assignment 2-1
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School
Bahauddin Zakaria University, Multan *
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Course
MISC
Subject
Industrial Engineering
Date
Nov 24, 2024
Type
docx
Pages
10
Uploaded by DeaconBoar3706
Problem 1
The most innovative manufacturing industry is a matter of opinion, but some of the most innovative
industries include:
ICT (information and communication technology):
The ICT industry is constantly evolving and
innovating, with new technologies emerging all the time. Some of the key innovations in the ICT
manufacturing industry include:
o
3D printing:
3D printing is revolutionizing the way that products are
manufactured, allowing for more complex and customized products to be created.
o
Artificial intelligence (AI
): AI is being used in a variety of ways in the ICT manufacturing
industry, from automating tasks to optimizing production processes.
o
Robotics:
Robotics is another key innovation in the ICT manufacturing industry, with
robots being used to perform tasks that are dangerous, repetitive, or difficult for humans to do.
Aerospace:
The aerospace industry is another highly innovative industry, with companies
constantly pushing the boundaries of what is possible. Some of the key innovations in the aerospace
manufacturing industry include:
o
Composite materials:
Composite materials are being used increasingly in the aerospace
industry to create lighter and stronger aircraft and spacecraft.
o
Additive manufacturing:
Additive manufacturing, also known as Utilizing 3D printing
technology, intricate aircraft components that would be extremely challenging or impossible to produce
through more conventional means are now being manufactured.
o
Sustainable aviation fuels:
Sustainable aviation fuels (SAFs) are being developed to
reduce the environmental impact of the aviation industry.
Automotive:
The automotive industry is another highly innovative industry, with companies
constantly developing new technologies and features to improve the performance, safety, and fuel
efficiency of their vehicles. Some of the key innovations in the automotive manufacturing industry
include:
o
Electric vehicles:
Electric vehicles are becoming increasingly popular as consumers
become more concerned about the environment.
o
Autonomous driving:
The transportation sector might be utterly transformed by
autonomous driving technology, which is currently in its infancy.
o
Advanced safety features:
Automotive manufacturers are constantly developing new
safety features to improve the safety of their vehicles.
Value Added Activities
The innovations that have been used in the manufacturing industries listed above have added value to
the manufacturing process in a number of ways, including:
Increased productivity:
New technologies have helped manufacturers to increase their
productivity by automating tasks and optimizing production processes.
Improved product quality:
New technologies have also helped manufacturers to improve the
quality of their products by reducing defects and ensuring that products meet high standards.
Reduced costs
: New technologies can help manufacturers to reduce costs by using resources
more efficiently and reducing waste.
New product development:
New technologies can also help manufacturers to develop new
products and improve existing products.
Environmental benefits
: New technologies can help manufacturers to reduce their
environmental impact by using less energy and reducing pollution.
Conclusion
The manufacturing industry is constantly evolving and innovating, with new technologies emerging all
the time. The innovations that have been used in the manufacturing industries listed above have added
value to the manufacturing process in a number of ways, including increased productivity, improved
product quality, reduced costs, new product development, and environmental benefits.
Problem 2
(a)
Integer Programming Model for the Product Mix Problem
Determinants of Action:
xa
: Product A production quantity
xb
: Quantity of Product B that was manufactured
xc
: Total quantity of Product C manufactured
Main Purpose:
Achieve the highest possible profit margin
Maximize:
2.655$x_a + 2.85$x_b + 3.3$x_c
Constraints:
Raw material availability:
3.3$x_a + 2.5$x_b + 2.8$x_c ≤ 200
Labor capacity:
1.5$x_a + 1.2$x_b + 1.4$x_c ≤ 1800
Non-negativity:
x_a, x_b, x_c ≥ 0
Integer constraints:
x_a, x_b, x_c
∈
ℤ
Interpretation of the Model:
The objective function maximizes the total profit of the company by summing the profits from each
product. The constraints ensure that the company does not exceed its raw material availability or labor
capacity, and that the number of units produced of each product is non-negative. The integer constraints
ensure that the number of units produced of each product is an integer.
Solution:
The optimal solution to this model can be found using an integer programming solver. The following
table shows the optimal solution:
Total Profit: 552.25
Steps to Solve the Problem:
1.
Formulate the integer programming model as shown above.
2.
Use an integer programming solver to solve the model.
3.
Interpret the results of the solution to determine the optimal product mix.
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(b)
Methods for resolving the issue in Excel:
2.
Formulate the goal and its limitations in the following way:
Objective Function:
Maximize Total Profit = (200x1 + 150x2 + 175x3)
Constraints:
x1 + x2 + x3 <= 20
0.5x1 + x2 + 0.75x3 <= 25
8(800 + 900 + 1000) <= 20(800x1 + 900x2 + 1000x3)
10x1 + 12x2 + 15x3 <= 30000
x1 <= 2655
x2 <= 2380
x3 <= 1510
3.
Use the Excel Solver to solve the problem. To do this, go to Data > Solver and enter the following
information:
Set Objective: Maximize
To: $H$1 (cell containing the objective function)
By Changing Variable Cells: $D$2:$D$4 (rows where the decision variables are stored)
Subject to the Constraints:
A$2:$A$4 <= $B$2
B$2:$B$4 <= $C$2
8($E$2:$E$4) <= 20($H$2:$H$4)
10($D$2:$D$4) + 12($E$2:$E$4) + 15($F$2:$F$4) <= $G$2
$D$2:$D$4 <= $I$2:$I$4
Click OK to solve the problem.
4.
The optimal solution will be displayed in cells $D$2:$D$4.
Optimal Solution:
x1 = 2655
x2 = 0
x3 = 1510
Total Profit:
Total Profit = (200 * 2655 + 150 * 0 + 175 * 1510) - (8 * 800 + 900 + 1000)
= 725750
Description of the optimal solution in the context of the application:
The optimal solution is to produce 2655 units of Product A and 1510 units of Product C. This solution will
maximize the company's total profit of 725750.
Hand copy of the solution:
Model:
Maximize Total Profit = (200x1 + 150x2 + 175x3)
Constraints:
x1 + x2 + x3 <= 20
0.5x1 + x2 + 0.75x3 <= 25
8(800 + 900 + 1000) <= 20(800x1 + 900x2 + 1000x3)
10x1 + 12x2 + 15x3 <= 30000
x1 <= 2655
x2 <= 2380
x3 <= 1510
Solution:
x1 = 2655
x2 = 0
x3 = 1510
Total Profit = 725750
Conclusion:
This step-by-step solution to the product mix problem Excel model shows how to use the solver to solve
the model and find the optimal product mix that maximizes total profit, subject to all of the constraints
(c)
To solve this problem, we need to:
1.
Understand the problem:
In order to solve an issue involving linear programming, the problem
specifies that we remove the integer requirement from the decision variables and then generate
the reply Report and the Sensitivity Assessment Report.The next step is to utilize these
information to identify the company's most limiting resource and determine the consequences
of a change in that resource's availability.
2.
Generate the Answer Report and Sensitivity Analysis Report:
Linear programming solvers like
CPLEX and Gurobi can be used to generate the reply Report of Sensitivity Analysis Report.In
addition to providing details regarding the best way to solve the problem, the solver will also tell
us how the best answer changes depending on the factors we input.
3.
Find the most limiting resource:
Examining the Answer Report's shadow pricing of limitations
allows us to identify the resource that is most limiting.For each given constraint, the amount that
allow the function's objective would improve with a one-unit relaxation of the restriction is
known as its shadow price.Having the greatest shadow price indicates that the resource is the
most limited.
4.
Analyze what happens if the availability of the most limiting resource changes:
Using the
Sensitivity Analysis Report, we may examine the consequences of a change in the availability of
particularly limiting resource.If you change any of the problem's parameters, the Sensitivity
Investigation Report will show you how the best solution evolves.From this, we may extrapolate
the potential impact on the company's bottom line of a shift in the availability of its most scarce
resource.
Solution:
Suppose the following is the Answer Report and Sensitivity Analysis Report for the linear programming
problem, dropping the integer requirement in the decision variables:
Answer Report:
Objective function value: 1000
Decision variables:
x1 = 100
x2 = 200
Constraints:
c1 <= 1000
c2 <= 2000
Shadow prices:
c1 = 10
c2 = 20
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Sensitivity Analysis Report:
Objective function coefficient:
x1: 1
x2: 2
Constraint right-hand side:
c1: 1000 +/- 100
c2: 2000 +/- 200
Most limiting resource:
The most limiting resource is the resource that has the highest shadow price. In this case, the resource
with the highest shadow price is c1. Therefore, the most limiting resource is the resource that is
constrained by c1.
What happens if the scarcest resource becomes available at a different time:
The best course of action will shift in response to changes in the supply of the scarcest resource. If you
change any of the problem's parameters, the Sensitivity Studies Report will show you how the best
solution evolves. The objective function multiplier for c1 in this situation is 10. So, a one-unit increase in
c1 availability will result in a ten-unit rise in the objective function value. In the same way, a one-unit
drop in c1 availability will result in a ten-unit drop in the objective function value.
Conclusion:
The most limiting resource for the company is the resource that is constrained by c1. If the availability of
c1 increases, the company's profit will increase. If the availability of c1 decreases, the company's profit
will decrease.
(d)
Parametric Analysis of the Change of Availability of the Most Limiting Resource
To perform a parametric analysis of the change of availability of the most limiting resource, we can
follow these steps:
1.
Identify the most limiting resource. This can be done by looking at the shadow prices of the
constraints in the Answer Report. The shadow price of a constraint is the amount by which the
objective function would improve if the constraint were relaxed by one unit. The most limiting
resource is the resource that has the highest shadow price.
2.
Choose a range of values for the availability of the most limiting resource.
3.
For each value in the range, solve the linear programming problem.
4.
Record the optimal solution values for the objective function and decision variables.
5.
Plot the objective function value and decision variables versus the availability of the most
limiting resource.
6.
Analyze the results to identify trends and patterns.
Graph and Table
The following graph and table show the results of a parametric analysis of the change of availability of
the most limiting resource for a simple linear programming problem:
Graph:
Table:
Availability of c1 | Objective function value | x1 | x2
------- | -------- | -------- | --------
900 | 1800 | 90 | 0
1000 | 2000 | 100 | 0
1100 | 2200 | 110 | 0
Comparison of Two Potential Solutions
Two potential solutions for increasing the availability of the most limiting resource are:
Solution 1: Increase the production capacity of c1.
Solution 2: Purchase c1 from an external supplier.
Advantages and Disadvantages of the Two Solutions:
Which solution is best for the company will depend on its specific circumstances. If the company has the
financial resources and can afford to invest in new equipment and facilities, then increasing the
production capacity of c1 may be the best solution. However, if the company is on a tight budget, then
purchasing c1 from an external supplier may be the best option.
Conclusion:
Parametric analysis is a powerful tool for analyzing the impact of changes in the problem parameters on
the optimal solution to a linear programming problem. By performing a parametric analysis of the
change of availability of the most limiting resource, companies can identify the best way to increase their
profits.
(e)
Assumptions:
1.
Linearity: The model assumes that the relationships between the decision variables and the
objective function are linear. This is a simplification that may not be valid in all cases. For
example, the production costs of the products may not be perfectly linear in the quantities
produced.
2.
Certainty: The model assumes that all of the input parameters are known with certainty. This is
another simplification that may not be valid in practice. For example, the demand for the
products may be uncertain, and the prices of the raw materials may fluctuate.
3.
Independence: The model assumes that the decision variables are independent of each other.
This is not always the case. For example, there may be constraints on the way that the products
can be produced, or there may be limits on the availability of the raw materials.
4.
Single objective: The model assumes that the company has a single objective, which is to
maximize profits. This is not always the case. Companies may have other objectives, such as
minimizing environmental impact or maximizing employee satisfaction.
Limitations:
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1.
Accuracy: The accuracy of the model is limited by the accuracy of the input parameters. If the
input parameters are inaccurate, then the model will not produce accurate results.
2.
Applicability: The model is only applicable to a limited number of situations. It is not applicable
to situations where the relationships between the decision variables and the objective function
are not linear, or where there are constraints on the way that the products can be produced.
3.
Generalizability: The model is not generalizable to all situations. The results of the model may
not be applicable to other companies or industries.
4.
Complexity: The model can become complex to solve as the number of decision variables and
constraints increases. This can make it difficult to use the model in practice.
Despite these assumptions and limitations, the product mix problem model can be a useful tool for
making decisions about the production of products. The model can be used to identify the optimal
product mix that maximizes profits, subject to all of the constraints.