02 CuNg t3 u( w.i "

yor" ,r,*ers in the D2 ans\rer book lbr this paper.

1.

The table shows the leasl distances, in km, betBeen fivc hidlng places, A, B, C, D and E.

uAgent Goodie hasto leave a secret message in cach oflhe hiding places. Helvill start and finishat A. and u,ishes to minimise the total distance travelled.

(a) Usc Prim's algorithm to find a milimrun spanning tree for this Dctwork. Make your order ofarc selection clear

(2)

(b) Use your answer to part (a) to dctcnninc an initial upper bound for thc lcngth ofAgent Goodie'sroute.

(1)

(c) Show that there are two nearest neighboLrr routes $hich start from A. State these routes andtheir lengths.

(3)

(d) State the better upper bound frofir vour answcrs to (b) and (c).(1)

(e) Staning by dclcting B, and a,11 ofits arcs, lind a lo*.cr bound 1br the length ofAgcnt Goodie'sroutc.

(4)

(t) Consider 1,our answers to (d) and (e) and hcrce state an optimal route.(1)

(Total 12 mark$)

B (l D E

l5 t9 25 20

B 15 l5 15 25

C 19 15 22 Il) 25 15 22 l8

20 25 lt 18

l. (a)

56

@o

AgrBC,C€,gD

(,

G./r re1-,,.,4)

\nr!6\ r.rpper bornrl ' %*z - ll2-

(c)

ri

ftR

r, !S- B -C-B - DrS t!

rE ls€-D-AE -C:A

-s+-+8

B C D F,

t5 19 25 20

B l5 15 l5 25

C 19 15 )) 1t

D 25 t5 2) l8

E 20 25 t1 t8

(d)

bdler'tg4.. hoo"A ' :lE

(c)

AC/ eglE D RMSa = .F8 tts(a]&) tlS{&D)

_i& _

C D F,

C Qs)D 25 22 ($)

AD

ftr.rc..a,tD

-'. lwc gessrlora-, so ltE t5 fl&to?t raaal soi.*ror -

(0

A -B.D-E-GA

Thc table sho\vs thc cost.points. A, B. C and D. roal eacl suppi] lloint andrequired.

in pound-s. of transporting one unit of stock from eaclr of four supplyeach of tluee dclnand points. 1. 2 and 3. lt also sho$,s thc stock ireklthe stock recluired at each dcmand point. A minimum cost solution is

) Supply

l0 20 l8

B l5 1 l3 11

C 21 t5 t2 2t

D 9 21 t8 12

Dsrrand 21 l8 20

(a) Use the noltli-rvest comer method to obtain an inirial sol lion(t)

(b) Taking Dl as the enrcring cell, usc the stepping stone mcthod to find an impl.ovcd solutioll.\4ake )oul route clear

(2)

Perform one ftlrther iteration of thc stepping stone method to obtain an impr.ovcd solution.You must make your method olcar-b), stating your-shador costs. improvcme;t indices. loLlte"<rtcring ucll and c\i ing c(ll

(4)

Delermine whcther your current solution is optirnal, giving a reason for \.orLr answer.(3)

(c)

(d)

(Total 10 marks)

2.

(a)

e-vltar r ^) c.-!l : Dto, .r

a-Y r&r ^! cc.lr : B i

t qoro *e..I s o luf(l,ra-

Cos{- ' 964*

S\or2o.r: Costr,

1 l Supply

10 11 20 l8

B l5 7 l3 t,l

C 2.1 t5 t2 21

D 9 21 IE t2.

Demand )-t IE l0

N\rc I ) 3 Suppl)

r8 t8

B 1 5 1,1

C l3 I 2.1

l) l2- t).

Demand t7 18 20

3 Supplyrr l8

B "-g t*g1.1

C r1-E I +9 21

I) +et IL .B 12

Dcmand 27 t8 20

2 3 Supply

r8 t8

B r+ 14

C tl l? 21

D q ? t2

Demand 2'7 llJ 20

rO 2-L

o

-l

2 3 Suppl1

6o) lt '1-O 18

B r3 r;) r3 1,1

C 2-* (s) 6) 21

I) (r) 2-t (,s) t2

Dcm;rnd 2'7 t 20

1 2 3 Supply

x -tl I t8

B zo Y 7 1,1

C 2.r x x 21

D x o x 1)

Demand 27 t8 20

l P\or rr\x-uaaJr tqAra.+ -

:- r.ot' op!!!!4 .

S|ee+..\, slorq L

.e-xr+( r.) CdA ' q3

\1p".*nA s!t+^ro!E 2.-

("st u {. u1

tortt

-E^.-D-*r Cr^-Q,o

-*

+-t

2. l Supplyt8-o +er 18

B r+14

C +-E tl *E 21

D '+6 "-i t2

Denand 2',7 l8 20

2 l Supply

l5 s t8

B rs 14

(l I 1o 2t

D l2 12

Demand 2',7 l8 20

I 2 l Supply

6.) 6n a6 18

R l3 @ l3 14

C 2* 60 (.) 21

D 6) zt rE 12

Demand 2',7 ti3 20

1 2. 3 Supply l,r'\F.o\ a-^,r.l-G} l nD rO.'r- -

,fF ieta.ts\\x- I *'Pfo,re rre.&r.adax .'. S6(()t\a^ t r

ol*tn44 .

(Total 10 marks)

x x t2 18

R q x 7 1,1

(l rI x * 2l

D x tl lr 12

Dcmand 21 18 20

r16D

The labelling procedure has been used and the rcsult drawn on Diaglam I ir

(c) Usc Diagram I to find the midimum flow through the network-flo\i-augmcnting roLlte you use, together with its llorv.

(d) Draw a maaimum 1lo$, pattem on Diagram 2 in vour answer book.

(e) Prole that the llow shown in (d) is maximal.

.12cut C

Figure 1

Figurc 1 shows a capacitated, directcd nchvork. Thc numbcr on each arc represents the capacity ofthat arc. The numbers in circles represent an initial tlow

(a) State the va1$e of tl're initial flou.(l)

(b) Statc the capacity ofclrt CL.

0)

31@

cut c

the answer-book.

Y()u musl Iist e2ch

(1)

(2)

(2)

(Total 10 marks)

(a) (b) Capacit) ofcul +l (sl.s*z*to+$+++tL)

- +ll

(c.)

(d.)

SftDGIT -I

": t\rn are.l- _ tw^x +lrr^) ilnc-ore.rvr

(^ru c*-t c+r, cF/ A0/ Bo/ oElEG, E1^)

SftCF+{T -3 SGEDF+TT.2

- ,h. ^ catr

H

Diagrrm 2

(e)

hrir cr,r,t p^Sgr^) +\cor1)\^- Satvrateclr otecS CH rCF rA0,BDrBE 4o.^.o,rds Sra6-.

t- -'

ogII

.t

mE\x flod . 53

^re- Ai ^dL ["wt a r ,r* .$* tlrro5\ CI as rt{^rrlt IDG- +12urarr.A r. .our.2 I

4. A tr.o-person zero-sum game is represcnted by the followillg pay-offmatrix for playerA.

B plays I B p1als 2 B plays 3

A plays 1 5 1 6

A plays 2 I 3

Aplays 3 1 I )

(a) Reducc lhe game so that playd B has only t\vo possible acrions.

(b) Write down the rcduccd pay-olfmatrix for playcr B.

(c) Find thc bost shategy lbr player ts and the value ofthe game to him.

(l)

(2)

(8)

(Total 11 marklt)

1.

ts plavs 2 B plays 3

A plays 1

A plays 2

A plays 3

a) (o!o-aa a. $ ia'*t."o*<-r G \.r.r,r,r.- 1 6tr elrc-{jvrr-\r- ra (alor.r..r= B r! ,r ^^.L!l., {,,tzrr.^- ttee-Co r..,spr )rtl'5 g{e.c- r.r. (.rlprrrn \ . : .

Co !.ra,rr.r- \ c\^ l:€- )c-l'r-t<A .Qree 1!<- cluae

:t

c) 6 9li1r ? Probt?I gt*Xs 3 9rob, l-P

rF ftet,{qtI vcg)' -*p+C(?). -l?*Lfi pruiLz vts), ? -3i-d . fo-xtarr.-$: v(e): p:l(....p) "39 2

v(s)

5

+

optrmal !o\et\e! n* n$4- \,llqre cho':r

t ftl *^'t fr3

.1o P+6': ?1 1-

6 irsp :lP::f,

? - f{ €v6s !Pta3t -geb, &

t ?taX1 - prota: fu

Y-- v(s) = ,;e-t r3

5. In soiving a three-\,ariable maliimising lioear prograrnming problem, the following tableau wasohtaincd aftcr thc first itcration.

Rasic r:rliahle t, t

1 2 0 0 8

I I 3 0 0 22

0 0 1 1t

0 0 0 I t5

(a) State which variable $'as increased lirst. giving a reason fbr your a11swer.

(1)

(b) Solve this linear proglam ing problem. \,Iake yoLrr method clearbv stating the rou, opcrations

,vou Llse.

(8)

(c) State the tinal valuc ofthc objective firnction andthefinal \dluesoieachva able.(2)

(Tot'at 11 rnarks)

b.v. x\v\ z r s t Value

r -l 2 0 I 0 I

')s -1 .' 0 0 1 22

z 11 I 0 0 I 11

P 2 -5 0 0 0 I2 t5

S" E;2"* *

S'ae:3 = +r,

0. = n i \ = l\

(b) You may not need to use all of these tableaux

b.v. x v z r s t Value Row Ops

rI o I o -l I o \+ +i lcr,uo(?-

7L I o o -3 z -t zo TA AL

z o o t -5 j -t 3+ +{ mo(zP o o o I I

6L

+5 +l r.r-r.^r P2

c) ?"t3A.=AO. $: 197 7=9*f=orS=Orb=6'

lwrcasg*'fo5r,lr ltl.lw&;,t ,

$=$:-!:-t.aa\--2Dx

: *:-1 = -!-q

'. z f,

b.v. v z r s I Value Row Ops

I -!L I o IL o .L

2- + (l +a

S/v.a o o -3

, I -tI _D -3 r.e..:R t

Z ,L o I -* o t

L ? - nc,^r RlP .L

L o o 51 o g l8 + B naro [.1

c.) Z w^s l^r-.c,aJeot {.rsl a.'r tt tS na/r) ot b'V'

6. Three workcrs, Haniet, Jason and Kathcrirre, are to bc assigned to thrce tasks, 1,2 and 3. Each\\'orkcr rnust be assigned to iust one task and each task must bc donc bv just one \\.oltc1.

The amount cach pcrson \\:ould eam. in pounds, [.hile assiened to each task is sho\\n in thc tablehelo$

Tosl, I Task2 Task.l

Hanict 251 2,13 257

Jason 211 141 255

Kalheline )49 )52 )16

The total incomc is to be maximised.

(a) N4odily the table so it can be used to find thc ma\imum income.

(b) Fonrulate the abo\e sitlration as a llnear proqrammlng problem. Youdecision variables and rrakc vour objective l'unction and constraints clcar

(l)

nlust deline your

(7)

(Total 8 marks)

6,

Task I Task 2 Task 3

Hariet 25t 243 257

241 247 255

Katherine 249 252 246

6l+olj, lo 2-

!) let XLi : ( t rt raprl^,-r i ts atloca"lgl tq t"+j, tO o r-,,rrsci e f H,5rl.r.1 ie t.rr!"!l -

. oLJc-c!.r*+ u lo tl!-,^ @!L tA ,l^".<-

C, 6 X*' . !t ILrq + l3Xrr rLQYsr +2Xrrt 8Xu r5&.ztlrDGr

.A)Ci, ,l tX 'r j ' t

txq tr t* 'r-; z \

-. Not.- - or\@- ,apt.rv\el o\\totu Aoi,\ ts +o.r.^cf . -0^r4

!4r-e!.la n....ol- ru]a€= o,e*rtorl !o q4\9! tv!A.\abL{e *.*dr aa^lz t.\zo.-r 2

7. Niscl has a business renling oLrt his fleet of bicycles lo tourists.

At the stirft ofeach ycar Nigcl must decide on one of two actions:

. K(ep r.. rlcet ol h,.).1(.. 'ri rr'_ing rriItcrrrr.e cu,t..

. Rcplacc his flcet ofbicycles.

The cost ofkeeping the flcct ofbiclclcs,the cost ofrcplacing thc tlcct ofbicyclesand thc annual income are dependent on the age ofthc flccr ofbiclrclesTable I sho\rs these amounts. in €1000s.

Age of fleet ofbicyclesI )ear

old2 ycars

old3 ycars

oldrl years

old

Cost of kccping (f 1000s) 0 2 l 8

Cost of rcplacing (11000s) 1 8 9 l0

Incomc (ll000s) lt 8 5 2 0

Table I

\igei has a re* fleet of bicyclcs norv and *'ishcs to maximise his total plofjt over the next fourYEAIS,

He is planning to sell l'ris business at the end ofthe lourth year.

The amoLrnt Njgel will receive $,ill dcpcnd on thc age ol his flcct of bicycles.

Thesc amounts, in ll000s. are sho\\'n in Table 2.

Age of fleet of bicvclcsar end of,1rh ycar

I "vcarold

2 vearsold

3 yearsold

,1 J,earsold

Amount r-eccived at cndof,lth year (II000s) 6

Table 2

Complete the table in thc ar]Swq book to detennine Nigcl s best straregy to rraximise his total pro{itover lhe next four ycars. You musl state the action he should take cach year (keep or replace) andhis total ploilt.

(Total 13 marks)

E NI)

TO'IAL FOR PAPER: 75 N{ARkS

7. You mdt ot need lo use all the tu)tts iti lhis ldb/(,

l

1,1

StageYalue(ProIi,

L+ 5-2.

State(Age of ileetof bicyclesat starl of

ycar)

Dcst.(Agc of fleetofbicyclesat end of

year)

(Sell)

I +2 3:0Rcplacc 6 r ll 9:8*

6+lt-8 = qIt_ +g-l : l1 *G +l\ -+: lO

f5-2=ll\l+t\-B:lq*cl+G-t,lb*Ir f tt -?c t5

lV+t-\:z\*[b f ll -a: zo2l +rt -O = 32*

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