Wdv^r€ 13 \lf.rT Write your answers in the D2 answer book for this paper.

1. Four workers, chris (c), James (J), Katie (K) and Nicky (lI), are to be allocated to four tasks, l, 2,3 and 4. Each worker is to be allocated to one task and each task must be allocated to one worker.

The profit, in pounds, resulting from allocating each worker to each task, is shown in the tablebelow. The profit is to be maximised.

1 2 t 4

Chris t27 116 111 113

James 225 208 20s 208

Katie 130 113 lt2 tt4Nicky 228 212 203 2t0

(a) Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises thetotal profit. You must make your method clear and show the table after each stage.

(8)

(b) State which worker should be allocated to each task and the resulting total profit made.(2\

(Total 10 marks)

1 ) 3 4

C r03 ll+ Ilg n+J 5 LL 1.S TLK 100 ul lzq lt6N a 18 1_+ 2.o

@soSrStract etdr{rom2-30

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Eeducr Cotorwrc

(b)

1 2 J 4

C o $ tb lq-J o r+ 20 r+K o tt 2t r6N o Lb 25 tg

A lraaa. A r\* optrua-0-

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Worker Task

Chris a, t'tta\James 3 ( ?rr1'lKatie + ilq)Nicky 0zr

2 3 4

e-- t-,J ( s ,)_ It

=K- .f 'v_

N ) rj 1 z

1 2 3 4

C *i o ,ry IJ O/ ,A o /b,K l''7 ,1 ht oN o ,{l' '8t 4

Maximum total profit: t bL3

2. The table shows the least distances, in km, between six towns, A, B, C, D, E and F.

Liz must visit each town at least once. She will start and finish at A and wishes to minimise thetotal distance she will travel.

(a) Starting with the minimum spanning tree given in your answer book, use the shortcut methodto find an upper bound below 810 l<rn for Liz's route. You must state the shortcut(s) you useand the length ofyour upper bound.

(2)

(b) use the nearest neighbour algorithm, starting at A, to find another upper bound for the lengthof Liz's route.

a)(c) Starting by deleting F, and all of its arcs, find a lower bound for the length of Liz's route.

(3)

confident contains the

(1)

(Total 8 marks)

(d) Use your results to write down the smallest interval which you areoptimal length of the route.

A B C D E F

A t22 2t7 t37 109 82

B 122 110 130 t28 204

C 217 110 204 238 135

D t37 130 204 98 2ttE 109 t28 238 98 113

F 82 204 135 211 113

A B C D E F

A t22 2t7 t37 109 82

B 122 110 130 t28 204

C 217 110 204 238 135

D 137 130 204 98 2tt

E 109 t28 238 98 113

F 82 204 135 211 113

)

(a)

It

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(c)

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I 2 J 4 Supply

A 22 36 I9 3t .tl

B 29 35 30 36 15

C 24 32 25 4l 20

D 23 30 23 38 30

Demand 30 20 30 20

Tablelbelowshowsthecost,inpounds,oftransportingoneunitofstockfromeachoffoursupplypoints, A, B, C and o, to roo, ai*J points 1' 2:3;d4' It also shows the stock held at each

suppty point and the stock re;"ttJ;;h ;"lnal'i poi"t' A minimum cost solution is required'

Table 1

Table 2 shows an initial solution given by the north-west comer method'

i;i; t ;h"*t some of the improvement indices for this solution'

Table 2 Table 3

(a) Explain why a zero has been placed in cell 83 in Table 2'(1)

(b) Calculate the shadow costs

in your answer book.

(c)Takingthemostnegativeimprovementindextoindicaletheenteringcell,statethestepping-'"' ;;;'?""*;; sh;ld b" ;;Jto obtain the next solution. You must state your entering cell

and exiting cell. (3)

and the missing improvement indices and enter them into Table 3

(4)

1 2 3 4

A 30 5

B 15 0

C 20

D 10 20

1 2 3 4

A x x

B x x

C 8 2 x 1

D 9., x x

(Total 8 marks)

I 2 J 4

A 30 5

B 15 0

C 20

D 10 20

(a)

Inttw( so\,rhon r^nsdewnqaa*z,duPrerl ce).rs = fntn-t.fo a zaco ts NqurMA.

Table 2

(b)

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A x x '12. -tB 3 x x -1C 8 ) x

.l

D 9 2 x x

Table 3

(c) You may not need to use all of these tables

1 2 3 4

A I 5-E +o

B ttao o€C ao

D lo ?t

gnterrnr C<,\\ : AB

9=o J

qxrhnor Ce,{rt ' A2

lnapnarcd .soluttolaI 2 3 4

A 30 5 oB t5C 79D lo 20

4. Robin (R) and Steve (s) pray a two-person zero-sum game which is represented by the followingpay-off matrix for Robin.

S plays 1 S plays 2 S plays 3

R plays 1 2 1 J

R plays 2 1 -t 2

R plays 3 -1 J 1

Find the best strategy for Robin and the value of the game to him.

(Total 9 marks)

4.

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a p{a,,y t erdtr,P ( Phr,y 3 Pob--l-P

tb Sptorst v(R)uS phds 2 v(0 'S pta,is S v(A),

zp -l(rf) , !?-t :r-

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-. (srrart) Pto,f t grob' tR shA,la 6,J,2 N ev€&tz- shAr$ pt i\3 gob=!vCOa?

A three-variable linear programming problem in r, y and z is to be solved. The objectivemaximise the profit, P.The following tableau is obtained.

is to

Basic variable x v z r s t Value

r 1

2_1

2 0 1 0 _12 10

s 1+ r1 0 0 I _12 5

z 1

, 1

z 1 0 0 I2 5

P -5 -10 0 0 0 20 220

(a) Starting by increasing y, perform one complete iteration of the Simplex algorithm, to obtain anew tableau, T. State the row operations you use.

(s)

(b) Write down the profit equation given by T.

(1)

a)

(Total 8 marks)

(c) Use the profit equation from part (b) to explain why T is optimal.

5.

Qzlsi-i---za

g =Si Ltr' L *

$zSil=b

(a) You mqt not need to use all of these tableaux

b.v. x v z r s t Value

r 1

2 _12

0 1 0 _12 10

s Q+.rl"a 0 0 1 _1

2

-----__-:_/

z 1

21

, 1 0 0 I2

5

P -5 -10 0 0 0 20 220

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

r +.s o o t I

5-9

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

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6. 11 11

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A

B

Figure I

Figure I shows a capacitated directed network. The number on each arc represents the capacity ofthat arc. The numbers in circles represent an initial flow.

(a) State the value of the initial flow.(1)

(b) on Diagram 1 and Diagram 2 in tle a.nswer book, add a supersource s and a supersink T. onDiagram 1, show the minimum capacities of the arcs you have added.

a)

(c) complete the initialisation of the labelling procedure on Diagram 2 in the answer book byentering values on the arcs to S and T and on arcs CD, DE, DG, FG, FI and GI.

(3)

(d) Find the maximum flow through the network. you must list each flow-augmenting route youuse, together with its flow.

(3)

(e) Show your maximum flow on Diagrani 3 in the answer book.(2)

(f) Prove that your flow is maximal.

Q)

(Total 13 marks)

(a) Value of initial no* 13

(b) and (c)

6"6

Diagram I

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Diagram 2

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Diagram 3

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56

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7, A two-person zero-sum game is represented by the following pay-off matrix for player A.

B plays 1 B plays 2 B play! 3

A plays 1 1 -1 2

A plays 2 3 -1

A plays 3 5 -1 0

Formulate the game as a linear programming problem for player A.inequalities. Define your variables clearly.

Write the constraints as

(Total 7 marks)

B plays 1 B plays 2 B plays 3

A plays 1 1 -3 2

A plays 2 1 't -lA plays 3 5 -1 0

+ + Jr, "^^q)

va,l^Q- +o alerke- rv,u) gaa,te-

6t gL 83 lct V = vor,lue of ^e.rd 5aMrlo * -

Alsibftz ?- + 3 lei * ph$ 1,2,3 wr{n prab =ffg q 3 + p,rprTfsur.c.rpectrugs.hlltfs>ro

ob3er*r..^.c- !s }o n^a^pry\l.rre- ?=! 2 P-V=O .

Su\eob +o i v -5?t-2(r-1Pr, s oV - fr -?r02 -34. sOV -6fi-9Qt-kfs € o

fi+pz+?afl

A factory can process up to five units of carrots each month.Each unit can be sold fresh or frozen o. .u*.a".The profits, in f 100s, for the numfg1.6f *ii, *ru, are shown in the table.The total monthly profit is to be maximis;.

- --'- *w o,vv

Use dynamic programmine ro:-*l:rmine how many of the five units shourd be sord fresh, frozenand canned in order to *uii,,i.. the monthry p."fi;. i;;;;. maximum monthry profit.

Number of units

Fresh

Frozen

Canned

(Total 12 marks)TOTAL FOR pApER: ZS tuams

Stage State Action Dest. Value

I o o b o+frc,rh t o tssr

L L 0 rsr3 t 0 nir(t q 0 tso {s s 0 E6*

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I o ts{-o 3qs *7 o L O *t9= tS

t t {S +t+t r 10 trL o ?O+O e ?o

1 o O -l@= r2Ot 2 {+S+gJ t ta9r2- :10+(6 - ils3 o K) +O t t(D

a o lh 6 rrlo = ISOt 3 Qs * rr,o . la' F

L L fO +63 ,15J3 I tUD+qS z n$+ o \toto . lt;/.,

s o s O -l- l*S : l+JI ct qs + tso = t1s 'rL 3 eo *l,tO : t1O

3 z lfiO +83 .lg4 t tzo +qS : t6JS o r3of9 = r3o

3 5 o 6 4- rqs : IQJC^NA t q 3'S +163 o l{tr,t)

2 3 +s+lt6 '.ao93 L t2S +qO = Lr$ *Ll I t55+ QS z M)s o lis+ O ' rQS

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