The Smalltown Fire Department currently has 4 old ladder companies and 4 alarm boxes. The For a minimization integer programming problem, the optimal Z value of its LP max * X5 *1; +41: 5.1.
Consider the following nonlinear programming problem:
zlxzxfxg 2
To promote onaeampus sarery, the U oft security department is in the process or installing
emergency telephones at selected locations The depamnent wants to install the minimum
number of telephones provtded that each of the campus main streets is served b
6.15(a.)Construct dual problem for primal problem.(b)Solve the dual graphically and use this solution
To identify the shadow prices for the resources in the primal problem.
6.114(a)The sum of the number of functional constraints and the number of variab
56:171 Operations Research
Exam 3
8:00~9:30 pm, December 10, 2012
NOTES:
1. Twopage formula sheet is allowed.
2. Write your answer on this exam paper. DO NOT use your own paper. If you need more
space, write on the backside of this exam paper.
3. The tes
56:178 Digital Systems Simulation
Exam 1
5:00~6:15 pm, March 5, 2014
NOTES:
1.
2.
3.
4.
5.
Open book, open notes.
Write your answer clearly on this exam paper. DO NOT use your own paper.
The exam is worth 25 points.
Partial credit may be given for partial
IE:3700 Operations Research
Instructor: Yong Chen
MWF 12:30  1:20pm
Fall 2016
1
My Background
B. E. in computer science, Tsinghua
University, China, 1998
Ph. D., 2003, Industrial and Operations
Engineering, University of Michigan
Now Professor in Dept.
Assumption of LPs
When we write a problem as a linear
program, we are making a few assumptions
about the underlying process
Proportionality: The contribution of a decision
variable to the objective function or any one of
the constraints is proportional to
Theory of the Simplex
Method
For any LP with feasible solutions and a bounded
feasible region:
If there is exactly one optimal solution, then it
must be a cornerpoint feasible (CPF) solution
If there are multiple optimal solutions, then at
least two must
No Feasible Solutions
Min Z = 2x1 + 3x2
s.t.
x1 + x2 4
x1 + 3x2 36
x1 + x2 = 10
x 1 , x2 0
An LP is infeasible if an artificial variable
is greater than zero in a final solution
x1
x2
s1
e1
a1
a2
RHS
Z
2M1
0
0
M
0
4M3
306M
s1
1/4
0
1
0
0
1/4
3/2
a1
Introduction to Excel Solver
86
Loading Solver
The Solver Addin is a Microsoft Office Excel addin
program that is available when you install Microsoft
Office or Excel. To use it in Excel, however, you need to
load it first.
Click the Microsoft Office Bu
IE:2500 Engineering Economy
Spring 2016
Homework #6
Rate of Return Analysis
Exercise 1 Fast Carriage Company is considering to purchase a large truck. The expected useful life of
the truck is 14 years. The cost and net cash inflow associated with the new
324
DIGITAL COMMUNICATIONS
Set I
u,(~h
o~,
Se1 II
Set Ill
FIGURE PS13
However, a lower error probability is possible with coherent detection of FSK if llf is
increased beyond 1/2T. Show that the optimum value of llf is 0.715/T and
determine the probabili
C'HAl'H.R
~
Of'TIMt;M
RECEIVERS
FOR THE
AODITrVE
WHITI: GAUSSIAN
NOISE
(HANNEI.
323
b p(u, <!>). where u = Va2 + b2 and <P =tan "b ] a;
c p(u ).
Note: In (b) it is convenient to define ()=tan ' (m,/ m,) so that
m, = v'm; + m7 cos e.
m,
=:
v'm~ + m~ sin 8.
322
DIGITAL COMMUNICATIONS
$i(J
s 1(r)
A
A
0
A
0
!r 2
T
A
FIGURE PS8
process with autocorrelation function ti>. (r> N06(r), Let /.,(t), m = 1, 2, . , M,
be a set of M orthogonal equivalent lowpass waveforms defined on the interval
0 ~ t ~ T. Define
N,.
CHAPTER
5:
OPTIMUM
RECEIVERS
FOR THE
ADDITIVE
WHITE
UAU~SIAN
NOISE
CHANNEL
)2)
53 This problem deals with the characteristics of a DPSK signal. a
Suppose we wish to transmit the data sequence
l 1 0 1 0 0 0 1 0 1 l 0
by binary DPSK Let s(t) =A cos (21ifrt
320
DIGITAL COMMt'NICATIONS
matched filter was first proposed by North (1943) for use in radar detection,
and is sometimes called the North filter. An alternative method for deriving
the optimum demodulator and detector is the KarhunenLoeve expansion,
wh
CHAPTER
):
OPTIM\:M
RECEIVERS
FOR THE ADDITIVE
WHITE
GAUSSIAN
NOISE
CHANNEL
319
it follows that
)gh)
R( No req
SNR per bit. Hence,
PR No
(5516)
if we have PR/ No and the required
data rate that is possible.
where (th/ No)req is the required
SNR per bit,
318
OllHTAL
cmtMl'NKATIONS
antenna pattern. For example, the 3 dB beamwidth of a parabolic antenna is a
pproxima le ly
(5512)
so that GT is inversely proportional to e~. That is, a decrease of the beamwidth by
a factor of two, which is obtained by doub
CHAPTER
5:
OPTIMUM
RECEIVERS
The factor
FOR THE ADDITIVE WHITE GAUSSIAN NOISE fHANNEL
Ls = 41Ul
(~)2
317
(557)
is called the freespace path loss. Jf other losses, such as atmospheric losses, are
encountered in the transmission of the signal, they may b
316
DIGITAL COMMlJNK .TIONS
I
I
, ,. I
I
f
,
I
I
I
 . .
, " 
. .
I
.
'"
',
\
.~
.
I
'
\
FlGURE 552
'
Isotropically radiating antenna.
l
',
'
\
'
'
"'
'
, Amcnna,
.
"'
,
.,.,_.,
\
"
\
\
t
I
t
I
'
I
,'
t
I
'
about 18.3 dB, or approximately 70 times the
UfAf'll
f< 5
CW!l.\H'~I Kl<Tfl"lRS FOR THE ADl)rJI\ f WHITF. (iAIJ.\SJA!\
a bit error for one hop (signal
repeater in the chain) is
transmission
JJ_5
:\OISI 'H.\'>:1'1
from one repeater
to the nc xt
Since errors occur with low probability, we may ignore
CHAPTER
5:
Ol"flMl .M RECEIVERS
FOR THE
ADDITIVE
WHITE
GAl sslA:1
:>;OISE
313
CHA:>;~U
,
I ;s;
.
.
~ .
Q.
"
=
z
'o"'
s,
:c
11)4
FIGURE 54l>
I
I
I
Pb=~
e
111
"lpl = 0.6 '
"
.
'
I"\. "\.
lpl,O.~
I
I() b

Probability of error for noncoherent
detection
314
DIGITAL COMMUNICATIONS
Trasmiued
signal
Received
r(t)
.<(I)
nGURE
551
Mathematical model of channel with attenuation
and additive noise.
Attenuation
a
= signal + n(t)
QJ(I)
Noist
n!)
where "th is the transmitted energy per bit and !Ne> is the power
312
mc;rrAL COMMUNICATIONS
arbitrarily small provided that the SNR per bit 'is _greafer than the Shannon
limit of 1;6 dB. The cost. (or increasing M is the bandwidth required to
transmit the signals. For M ary FSK, the frequency separation between
adjac
CHAPTER
5:
OP'Tl\.ll'~
RECEIVERS
FOR THE
ADDITIVE
WHITE
GAU~SIAN
NOISE
Substitution of this result into (5442) and integration
probability of a correct decision as
Pc =
M
L (
exp
1 .)" ( M  1 )  l
[
3 1
CllA'l~tl
over x yields

(5445
l
n ~'
the
n
CHAPTER
S:
OPTIMUM
RECEIVERS
FOR THE ADDITIVE
WHITE
GAUSSIAN
NOISE
3()9
CHANNEL
cos2rif.t
!~()di
)(
sin2it/,r
J~(d I
cosh(f.+ll/)1
0
l' ( )dr
sin21c(f. + Af)t
Envelope
or
Received
Output
JI ( )df
squarelaw
signal
decision
0
detector
c<Y.2llcfw_/,.+(M
l)
J0
DIGITAL COMMUNICATIONS
Let us make a change in variables in the joint pdfs given by (5435) and
(5436). We define the normalized variables
R,. ='=me~m'2
_s
2
v'
+
(T
(5437)
Clearly, rm('=: CTRm cos E>m and 'ms= uRm sin em. The Jacobian of this t