sequence, s e t t i n g t h e penalty rates and
obtaining          a potential schedule.                    (Note: S i n c e t h e Q A model
does not include                      the         trlimitingIr     and       morale-rdated
constraints,             changes i n t h e p e n a l t y rates o r d e s i r e d goals
for t h e s e c o n s t r a i n t s a f f e c t o n l y t h e L P p o r t i o n o f t h e
model.)         C y n a m i c i n t e r a c t i v e u s e of t h e m o d e l i s a n i n h e r e n t
p r o p e r t y cf t h e m o d e l i n g a p p r o a c h .



                                                 70
                                             --
                            I n f l u e n c e of --- --- a k p o i n t s &.
                            --I-                 More B r e
                           ---
                           P e n a l 9 _Cost S t r u c t u r e fog r o t a 1
                                    -a y
                                    Aw      Hornegort (AHPL g&Cg

Two A l t e r n a t i v e s f o r D i s t r i b u t i n g T o t a l AHP T i m e

S e t A:     S h i p 1:          f o u r u n i t s u n d e r d g s i r e d b a l a n c i n g amount
             s h i p 2:          one o v e r
S e t B:      S h i p 1:         two under
              S h i p 2:         three over




                                                     -6             -3     0        3        6
                                  DEVI AT1 ON
                                                                                    DEVI AT1 ON



                                           Total Penalty

S e t A:         4 + 1 = 5                                     A:         6 + 2/3       = 6 2/3
      B:         2 + 3 = 5                                     B:        4/3 + 2        = 3 1/3


No p r e f e r e n c e b e t w e e n                           Cost structure prefers
t h e s e t s exist from                                       B o v e r A. B i s
Cost structure.                                                more c e n t r a l a b o u t
                                                               d e s i r e d t o t a l amount.

                                            Figure        13




                                                  71
                                      B a l a n c u of C o s t s
                                      v-7-




                                -----
                                B e t w e e n Two C o m p o n e n t s

Situation:              Schedule         10 weeks A l p a t
                                          2 weeks Ocean
                        Ignore all penalties except Cruise
                        l i m i t a n d T r a n s i t i o n costs.
                        C r u i s e l i m i t i s 1 0 weeks.
                        A l p a t a n d O c e a n a r e AHP m i s s i o n s .

Alternatives:
        A
        B
                        Inport/Ocean/Alpat/In
                        Inpcrt/Ocean/Inport/A                Eo ar tt / I n p o r t
                                                              p




        Inport                0                10                10             Cruise penalty
        Alpat                10                 0             1000              2 0 0 0 / u n i t over 1 0
        Ocean              1000                10                0

 Ccsts:       Transition + Cruise                = Total
        A:            30          +   4000       = 4030
        B:          1030        +        0       = 1030
    B is p r e f e r r e d b u t a s d i s c u s s e d i n a p r e v i o u s C h a p t e r ,
InFort/Ocean/Inport               is a n i n f s a s i b l e s e q u e n c e . T h e p e n a l t y
cost of O c e a n / I n p o r t s h o u l d be m o r e - t h a n t h a t n e c e s s a r y t o
link O c e a n / A l p a t e v e n i f cruise l i m i t i s v i o l a t e d .

---s t -------u r e
Co     Struct           gL
                                  T r a n s i t i o n Costs
                        -
                        Inport           AlEBt              Ocean
        Inport                0                10               10              Cruise penalty
        Alpat                10                 0             1000              2000/unit o v e r 10
        Ocean              7000                20                0

 Costs:       Transition + Cruise                = Total
        A:           30         +     4000          = 4030
        B:         7030         +          0     = 7030
                   referred, ag desired b y t h e g u i d e l i n e s .        T h e cost
of Ocean/A p a t was r a i s e d a b o v e t h e i n c r e a s e i n c r u i s e
      A
 e n a l t y c a u s e d b y a d j o i n i n g 2 w e e k s o f Ocean a n d 1 0 w e s k s of
Lpat    .
                                         Figure        14

                                                        \




                                                72
      The model c a n b e used f o r b o t h                  strategic         and      tactical
decisions,            Strategic          "what       i f # ' q u e s t i o n s can be e x a m i n e d
for t h e i r e f f e c t s on scheduling (for example, changes i n t h e
n u m b e r of s h i p s a v a i l a b l e , t h e c r u i s e l i m i t f o r a new c l a s s
of s h i p , or new a r e a s of r e s p o n s i b i l i t y for t h e Coast
Guard). Dparric s c h e d u l e changes, due t o r e q u i r e m e n t c h a n g e s
o r u n a n t i c i p a t e d s h i p n o n - a v a i l a b i l i t y , c a n be e v a l u a t e d and
t h e least lfuFsettingf* s o l u t i o n found.



B.     QUALITY OF SCHEDULES PRODUCED



       B e c a u s e cf t h e i m p o r t a n c e of      managerial           judgement          and
personal           preferences,              no     single         is available to
                                                                 ROE
q u a n t i t a t i v e l y measure t h e q u a l i t y of t h e s c h e d u l e s p r c d u c e d
by t h e a n a l y t i c model.

    The   sc3edules   produced   completely satisfy   the
restricticns   governing   desired  and undesired mission
transitions,              No t r a n s i t i o n s w i t h h i g h , i n f e a s i b l a c o s t s a r e
present.            Almost a l l r e q u i r e m e n t s a r e s c h e d u l e d t o meet
t h e i r s p e c i f i e d s t a r t a n d f i n i s h times.         Those t h a t a r E n o t
A l p a t m i s s i o n s a r e u s u a l l y o n e o r two w e e k s d i s p l a c e d from
t h e d e s i r e d times.         T h e A l p a t missions,           having t h e Patrol
requirement,             are e x c e p t i o n s .         As e x p l a i n e d , t h e P a t r o l
r e q u i r e m e n t i s i m p l i c i t l y m o d e l e d by s e t t i n g h i g h p r i o r i t y
( c a u s i n g a h i g h e r p e n a l t y r a t e ) on f u l f i l l i n g t h e s p e c i f i e d
s t a r t / f i n i s h times. A b o u t 7 0 % of t h e A l p a t m i s s i o n s a r e
scheduled          t c s a t i s f y t h e times, b u t t h e r e m a i n i n g A l p a t
m i s s i o n s v a r y from b e i n g d i s p l a c e d 1 week t o a maximum of 8
weeks.


       T h i s r e s u l t for a P a t r q l r e q u i r e a e n t m i s s i o n - t y p eis n o t
realistic.            To     o k t a i n e v e n t h i s r e s u l t , i t is n e c e s s a r y t o



                                                  73
exFress t h e c v e r a l l r e q u i r e m e n t a s many d e t a i l e d , i n d i v i d u a l
r e q u i r e m e n t s with exact d u r a t i o n s ( e q u a l iower/upper bounds)
a n d s p e c i f i c s t a r t / f i n i s h times.        T h i s l e v e l of i n p u t d e t a i l
is a n u n d e s i r a b l e f e a t u r e . T h i s r e s u l t a n d t h e n e c e s s a r y
d e t a i l o f d a t a are i m p l e m e n t a t i o n i s s u e s , a n d not model
shortcomings.               model.           An e n h a n c e m e n t d i s c u s s e d i n C h a p t e r
VIII w i l l c c r r e c t t h i s p o o r p e r f o r m a n c e .

       with      the      exception          of t h e P a t r o l t y p e of m i s s i o n , t h e
p r e s e n t model p r o d u c e s g o o d                  quality         solutions.          The
g u i d e l i n s s t h a t a r e e x p l i c i t l y modeled a n d t h o s e t h a t c a n b e
i m g l i c i t l y m o d e l e d by t h e s c h e d u l e r t h r o u g h t h e i n p u t d a t a
c a p t u r e a l l o f t h e major c o n s i d e r a t i o n s of t h e o v e r a l l
guidelines.             T h i s model a l s o meats t h e o b j e c t i v e o f           quickly
g e n e r a t i n g s e v e r a l p o t e n t i a l s c h e d u l e s . Minor v a r i a t i o n s i n
some c o n t r o l p a r a m e t e r s ,           p a r t i c u l a r l y t h e minimum         and
maximua d u r a t i o n s f o r I n p o r t m i s s i o n s , c a n c a u s e a l t e r a t i o n
of t h e s c h e d u l e s .      For t h e C o a s t G u a r d p r o b l e m , t h i s a b i l i t y
t o easily           generate         alternatives            is     a     highly beneficial
property.

     .The       P o d e l , i n summary, p r o v i d e s g o o d q u a l i t y s o l u t i o n s ,
generates alternative schedules easily,                                      and i s    highly
responsive t o managerial controls.                                   T h i s model c a n be a
p o w e r f u l m a n a g e r i a l t o o l f o r s c h e d u l i n g t h e High E n d u r a n c e
Cutters.

      The use          of t h i s model f o r H e d i u m E n d u r a n c e C u t t e r
s c h e d u l i n g ( n o t t h e d i r e c t f o c u s of t h i s s t u d y ) a p p e a r s more
i n doubt:          principally because shorter s h i p endurances cause
s h o r t e r p a t r o l l e n g t h s ; more m u l t i - s h i p r e l a t i o n s h i p s e x i s t
f o r Search a n d R e s c u e c o v e r a g e ; R e s e r v e t r a i n i n g c r u i s e s a r e
h i g h l y c o m F l i c a t e d ; a n d l a r g e r r e l a t i v e l e n g t h s of t r a v e l
time t o p a t r c l t i m e e x i s t .




                                                  74
C.    QA      /    LP RESULTS



      The         size    a n d c o m p u t a t i o n a l d i f f i c u l t y of t h e Q u a d r a t i c
A s s i g n m e n t F r o b l e m is d e t e r m i n e d     by   the     time r e s o l u t i o n .
The t i m e r e s o l u t i o n           (S) is t h e u n i t t h a t d e t e r m i n e s t h e
n u m b e r o f f i n i t e t i m h - u n i t s t o b e s c h e d u l e d for e a c h s h i p ,
a n d t h e number of m i s s i o n - u n i t s f o r e a c h r e q u i r e m e n t .            The
r e s u l t s disFlayed i n Figure 9 (Chapter V I )                                     show     the
c o m p u t a t i c n a l p e r f o r m a n c e s of t h e G r a v e s - W h i n s t o n a n d CGaET
m e t h o d s f c r time r e s o l u t i o n s o f 2 0 , 1 5 , 1 0 a n d 5 d a y s .


       C o m p a r i s c n of t h e s e m e t h o d s a n d / o r r e s o l u t i o n s c a n n o t b e
b a s e d c o m F l e t e l y on t h e f i n a l        QA  solution        cost.         B E t ween
resoluticns,             t h e number of                mission-unit        t o mission-unit
t r a n s i t i o n s is d i f fexent, causing d i f f e r e n c e s i n t r a n s i t i o n
cost.           B e t w e e n m e t h o d s , t h e f i n a l c o s t i s a n i n d i c a t i o n of
t h e d e g r e e o f c o m p l i a n c e of t h e s c h e d u l e t o t h e d e s i r e d
mission sequences and s t a r t / f i n i s h                      times.         It i s n o t a n
i n d i c a t i o n of t h e q u a l i t y of t h e o v e r a l l s o l u t i o n t h a t w i l l
r e s u l t a t t h e e n d of t h e L P s i n c e t h e i i l i m i t i n g l t c o n s t r a i n t s
a r e n o t modeled i n t h e QA.                 Also, t h e c o n t r i b u t i o n o f t h e Q A
i s o n l y t h e m i s s i o n s e q u e n c e e x t r a c t e d from i t s ( t i m e - u n i t ,
m i s s i o n - u n i t ) Fairs, n o t t h e a d d i t i o n a l t i m i n g i n f o r m a t i o n .


       T h e C G N E T m e t h o d u s u a l l y p r o d u c e s a l a r g e r n u m b e r of
s e p a r a t e m i s s i o n s t h a n t h e - G-W    method.         Most of            the
additional           f f i i s s i o n s a r e I n p o r t w h i c h is b e n e f i c i a l l a t e r i n
t h e lineighboringi' sequence searches.                      more I n p o r t s
                                                                    Having
causes l a r g e r t r a n s i t i o n cost reductions t o be possible.


      Q u a l i t a t i v e e v a l u a t i o n of t h e m o d e l ' s f i n a l schedule
shcws t h a t t h e m e t h o d of s o l u t i o n of t h e Q A h a s n o i m p a c t o n
t h e f i n a l quality.            Each method p r o d u c e s a s u p e r i o r f i n a l



                                                   75
s c h e d u l e f o r a b o u t h a l f o f a l l cases. CGNET, w i t h i t s good
c o m p u t a t i o n a l p e r f o r m a n c e , i s t h e recommended method.


         C o n t r a r y t o e x p e c t a t i o n s , t h e 20 d a y r e s o l u t i o n p r c v i d e s
l l h i g h e r t l q u a l i t y o v e r a l l s c h e d u l e s t h a n t h e smaller ( 5 day,
10 d a y ) r e s o l u t i o n s ; t h e 5 d a y r e s o l u t i c n p r o d u c e s t h e worst
schedules.               The s c h e d u l e s from t h e f i n e r r e s o l u t i o n s h a v e
greater difficulty i n satisfying the transition restrictions                                               .
a n d r e d u c i n g t h e v i o l a t i o n s of t h e d e s i r e d s t a r t / f i n i s h
times.


        The c o m p u t a t i o n time f o r e a c h LP s o l u t i o n i s              primarily
d e t e r m i n e d b y t h e number of m i s s i o n s .              Hith              about 145
m i s s i o n s , a n LP r e q u i r e s , on t h e a v e r a g e ,       0.5 s e c o n d s for
problem g e n e r a t i o n ,           4.5   seconds for solution,                     a n d 100
pivots.          T h e LP c a l l i n g s t r a t e g y i m p l e m e n t e d is of v i t a l
importance           -    t h e most s i g n i f i c a n t i m p r o v e m e n t s a r e l o c a t e d
and made first. I n t h i s way, t h e t o t a l number o f i n d i v i d u a l
LP        solutions           i s minimized.              I n a l l experiments, t h e
s c h e d u l e s w i t h t h e lowest pseudo-cost                r e q u i r e fewer t o t a l
c a l l s t o t h e LP, a n d h a v e t h e h i g h e s t h i t r a t i o s .                  This
r e s u l t means t h a t a f t e r t h e f i r s t 2 0 t o 30 s i g n i f i c a n t
i m p r o v e m e n t s , e x t e n d e d c o m p u t e r time a n d more LP c a l l s a r e
not particularly effective.

       To      summarize t h e c o m p u t a t i o n a l p e r f o r m a n c e , t h e b e s t
s c h e d u l e s a r e o b t a i n e d b y t h e CGXET method for t n e Q A w i t h
20 d a y t i n e r e s o l u t i o n .     C o m p u t a t i o n time f o r t h i s iaethod is
about 52 seconds.                 With v a r i o u s b o u n d s on I n p o r t m i s s i o n s ,
t h e L P p o r t i o n of t h e model r e q u i r e s an a v e r a g e of 7
m i n u t e s , 36 s e c o n d s .     Thus,     good p o t e n t i a l s c h e d u l e s a r e
o b t a i n e d b y t h i s a n a l y t i c model i n a n a v e r a g e 8 . 5 m i n u t a s
u s i n g 70 ic words o f memory.




                                                   76
A.     APPXOACH




       Vith      the      o v e r a l l s u c c e s s of t h e h y b r i d a p p r o a c h , major
c h a n g e s i n t h e modeling a p p r o a c h are               not      necessary.             with
t h e e x c e p t i o n of t h e PATaOL r e q u i r e m e n t , t h e items d i s c u s s e d
in this         ChaFter           a r e i d e a s f o r p o s s i b l e i m p r o v e m e n t of
c o m p u t a t i c n a l p e r f orrnance a n d a d d i t i o n a l m a n a g e r i a l c c n t r o l s
t h a t would b e u s e f u l i n a p r o d u c t i o n l e v e l implementation.


       Improvement                (fewer t o t a l LP c a l l s ) w o u l d r e s u l t i f t h e
l t l i m i t i n g f * a n d morale-relat e d c o n s t r a i n t s c a n b e c o n s i d e r e d
i n t h e QA.                 T h e i n c l u s i o n of t h e s e c o n s t r a i n t s r e q u i r e s a
m e c h a n i s m t o c o n d i t i o n e a c h a s s i g n m e n t ( o n t h e t o t a l number
of         AHP t i m e - u n i t s        previously assigned each s h i p ,                           for
example).                   The     Graves- Whinston              method,           which            uses
probability                  in     determining             t h e next assignment, can
p o t e n t i a l l y ke extended.                So f a r , t h e n e c e s s a r y t h e o r e t i c a l
d e v e l o p m e n t s i n t h i s a r e a h a v e n o t b e e n made.

        The L i n e a r P r o g r a m m i n g m o d e l n e e d s e x p l i c i t m o d e l i n g of
PATROL          requirement missions.                       A new t y p e of p e n a l t y o r
n o n - v i o l a b l e c o n s t r a i n t l i n k i n g t h e e n d of o n e s h i p ' s p a t r o l
a n d t h e b e g i n n i n g of t h e n e x t s h i p ' s s h o u l d b e a d d e d . (The
n e x t s e c t i o n g i v e s f u r t h e r d i s c u s s i o n of t n i s p o i n t . )      Also,
a d d i t i o n a l e q u a t i o n s f o r A H P b a l a n c i n g f o r e a c h y e a r a s well
a s t h e whcle s c h e d u l e h o r i z o n may b e n e e d e d .               A difficult
d e c i s i o n i s where t o s p l i t t h e m i s s i o n s e q u e n c e i n t o t h o s e
A H P m i s s i c n s o f t h e f i r s t y e a r a n d t h o s e of t h e s e c o n d y e a r



                                                   77
    p r i o r t o t h e m i s s i o n d u r a t i o n s b e i n g known.

          T h e i n t r c d u c t i c n of human i n t e r a c t i o n a t s e v e r a l    points
    may a l s c b e w o r t h w h i l e .         Visual i n s p e c t k o n of t h e s e q u e n c e
    from t h e Q A s o l u t i o n , a s well a s p e r i o d i c i n s p e c t i o n of t h e
    p r c g r e s s of t h e LP c a l l i n g s t r a t e g y a n d s e q u e n c e s e a r c h e s
    may i m p r o v e c o m p u t a t i o n times a n d s o l u t i o n q u a l i t y .         The
    a d d i t i o n of a p o s t p r o c e s s o r t o i n f o r m t h e s c h e d u l e r of t h e
    s t a t u s o f t h o s e g u i d e l i n e s not a n a l y t i c a l l y m o d e l e d w o u l d
    a l s o b e a p p r o p r i a t e f o r a p r o d u c t i o n l e v e l program.



    B.    I!IPLXl4ENTATION



            S e v e r a l i m p l e m e n t a t i o n iaodif i c a t i o n s      w i l l   increase
    managerial c c n t r o l and f u r t h e r snhance computation speed. A
    more c o m p l i c a t e d s e l e c t i o n s c h e m e f o r d e c i d i n g w h i c h o n e of
    m u l t i p l e m i s s i o n s t o r e t a i n f o r r e q u i r e m e n t s t h a t c a n n o t be
    s p l i t w o u l d r e m o v e a t l e a s t 10 t o 15 L P c a l l s .                      Also,
    r e l o c a t i c n of i n f e a s i b l y a s s i g n e d m i s s i o n s t o t h e b e s t
    r a t h e r t h a n t h e f i r s t l o c a t i o n w o u l d be a n i m p r o v e m e n t .      As
    s h o w n by P i g u r e 1 4 ( C h a p t e r V I I ) , a c h a n g e t o a inore c o n v e x
    F e n a l t y s t r u c t u r e f o r t o t a l AHP time w o u l d l e a d t o k e t t e r
    workload b a l a n c i n g between t h e s h i p s .                     T r a v e l time c a n be
    e x p l i c i t l y modeled i n t h e LP m o d e l .                     T h e t r a v e l times
    b e t w e e n a r e a s f o r each m i s s i o n t r a n s i t i o n a r e k n o w n . F h e
    l o w e r / u p p e r b o u n d s on m i s s i o n d u r a t i o n s a n d t h e d e s i r e d
    t o t a l time f o r each ' s h i p c a n b e d y n a m i c a l l y c a l c u l a t e d
    d u r i n g t h e g s n e r a t i o n of e a c h LP a t n o i n c r e a s e i n LP
    s o l u t i o n time.            A l l of        t h e above l i s t e d ideas are easy
-   modifications t o t h e present implementation.

           A more d i f f i c u l t i m p l e m e n t a t i o n m o d i f i c a t i o n i s t h e
    e x p l i c i t m o d e l i n g of m i s s i o n s w i t h t h e PATROL r e q u i r e m e n t i R
    t h e LP model. T h e a l t e r n a t e n e t w o r k f o r m u l a t i o n d i s c u s s e d


                                                    78
i n C h a p t e r VI most e a s i l y accomodates t h i s c h a n g e .           In t h i s
formulation,             each m i s s i o n ' s s t a r t i n g time a n d d u r a t i o n a r e
a v a i l a b l e as primary v a r i a b l e s .              The l i n k i n g   of       one
m i s s i o n ' s f i n i s h t o a n o t h e r mission's s t a r t is e a s i l y and
d i r e c t l y acccmplished. T h i s enhancement w i l l remove t h e
o n l y d i f f i c u l t y with t h e q u a l i t y of t h e f i n a l p r o p o s e d
s c h e d u l e s , a n d a l s o ' remove t h e u a d e s i r e d d e t a i l p r e s e n t l y
r e q u i r e d i n t h e i n p u t s p e c i f i c a t i o n of PATROL r e q u i r e m e n t
missions.




                                              79
                                           APPENDIX A


                       SANPLE PROBLEM A N D RESULTANT SCHEDULE




Problem :
---v--7       Two s h i p s a r e a v a i l a b l e f o r a 7 week s c h e d u l e
  horizon t o f u l f i l l t h e s e requirements:
     1.    S e v e n w e e k s o f A l p a t w i t h PATROL r e i u i r e m e n t ;
     2.    Two w e e k s M a i n t f o r S h i p T w o b e t w e e n w e e k s 5 a n d 6 ;
     3.    Two w e e k s of O c e a n b e t w e e n w e e k s 2 a n d 5 ; a n d
     4.    Three w e e k s o f I n p o r t .

          Alpat       and   Ocean a r e Away H o m e p o r t m i s s i o n s . O n l y S h i p
   One c a n f u l f i l l t h e O c e a n r e q u i r e m e n t i n t h i s p e r i o d . The
   l a s t m i s s i o n o n t h e p r e v i o u s s c h e d u l e f o r S h i p One i s
   I n p o r t ; S h i p Two, Ocean.       T h e time r e s o l u t i o n for t h e Q A
   m o d e l i s o n e week.          The f o l l o w i n g list c o n t a i n s t h e c o s t
   p e n a l t i e s and desired goals:
     1.    I n f e a s i b l e assignment c o s t                                            700.
     2.    Time penalty cost rate                                                             50.
     3.    A9P p e n a l t y c c s t r a t e                                                 600.
     4.    Cruise penalty cost rate                                                      2000.
     5.    AHP d e s i r e d g o a l p e r s h i p                                   4 weeks
     6.    Cruise l i m i t                                                          5 weeks

   Transiticn Costs:
                            INPORT        ALP AT         OCEAN           t3AINT
           INPOBT             0            10             10              10
           ALPAT             10             0            700             700
           OCEAN            700            10              0             700
           H A 1 NT          10           700            700                0




                                                80
                 I n i t i a l D-Matrix of F i x e d C o s t s

  0          0         0           350              350           350   350             110    700
  0          0         0              0             3 50          350   3 50             50    700
  0          0         0              0             350           350   350                0   700
  0          0         0           350                 0          350   3 50               0   700
  0          0         0           3 50                0            0   350                0   700
   0         0         0           3 50             3 50            0     0              50    700
   0         0         0           350              350           350     0             100    700
70 0     700        700            350              3 50          350   3 50            706    700
  0        0          0               0             350           350   350             700    1.5 0
  0          0        0               0             3 50          350   3 50            70 C   100
  0          0        0            350                 0          350   350             700     50
  0          0        0            3 50                0            0   3 50            700        0
  0          0        0            350              350             0      0            700        0
  0          0        0            350              350           350     0             700     50


       I l l u s t r a t e d Cost Components:
       1 .     I n f e a s i b l e Assignment
       2.      T r a n s i t i o n frcm p r e v i o u s s c h e d u l e
       3.      Time Penalty
       4.      Single Hissicn Structure
       5.      PATZOL S t r u c t u r e

      A s s i g n m e n t Map :
           1   2   3           4      5        6    7        8      9 1 0 1 1 1 2 1 3 1 4
           1 1 1 1 2           7      2        8    9        4      3   5   6 1 4 1 3 1 0

   Assicrn F i x e d Cost                  -        I0
   T r a n s i t i o n Cost    ~           = 1460
   Total                                   = 1470
   E n t e r Switch         1470            Leave Switch                   410

   A s s i g n m e n t Map:
           1   2   3           4      5        6    7        8      9 1 0 1 1 1 2 1 3
           1 1 1 1 2           6      7        8    9        i)     5 1 0   2 1 4 1 3




                                                        81
                                                                    S h i p Two
                                                             M i s s i o n s Weeks
     Inport           1                                      Alpat            1?2,3,4
     Ocean            2,3                                    Inport           5
     Alpat            4f5,6                                  Haint            p 7
                                                             Inport

     T r a n s i t i o n Costs            =  20              T r a n s i t i o n Cost        =     40
     Penalties: Horizon                   =  50              P e n a l t i e s : Horizon     =     50
                      AHP                 = 600                                 Time         =     50
                                                                                 (Main t )   ----
                                               670                                                140
                                                     T o t a l = 810
--------- 2
Schedule


     -----Sohni p
     Missi
                     One
                      &Je&
     Inport           1,2                                    Alpat            p 3 , 4
     Ocean            3,4                                    Inport
     Alpat            5,6?7                                  Maint.           6f7

     T r a n s i t cn C o s t             = 20               T r a n s i t i o n Cost        =     30
     P e n a l t y : AHP                      ----
                                          = 600              P e n a l t y : T i(Main t )
                                                                                 me          =   ----
                                                                                                   50
                                            620                                                    80
                                                     T o t a l = 700.




                                F I FlAL SC!IEDULE

                                                   WEEK

                          1         .-
                                    3          3         4       5        6         7
     S t i I P ONE         1N   wivr      I    OCEAN-        I        ALPAT -
                                                                      r                 I
     SHIP     TWO     b
                                m
                                       ALPAT
                                          r          -           IN   1   MAINT
                                                                                        I




                                                   82
                                      L I S T O F REFERENCES




1.   B r a d l e y , G.     H.,    Brown,        G.      G.,       and Graves,       G.  W.,
     "Design            a n d I m p l e m e n t a t i o n of L a r g e S c a l e Erimal
     TransshiFment Algorithms",                         Naval P o s t s r a d u a t e School
     T e c h n i c a l Note
     - - - -
      - - I                        (NPS55BZBW76091) , S e p t e m b e r 1 9 7 6 , a n d
     W o r k i n g P a p e r No.         260,         W e s t e r n Yanagement S c i e n c e
     I n s t i t u t e , U C L A , November, 1 9 7 6 .

2.   Q r o u n , G.G.      a n d Graves,        G.W.,          "XS: An A d v a n c e d        Design
     M a t h e m a t i c a l Programming System,"                    (In Preparation)             .
3.   Conway, R. W . ,            Maxuell,           W.   L.,        and          Miller,     L.       W.,
     ----o r y of S c h e d u l i ? q ,
     The                                       Addison         -   W e s l e y , R e a d i n g , Bass,
     1967,

4.   Geoffrion,            A.    M. a n d G r a v e s , G.     W.,   I* S c h e d u l i n g

     P a r a l l e l Production Lines with                 Changeover             Costs:
     P r a c t i c a l A p p l i c a t i o n of a Q u a d r a t i c A s s i g n m e n t / L P
     Approachi1, Q E g r a t i o n s R e s e a r c h ,              v.           24,   No.    4,        p.
     595-610,   July, 1976.

5.   G r a v e s , G.  W., I* A C o m p l e t e C o n s t r u c t i v e A l g o r i t h m f o r
     t h e General Mixed L i n e a r P r o g r a m m i n g P r o b l e m " ,             Naval
     -----
     R e s e a r c h Loa&&s     G g r t e r l y , v.        1 2 , No. 1, p .             1-34,
     March 1965.

6,   G r a v e s , G.   W. a n d W h i n s t o n , A. B . ,           It    An             for
                                                                                   Algorithm
     t h e Q u a d r a t i c A s s i g n m e n t Problem",                 Manaqement Sciencs,
     v.    17, p.          453-471,         March, 1 9 7 0 .

7.   Koopmans,            T.     C.     and     Beckmann,             M.         J.,     Assignment
     Problems and               the     Location         of        Economic            Activitiest1,
     ------m e t r i c a ,
     Econc                      v.     25,    No.    1, p.         52-75,          1957.



                                                03
 8.   Lawler, E . W.,       "The Q u a d r a t i c A s s i g n m e n t Problein: A
      Brief Review , I 1       i n C q m b i n a t o r i a l Prosramminq; l e t h o d s
      ---
      a n d Q E A S t i o n s , 5 . Roy ( e d . ) , D .       Reidel, Dordrecht,
      H o l l a n d , p.   351-360, 1 9 7 5 .

 9.   P a n w a l k a r , S. S . , a n d    ISkander,     W.,           A   Survey    of
      Scheduling XulesI',              opegamon~     R e s e a r c h , v.   2 5 , No. 1,
      p.     45-61,        J a n u a r y , 1977.

10.   Prabhakar, T . ,  I* A P r o d u c t i o n S c h e d u l i n g P r o b l e m w i t h

      Sequencing Considerations",               Managemens Sccncs, v.
      21, p.  34-42,   1974.




                 \




                                              84
                                INITIAL DISTRfaUTION LIST



                                                                                       No.    Copies

1   .   Commandant (G-PTE-1/72)                                                                    2
        U.   S. Ccast G u a r d ,
        B a s h i n g t o n , D . C. 2 0 5 9 0

2.      Defense Documentation C e n t e r
        Cameron S t a t i o n
        A l e x a n d r i a , V i r g i n i a 22314

3.      Library, Code 0 1 4 2
        Naval P o s t g r a d u a t e School,
        Monterey, C a l i f o r n i a 93940

4.      Department Chairman, Code 52
        D e p a r t m e n t of C o m p u t e r S c i e n c e
        Naval P o s t g r a d u a t e School,
        Monterey, C a l i f o r n i a 93940

5.      D e p a r t m e n t C h a i r m a n , C o d e 55                                           1
        D e p a r t m e n t of O p e r a t i o n s R e s e a r c h
        Naval P o s t g r a d u a t e School,
        Honterey, California 93940

6,      Assoc. P r o f e s s o r G.G.Brown,                Code 55BW       (advisor)               1
        D e p a r t m e n t of O p e r a t i o n s R e s e a r c h
        Naval P o s t g r a d u a t e School,
        U o n t e r e y , C a l i f o r n i a 93940

7.      Assoc.      P r o f e s s o r G. H. B r a d l e y , C o d e 5 2 ( c o - a d v i s o r )    1
        D e p a r t m e n t of C o m p u t e r s c i e n c e
        N a v a l P o s t g r a d u a t e School,
        Monterey, C a l i f o r n i a 93940



                                                  85
 8.   Professor G . W. G r a v e s ( c o - a d v i s o r )          1
      G r a d u a t e S c h o o l of Hanagement,
      U n i v e r s i t y of C a l i f or n i a ,
      Los A n g e l o s , C a l i f o r n i a 90024

 9.   Lt, C h a r l e s E . S i b r e , USCG                        1
      1320 Dermond Road
      Drexel H i l l , P e n n s y l v a n i a 1 9 0 2 6

70.   Commander, U.S.Coast Guard P a c i f i c Area ( P - o s r )   1
      6 3 0 Sansome S t r e e t
      San F r a n c i s c o , C a l i f o r n i a 9 4 1 2 6

11.   Commander, U. S.Coast Guard A t l a n t i c Area (A-osr)      1
      Governors I s l a n d
      New York, N e w Y o r k 1 0 0 0 4




                                               86

				
DOCUMENT INFO