Dynamic programming is both a mathematical optimization method and a computer programming method. .in computer science, if a problem can be solved optimally by breaking it into sub-problems .

1. Nadar Saraswathi college of
arts and science Theni
Department of CS and IT
Presented by:
S.Swetha
I M.sc(IT)
Dynamic programming

2. Title:
Reliability design.
The Traveling salesperson
. Flow shop schedulin
g.

3. Reliability design
• In reliability design, the problem is to
design a system that is composed of
several devices connected in series. I
f we imagine that r1 is the reliability o
f the device.

4. Reliability design
• Solve a problem with a multiplicative
optimization function
• Several devices are connected in series
• ri be the reliability of device Di
• Reliability of the entire system
• Duplicate : multiple copies of the same
device type are connected in parallel use
switching circuits

5. Reliability design
n devices Di, 1<=i<=n, connected in series
Multiple devices connected in parallel in each stage

6. Multiple copies
stage in contain mi copies of Di
P(all mi malfunction) = (1-ri)mi
Reliability of stage i =1-(1-ri)mi

7. Reliability design
• Maximum allowable cost of the system
Maximize
Subject to
Mi >=1 and integer, 1<=i<=n
• Assume ci>0
ui =







  i
n
j
i c
c
c
c /
)
(
1

8. Travelling salesperson
• In the traveling salesman Problem, a
salesman must visits n cities. We can
say that salesman wishes to make a t
our or Hamiltonian cycle, visiting eac
h city exactly once and finishing at th
e city he starts from. There is a non-
negative cost c (i, j) to travel from the
city i to city j.

9. The traveling salesperson problem
• mail pickup
– n+1 vertex graph
– Edge <i,j> distance from i to j
– Tour of minimum cost
• Permutation problem
– n! different permutation of n object while
there are 2n different subset of n object
n! > O(2n)

10. The traveling salesperson problem
• Tour : simple path that starts and ends at v
ertex 1
• Every tour : edge<1,k> for some k v-{1} e
ach <k,1>
• Optimal tour : path(k,1)
– Shortest k to 1 path all the vertices in V-{1,k}
• Let g(i,s) be the length of a shortest path starting
at vertex i, going through all vertices in S and term
inating at vertex 1
})}
,
1
{
,
(
{
min
})
1
{
,
1
( 1
2
k
V
k
g
c
V
g k
n
k






})}
{
,
(
{
min
)
,
( j
S
j
g
c
S
i
g ij
S
j






11. The traveling salesperson problem
• Directed graph and edge length matrix c

12. The traveling salesperson problem
• Thus g(2, ) = c21 = 5, g(3, ) = c31 = 6, and g(4, ) = c41 = 8. We
obtain
g(2,{3}) = c23 + g(3, ) = 15 g(2,{4}) = 18
g(3,{2}) = 18 g(3,{4}) = 20
g(4,{2}) = 13 g(4,{3}) = 15
g(2,{3,4}) = min{c23+g(3,{4}),c24+g(4,{3})} = 25
g(3,{2,4}) = min{c32+g(2,{4}),c34+g(4,{2})} = 25
g(4,{2,3}) = min{c42+g(2,{3}),c43+g(3,{2})} = 23
g(1,{2,3,4}) = min{c12+g(2,{3,4}),c13+g(3,{2,4}),c14+g(4,{2,3})}
= min(35,40,43}
= 35


 

13. The traveling salesperson problem
• Let N be the number of g(i,s), that have to be com
puted before g(1,V-{1}) i, computed for each valu
e of |s|
• n-1 choices of i
• The number of distinct sets of S of size k not inclu
ding 1 and i

i 






 
k
n 2
2
2
0
2
)
1
(
)
2
)(
1
( 







  n
n
k
n
k
n
n
N
)
2
(
lg 2 n
n
orithm
a 

)
2
(
)
( n
n
O
need
space 

14. Flow shop scheduling
• Flow-shop scheduling is a special ca
se of job-shop scheduling where ther
e is strict order of all operations to be
performed on all jobs. Flow-shop sch
eduling may apply as well to producti
on facilities as to computing designs.

15. Flow shop scheduling
• The process of a job requires the perf
ormance of several distinct tasks.
• In a general flow shop we may have n
jobs each requiring m tasks T1i,T2i,...
.......Tmi,1≤i≤n, to be performed.
• Tji is to be performed on processor pj
, 1≤j≤m.

16. Two jobs have to be scheduled on thre
e processors. The task time are given b
y the matrix j
j= [2 0]
[3 3]
[5 2]