practice exercise for final exam[1]

7/24/2019 Practice Exercise for Final Exam[1]

1/10

MTS254 Introduction to Management Science

Solution of Practice Exercise for Final Exam

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! = ?GG

= ! G

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? = !GG

= ? !GG

= @ !>G

@ = >G


2/10

!

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Max x41

Subject to

x41=x

13+x

12

x12+x

32=x

23

x13+x

23=x

32+x

34

x34 =x41

x13! 200,x

12 !150,x

32 !100,x

23!100

x34 !150

All variables " 0

!" $%& '(& )*++*,-./ -.)*012'-*.

H,'(- )*( -()6%2I ,- D,1F2( .3%'(J K*.) ,/ )*( 92,),9.& 0.)*L


3/10

H,'(- )*( -()6%2I ,- D,1F2( .3%'(J )*( 92,),9.& 0.)* ,/

8-B-;-D-MN

3 !

45

! 6

784

78 74

7478

74 95

9574

6 75

!"79

6 78

786

5 6

65

3 5

53

3 !

45

! 6

784

78 74

7478

74 95

9574

6 75

7579

6 78

786

5 6

65

3 5

53


4/10

8" :2%'-++& ;*++&/& -% /*-./ '* (2%& .&, >*1=+'@ 2.A %'2))B C(&0& 20& 2 '*'2+ *)

53 =+'@ 2.A 93 %'2))B C(& >*%' *) '(& ?2%-> *1%H '(& .=1?&0 *) A&+=I& >*1*1=+'@B C(& ;*++&/& )&&+% '(2' -' 1=%'

(2%& 2' +&2%' 5 A&+=I& >*1=+'@N -) *1=+'@B O'2)) 1&1?&0% A* )&&+ %*1&,(2' P(-.&% )*0 '(& %'2)) 2% +=%'(2.9*,('(+(-) %5 +.C,+F+ /).55 4(&FC( 9%+0F)(2/

4A-=F-4(2.9*,('(+(-) %5 )%).& /).55 9%+0F)(2/

4A+=%'(2.9*,('(+(-) %5 )%).& /).55 9%+0F)(2/

4R-=F-4(2.9*,('(+(-) %5 3F41()

4R+=%'(2.9*,('(+(-) %5 3F41()

It might not be clear but the following is the goals of the problem:

Goal 1: At least 5 faculty deluxe computers

Goal 2: At most 20 faculty deluxe computers (as many as 20 so cannot exceed 20)

Goal 3: Staff deluxe computer is at most 50% of faculty deluxe computerGoal 4: At least total 30 faculty computers. More than 30 is ok but less than 30 is not ok

Goal 5: At least 10 staff deluxe computers

Goal 6: At least total 20 staff computers. More than 20 is ok but less than 20 is not ok

Goal 7: Used at most 100,000 budget. More than 100,000 is not ok but less than 100,000 is ok


5/10

S,-,+,T( 4!-+4?++4=++4@-+4>-+4A-+4R+

$F3U(9) )%7 DB+ 4!-- 4!+=> B(&FC( 5.9F&)Q

DB+4?-- 4?+=?G B(&FC( 5.9F&)Q

-GN>DB +$B+4=--4=+=G B(&FC( 2.),%

DB+ DP+4@--4@+==G E%).& 5.9F&)Q$B+4>- -4>+=!G B(&FC( $).55

$P+ $B+4A--4A+=?G E%).& /).55

=>GGDB+?GGGDP+?GGG$P+=>GG$B+4R--4R+=!GGGGG 8F41()

.&& '.2,.3&(/ !G

5) Consider the following network model. The transportation costs are shown

above the arcs. The supply capacities are shown in the left of plant nodes and

the demands are shown on the right of retail nodes. Develop the LP model tofind how to satisfy the demand with minimal cost.


6/10

6) Consider the network below. Develop the LP model to find shortest path

from node 1 to node 6.


7/10

7) The bank head quarter is trying to locate the minimal number of branches to

serve all 20 counties below. If a branch is set up at a county, it can serve all

adjacent counties as well. The list of adjacent counties is shown below. Develop

LP model to find the minimal number of branches to cover all counties.


8/10

8) A company has 3 clients to serve. There are 3 available project leaders tocomplete the projects. The table below shows the completion of the project is

the particular project leader leads the project. Develop the LP model to find how

to assign the project leader to client such that the total completion time is

minimized.


9/10

9) Consider the project relationship diagram below. The normal time and crash

time and related crash costs are shown in the table below. Develop the LP

model to reduce project completion time to 12 weeks.


10/10

xi = earliest start time of task i (not the same as earlier solution)

yi= crash time of task i

Minimize1000ya+2000y

b+1000y

c+1000y

d+1000y

e+500y

f + 2000y

g+3000y

h

subject to

Task A and B no constraint

xc !x

a+ (2 "y

a)

xd !x

b+ (3"y

b)

xe!x

c+ (2 "y

c)

xf !x

c+ (2 "y

c)

xg !x

e+ (4 "y

e)

xg !x

d+ (4 "y

d)

xh !xg+ (5"yg)

xh!x

f + (3"y

f)

xh+ (2 "y

h) #12

ya #1;y

b# 2;y

c#1;y

d #1;y

e# 2;y

f #1;y

g # 3;y

h#1;

All variables! 0

practice exercise for final exam[1]

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