Homework #2 2602327 Quantitative Business Analysis
NX ,
:S
,
Xz
:
8
1) Graphically solve the following problem
isoprofit ;
401--841-5×2
​Maximize profit = 8X
1
+ 5X2
✗ 1=10
​subject to
✗ 2:10
X1 + X2 ≤ 10
​
X1 ≤ 6
-
M=4
✗
②
slope
:&
2=2
2X1 + 4X2 ≥ 8
✗ 1=2
Xz :3
3X1 -2X2 ≥0
slope
:
-1
,
✗ 1=8
xi "
X1 + 2X2 ≤ 8
c-
​
-
61-2×2=8
⑤
2×2=2
X1, X2 ≥ 0
a) What is the optimal solution? 16,11
,
profit
:S }
b) What is the range of optimality for this problem?
'
Xz :|
81611-5117
c) What is the range of the change in right-hand side for all constraints?
d) What is value of the dual price for all constraints?
e) What would be happened if the coefficient of X1 of the objective function change from 8 to 2
f) What would be happened if the coefficient of X2 of the objective function change from 5 to 9
g) Change the right-hand side of constraint 1 to 11 (instead of 10) and resolve the problem. How
much did the profit increase or decrease as a result of this?
h) Change the right-hand side of constraint 2 to 7 (instead of 6) and resolve the problem. How much
did the profit increase or decrease as a result of this? Looking at the graph, what would happen if
the rigth-hand side value to go to 4?
i) Change the right-hand side of constraint 5 to 10 (instead of 8) and resolve the problem. How much
did the profit increase or decrease as a result of this?
(b)
-
E- Cx ,
so
E
tz
-
Cxz
let
let cx , :S
-
e-
•
I
¥
-
Cxz
NY
§
t
,
-164
,
Cxzt
( ✗ 270
-
-
t
-
cx
t
•
ca
-
of
Range
Cx
-
t
N
,
Cx
16
,
-
Cx
change
1- ✗
in
)
tower limit at
✗
upper limit
,
=
point
12,37
2
:
upper
limit
=
-
N
point
16,11
:
16
upper limit at
18,01=8
lower limit at
61-2101--6
=
at
point
16,47
point
16,01
61-2147=14
constraint
lower limit
lower limit
5th constraint
constraint
grd
RHS
3161 -2117
N
=
N
-
point
21611-4111
=
16,11
16
,
E
-1g
-
2.5
Cx , >12.5
E N
Upper limit at
2=7
upper limit
2nd
Cx
4th constraint
lower limit at point 16,1
,
-
,
!
2.SE Cx , EN
1st constraint
X
¥
-
5
Of Cx , £1b
(C)
:S
,
d)
✗
sub
7
:
,
✗
2×2--8
t
,
5
sth constraint
Dual price of
t
⑤
-
t
sub
2×2--8
it
:
constraint
②
-
into
2
2nd
price of
Dual
:b
,
-
1-2×2--9
,
⑤
-
into
2
②
s
61-2×2=9
;
✗
✗ it
,
i.
Profit
:
8171
dual price
i.
-
-
Dual price of
:
The
:
profit
of
hl
It
:
it
It
'
.
.
.
Profit
.
:
constraints ①
solution
because
is
8167 t 5
dual price
(y)
:X
?
,
:b
55.5
=
t 2. s
:
they
are
non
-
binding
constraints
.
changed
optimality
out of the range of
is
2
=
because
:O
,
optimal
.
.
changed
will be increased
the
③ ④
,
changed
won't be
It will not affect
g)
7
58 s
The profit will be decreased
.
:
,
tss
will be
f) The decision
.
sly )
/
The decision
e)
+
:
X
=
a
by
profit
because g
19-57×1
because
=
is
in
the range of
optimality
.
4
constraint ①
is a
non
-
binding
constraint and the
change
is in the
RHS
is
binding
Total
is
profit
binding
Total
profit
constraint
and the
will be
increased
constraint
and the
will be increased
change
by
the
range
the
range
.
As
a
result
As
a
result
,
we can use
dual
price to calculate
we can use
dual
price to calculate
(7-6115.5)=5.5
change
by
is within
is within
(10-8112.5)
=
5
.
,
range
✗
2
n
"
g
t
i.
f
-
I
-
.
"
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