Practice MidT
xlsx
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School
Bethel University *
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Course
622
Subject
Industrial Engineering
Date
May 24, 2024
Type
xlsx
Pages
23
Uploaded by jhilary
Minutes per garment
Machine
Shirt
Pant
Cutting
3
6
1413
Sewing
6
2
1218
Assembly
5
4
1317
Minutes Available
A factory makes shirts and pants. Each shirt yields a profit of $2, while each pant gives a profit of $3. They can sell at most 190 pants. Each garment spends time on three machines as shown in the table. The number of pants must be at least 30% of the total number of garments made.
1) Formulate an algebraic model to this scenario. Make sure the objective function and constraints are in standard form (linear combination of coefficients and variables on the left hand side and no variables on the right hand side).
2) Insert an excel graph on this sheet and add the constraints, an isoprofit line, and highlight the feasible region.
A company makes two models of cars, called the Armitage and the Bronny models. After deducting all direct costs, each Armitage car gives a profit of $1020, while each Bronny car gives a profit of $360. There is a contractual obligation to produce at least 25 Armitage cars per week. Every car goes through three assembly-lines. Every Armitag
car takes 50 minutes in Assembly-line 1, 90 minutes in Assembly-line and 25 minutes in Assembly-line 3. Every Bronny car takes 40 minutes
in Assembly-line 1, 45 minutes in Assembly-line 2, and 75 minutes in Assembly-line 3. Every week, Assembly-lines 1, 2, and 3 are open 80, 108, and 100 hours respectively. They want the production of Armitag
cars to be no more than 80% of the total production.
1) Formulate an algebraic model to this scenario. Make sure the objective function and constraints are in standard form (linear combination of coefficients and variables on the left hand side and no variables on the right hand side).
2) Insert an excel graph on this sheet and add the constraints, an isoprofit line, and highlight the feasible region.
Armitage
Bronny
Available
Assy 1
50
40
minutes
80
Assy 2
90
45
minutes
108
Assy 3
25
75
minutes
100
Min prod
25
cars
Max prod
80%
ttl cars
Profit
$1,020 $360 ge 2, s ge
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hours
hours
hours
A brewery producing two types of beer has an algebraic model as shown. 1) Given the algebraic formulation, set up an Excel solver solution of the model an
find the optimal values for X1 and X2.
2) Describe your optimal model includeing final values and binding constraints.
3) Extra Credit: how would you calculate the actual proportion that resulted in the
proportion constraint. (as it would appear in a problem description before transformation into standard form)?
\begin{array}{l}
x_{1} \text{= the number of litres of lager made x_{2} \text{= the number of litres of ale made ea
\end{array} \\
\\
\begin{array}{rl}
\text{Maximize } & 1.2x_{1} + 0.9x_{2} \\
\text{subject to } \\
\text{production of lager } & 1x_{1} + 0x_{2} \g
\text{production of ale} & 0x_{1} + 1x_{2} \ge 3
\text{total production } & 1x_{1} + 1x_{2} \le 80
\text{process 1} & 4x_{1} + 2x_{2} \le 22000 \\
\text{process 2} & 2x_{1} + 6x_{2} \le 38000 \\
\text{process 3} & 3x_{1} + 8x_{2} \le 50000 \\
\text{proportion} & 1x_{1} - 0.7x_{2} \le 0\\
\end{array}
nd e
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each day} \\
ach day} \\
ge 2000 \\
3000 \\
000 \\
A worldwide shortage of microchips has prompted an entrepreneur to open a microchip assembly plant. The plant will produce two kinds of microchips, called M
and M2. The operations at the plant has an algebraic model as shown. 1) Setup the initial tableau to run the simplex method. Do not iterate the tableau,
identify the incoming and exiting variables.
M1 , \begin{array}{l}
x_{1} \text{= the number of M1 chips m
x_{2} \text{= the number of M2 chips m
\end{array} \\
\\
\begin{array}{rl}
\text{Maximize } & 25x_{1} + 75x_{2} \text{subject to } \\
\text{assembly-line 1} & 5x_{1} + 10x_{
\text{assembly-line 2} & 12x_{1} + 8x_{
\text{assembly-line 3} & 8x_{1} + 14x_{
\text{total production} & 1x_{1} + 1x_{
\text{production M1 chips} & 1x_{1} + 1
\text{production M2 chips} & 0x_{1} + 1
\text{proportion} & 0.8x_{1} - 0.2x_{2}
\end{array}
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made each week} \\
made each week} \\
\\
{2} \le 102000 \\
{2} \le 105000 \\
{2} \le 136000 \\
{2} \le 12000 \\
1x_{2} \ge 3000 \\
1x_{2} \ge 7500 \\
} \ge 0\\
A farmer owns 500 hectares of land in an arid region. The government gives him u
to 1,000,000 cubic metres of water for irrigation each year. In addition, he may purchase up to an additional 800,000 cubic metres of water per annum at a cost o
$0.20 per cubic metre. He grows corn, peas, and onions. The net revenue per hectare of each commodity (excluding the cost of purchased water, if any) and th
water requirement in cubic metres per hectare are shown in the table.
He wishes to diversify his crop in case one commodity suffers an unanticipated fa
in price. Therefore, no commodity may occupy more than 50% of the total area planted, nor may any commodity occupy less than 10% of the total area planted.
1) Find the optimal solution using excel Solver.
2) Describe your findings: objective value, optimal solution, and constraint observations.
3) Remember all constraints must be in standard form.
Commodity
Corn
$600 Peas
$1,200 Onions
$900 Free Available
Purchase Available
$0.20 Ttl land (hectares)
500
Min land / crop
10%
Max land / crop
50%
Revenue per Hectare
Purchase cost / m
3
up of he all
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4000
8000
2000
1000000
800000
Water Requirement
(cubic metres per hectare)
A metalworking company buys sheet metal from which they make swings and slides for children's playgrounds. They then outsource the rustproofing of the swings and slides, and sell the finished products. They buy the metal at a cost of $
per kilogram (kg). Each swing requires 3 kg of metal, while each slide requires 6 k
Each product spends time in three operations: cutting; polishing; and assembly. Th
times in minutes per unit are in the table.
Each day, the shop is available for 6.5 hours of productive time. There are four cutting machines, one polisher, and one person to do the assembly. However, up t
an extra 80 minutes of assembly time can be purchased for $2 per minute. The rust-proofing firm charges $30 per hour. When rustproofing swings, they can rust-
proof 5 swings per hour; when rust-proofing slides, they can rust-proof 16 slides p
hour. The metalworking company sells its products to a wholesaler at $190 per swing and $75 per slide. The market requires that at least two slides be made for every swing made. We define all variables are on a daily basis as shown.
1) Find the optimal solution using excel Solver.
2) Describe your findings: objective value, optimal solution, and constraint observations.
3) Remember all constraints must be in standard form.
Cutting
Polishing
Swing
25
12
Slide
18
10
$5 kg. he to -
per
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Assembly
16
11
The production manager of a company needs to determine next month’s production plan for the company’s ten products. Each product is measured (an integer) per month. The products use six resources: assembly line 1; as
line 2; painting; dryers; packaging; and storage. Each month, the produced are sent to storage in the warehouse. At the end of each month, all content
storage
are sent to the customers. The first five resources are each measured in ho
while storage is measured
in cubic metres (m
3
). The requirements per unit for each product, and the a
of each resource available are shown in the table.
i. There should be at most 4,500 units produced.
ii. There should be at least two units of product 3 for every unit of the comb
production of products 6 and 8 produced.
iii. The total production of product 4 should be no more than the combined production of products 2 and 7.
iv. The combined production of products 1 and 5 must be at most twice the
production of product 9.
1) Find the optimal solution using excel Solver.
2) Describe your findings: objective value, optimal solution, and constraint observations.
3) Remember all constraints must be in standard form.
Product
1
Assembly 1
2
Assembly 2
0
Painting
0
Dryers
0
Packaging
0.5
Storage
0.25
Profit/unit
$2.10
Storage cost
$2.50
in units ssembly units ts in ours, amount bined e
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2
3
4
5
6
7
8
1
0.5
0.75
1.5
0.25
0
0
0
0.3
0.45
0.5
0.65
1
0.8
0.2
0
0.4
0.5
0.65
1.5
0.1
0.3
0
0.8
0.2
0
1
0.3
0.1
1
0.2
0.1
0.65
0.1
0.2
0.1
0.5
0.45
0.4
0.25
0.1
0.1
$3.20
$1.60
$4.80
$1.20
$4.30
$3.50
$1.80
per m
3
9
10
Available
0
0
2100
hours
2
3
1500
hours
0.15
2
1000
hours
0.2
1
1000
hours
1
0.5
1600
hours
2
0.3
1300
cubic meters
$5.50
$3.90
A car assembly plant produces both sedans and SUV’s. The operations at th
have been algebraically modeled as shown. 1) Setup the initial tableau using the matrix method as shown in class. Find
entering and exiting variable.
2) Iterate the tableau ONE time. Identify the entering and exiting variable.
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he plant d the \begin{array}{l}
x_{1} \text{= the number
x_{2} \text{= the number
\end{array} \\
\\
\begin{array}{rrcrr}
\text{Maximize } & 300x_
\text{subject to } \\
\text{assembly-line 1} & 6
\text{assembly-line 2} & 1
\text{assembly-line 3} & 5
\text{total production} & 1
\text{production of sedans
\text{production of SUV's}
\text{proportion} & -.65x_
\end{array}
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r of sedans made each week} \\
r of SUV's made each week} \\
_{1} & + & 400x_{2} \\
6x_{1} & + & 9x_{2} & \le & 13500 \\
10x_{1} & + & 7x_{2} & \le & 14400 \\
5x_{1} & + & 8x_{2} & \le & 11700 \\
1x_{1} & + & 1x_{2} & \le & 1800 \\
s} & 1x_{1} & + & 0x_{2} & \ge & 450 \\
} & 0x_{1} & + & 1x_{2} & \ge & 630 \\
_{1} & + & 0.35x_{2} & \le & 0\\
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