3315 Fall 2023 Homework-1
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School
San Jacinto Community College *
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Course
3315
Subject
Industrial Engineering
Date
Jan 9, 2024
Type
Pages
5
Uploaded by SuperHumanIron7208 on coursehero.com
IE 3315
Homework
1.
A farmer has 3 farms to grow 3 crops. The farms are 500, 800, and 700 acres, and the water allocation for each
farm is 600, 800, and 375 acre-feet, respectively. The crops suited for this region include sugar beets, cotton and
corn. These crops differ primarily in their expected net return per acre and their consumption of water. The USDA
has set a maximum quota for the total acreage that can be devoted to each of these crops. Each can be grown on
any of the 3 farms, and multiple crops can be grown on a farm. At least 50% of each farm must be used. Formulate
the problem to maximize the net return.
2.
An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A
contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon and 3% carbon. The
costs per ton for scraps A and B are $100 and $80, respectively. The specifications of the alloy are as follows:
•
The aluminum content must be at least 3% and at most 6%.
•
The silicon content must lie between 3% and 5%.
•
The carbon content must be between 3% and 7%.
Formulate a linear program that can be used to determine the amounts of scrap A and B that should be used to
minimize the cost of creating 1000 tons of the special alloy.
3.
A commercial store plans to advertise in different outlets. There are three possible outlets to choose from: TV,
radio, and magazines. Each ad on TV, radio, and magazines reaches the following number of people (in millions).
The advertising budget is $10,000. There cannot be more than 5 TV ads total. The total number of ads
should be between 15 and 25. Formulate an integer program to maximize the number of people reached.
4.
Solve the following LP problem manually using the simplex tableau approach.
Maximize z = 2x
1
+ 5x
2
s.t.
x
1
+ x
2
<
12
3x
1
+ x
2
< 1
x
1
, x
2
> 0.
Maximum quota
(acre)
Water
consumption
(acre-foot / acre)
Net return ($/acre)
Sugar beets
600
3
1,000
Cotton
500
2
750
Corn
400
1
250
Outlet
# people who see ad
Cost
TV
20
600
Radio
12
300
Magazine
9
500
5.
Solve the following LP problem manually using the simplex tableau approach.
Minimize z = 2x1
–
4x2
s.t.
x
1
+ x
2
< 10
2x
1
+ x
2
< 16
x
1
, x
2
> 0.
6.
Two linear programming problem solved by GAMS are due as stated in the notes.
A free trial license is available
at
https://www.gams.com/try_gams/
. Turn in a screen shot of the GAMS solution. The first problem is as follows.
7.
Two linear programming problem solved by GAMS are due as stated in the notes.
A free trial license is available
at
https://www.gams.com/try_gams/
. Turn in a screen shot of the GAMS solution. The second problem is as
follows.
8.
Two integer programming problems solved by GAMS are due as stated in the notes. The first problem is the
following IP.
Minimize
z = 3x
1
+ 2x
2
s.t.
x
1
+ 2x
2
≤ 11
4x
1
+ x
2
≤ 8
x
1
, x
2
≥ 0
x
1
, x
2
integer
9.
Two integer programming problems solved by GAMS are due as stated in the notes. The second
problem is the
following binary IP.
Minimize
z =
−
x
1
+ 2x
2
s.t.
−
x
1
+ 9x
2
≤ 8
−
4x
1
+ 2x
2
≤16
x
1
, x
2
∈
{0,1}
10.
Minimize
f (
x) = |x+3| + x3
s.t.
x
∈
[-2,6].
Minimize
𝑧𝑧
= 2
𝑥𝑥
1
−
4
𝑥𝑥
2
+ 3
𝑥𝑥
3
s.t.
5
𝑥𝑥
1
−
6
𝑥𝑥
2
+ 2
𝑥𝑥
3
≥
5
−𝑥𝑥
1
+ 3
𝑥𝑥
2
+ 5
𝑥𝑥
3
≥
8
2
𝑥𝑥
1
+ 5
𝑥𝑥
2
−
4
𝑥𝑥
3
≤
4
𝑥𝑥
1
,
𝑥𝑥
2
,
𝑥𝑥
3
≥
0.
Maximize
𝑧𝑧
= 3
𝑥𝑥
1
+ 2
𝑥𝑥
2
s.t.
4
𝑥𝑥
1
− 𝑥𝑥
2
≤
5
4
𝑥𝑥
1
+ 3
𝑥𝑥
2
≤
12
4
𝑥𝑥
1
+
𝑥𝑥
2
≤
8
𝑥𝑥
1
,
𝑥𝑥
2
≥
0.
11.
Determine if
f
(x)
=
−
x + x
2
is convex, concave, or neither for on R
1
.
12.
Solve the following NLP.
2
2
1
2
1
2
1
2
1
2
1
2
1
2
minimize
s.t.
1
cos(
)
,
0.
x
x
x
x
x
x
x
x
π
+
+
=
≤
≥
2
2
1
2
1
2
1
2
1
2
1
2
1
2
minimize
s.t.
1
cos(
)
,
0.
x
x
x
x
x
x
x
x
π
+
+
=
≤
≥
(13,14) Two nonlinear programming problems solved by GAMS are due as stated in the notes. These problems must
be formulated by the student, so each student will have different problems. A copy of your problems and a
screen shot of the computer output with the solutions are required. Screen shots work.
(
)
1
2
2
2
1
2
1
2
2
2
1
2
1
2
13
13
7
7
(15)
maximize
2
5
s.t.
2
5
13.
Answer:
,
(16)
minimize
2
s.t.
4
3
+
4.
x
x
x
x
x
x
x
x
x
x
+
+
=
+
+
=
≤
(
)
Answer:
0.89,
1.79
−
−
(
)
1
2
2
2
1
2
1
2
2
2
1
2
1
2
13
13
7
7
(15)
maximize
2
5
s.t.
2
5
13.
Answer:
,
(16)
minimize
2
s.t.
4
3
+
4.
x
x
x
x
x
x
x
x
x
x
+
+
=
+
+
=
≤
(
)
Answer:
0.89,
1.79
−
−
(
)
1
2
1
2
1
2
1
2
1
2
(17)
maximize
s.t.
4
,
0.
Answer:
2, 2
(18)
maximize
s.t.
4.
Answer:
x
x
x
x
x x
x
x
x
x
⋅
+
≤
≥
⋅
+
≤
unbounded objective function
(
)
1
2
1
2
1
2
1
2
1
2
(17)
maximize
s.t.
4
,
0.
Answer:
2, 2
(18)
maximize
s.t.
4.
Answer:
x
x
x
x
x x
x
x
x
x
⋅
+
≤
≥
⋅
+
≤
unbounded objective function
19. Acme Appliance must determine how many washers and dryers should stocked. It costs Acme $350 to purchase
a washer and $250 to purchase a
dryer. A washer requires 3 sq. yd of storage space, and a dryer requires 3.5
sq. yd. The sale of a washer earns Highland a profit of $200, and the sale of a dryer $150. Acme has set the
following goals (listed in order of importance):
Goal 1:
Highland should earn at
least $30,000 in profits from the sale of washers and dryers.
Goal 2:
Washers
and dryers should not use up more than 400 sq. yd. of storage space.
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