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Bahauddin Zakaria University, Multan *
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Industrial Engineering
Date
Nov 24, 2024
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docx
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Uploaded by DeaconBoar3706
Problem 1:
Among the world's most inventive industrial sectors, the automobile industry stands out. Technologically,
operationally, and environmentally, the automobile industry has come a long way in the last few
decades.
The introduction of cutting-edge production methods is largely responsible for the automobile industry's
recent period of unprecedented innovation. Robots and automation, for instance, have contributed to
more efficient and high-quality production. Carbon fiber & aluminum are only two examples of the
modern materials that have contributed to lighter and more fuel-efficient automobiles.
New vehicle technologies are another example of an innovative shift in the car business. One way that
transportation's negative effects on the environment have been mitigated is by the rise of hybrid or
electric automobiles. Furthermore, autonomous vehicle technology may soon change the way people
travel forever.
Here are a few concrete instances of developments in the car industry:
Lean manufacturing: The goal of the production technique known as "lean manufacturing" is
to increase efficiency while decreasing waste.Lean manufacturing methods have been widely
used by the car sector.
Production methods that use just-in-time (JIT) delivery of materials and components to the
production line are known as just-in-time manufacturing (JIT).Inventory expenses may be
reduced and efficiency can be improved using this.
Technology that helps with both product design and production is known as computer-aided
design (CAD) or computer-aided manufacturing (CAM).Accuracy and efficiency in the production
of automobiles have been enhanced by computer-aided design and computer-aided
manufacturing.
Welding, painting, and assembly are just a few of the many automotive-related jobs that make
use of robotics and automation.As a result, the production process becomes more efficient and
of higher quality.
New materials:
New materials, such as carbon fiber and aluminum, are used to make vehicles
lighter and more fuel-efficient.
Hybrid and electric vehicles:
Hybrid and electric vehicles are more environmentally friendly than
traditional gasoline-powered vehicles.
Self-driving cars:
The way we travel might be drastically altered by self-driving automobiles.
The production process has been greatly enhanced by the developments in the automobile sector. One
strategy that has contributed to better quality and lower costs is lean manufacturing. Efficiency and
inventory costs have been enhanced by the use of JIT production. Manufacturing processes have
become more precise and efficient because to computer-aided design (CAD) and computer-aided
manufacturing (CAM). Robots and automation have made manufacturing more efficient and of higher
quality. Vehicles are now softer and more fuel-efficient thanks to the usage of new materials. A lessening
of transportation's negative effects on the environment has been aided by the advent of electric and
hybrid vehicles. The transportation industry stands to be dramatically altered by the advent of
autonomous vehicles.
Finally, when it comes to production, few sectors are as inventive as the automobile business. The
production process has been greatly enhanced by the developments in the automobile sector. Thanks to
these advancements, the car industry is now more efficient, environmentally friendly, and of higher
quality.
Problem 2:
(a)
Integer Programming Model for the Product Mix Problem
Objective function:
Maximize:
Profit = Revenue - Cost of Raw Materials - Cost of Labor
where:
The total revenue is calculated by adding up the product prices and quantities for each product.
Add the following to the cost of raw materials: stainless steel per kilogram * amount of stainless
steel + rustproofing per liter * quantity of rustproofing.
LabourCost = (Assembly department labourcost per hour * Assembly department labourhours) +
(Chemical processing department labourcost per hour * Chemical processing department
labourhours)
Decision variables:
Product A quantity
Product B quantity
Product C quantity
Stainless Steel quantity
Rustproofing quantity
Assembly department labourhours
Chemical processing department labourhours
Constraints
:
Stainless Steel constraint:
Stainless Steel quantity >= Product A quantity * Stainless Steel requirement per Product A + Product B
quantity * Stainless Steel requirement per Product B + Product C quantity * Stainless Steel requirement
per Product C
Rustproofing constraint:
Rustproofing quantity >= Product A quantity * Rustproofing requirement per Product A + Product B
quantity * Rustproofing requirement per Product B + Product C quantity * Rustproofing requirement per
Product C
Assembly department labourhours constraint:
Assembly department labour hours >= Product A quantity * Assembly department labourhours per
Product A + Product B quantity * Assembly department labourhours per Product B + Product C quantity
* Assembly department labour hours per Product C
Chemical processing department labour hours constraint:
Chemical processing department labour hours >= Product A quantity * Chemical processing department
labour hours per Product A + Product B quantity * Chemical processing department labour hours per
Product B + Product C quantity * Chemical processing department labour hours per Product C
Non-negativity constraints:
Product A quantity >= 0
Product B quantity >= 0
Product C quantity >= 0
Stainless Steel quantity >= 0
Rustproofing quantity >= 0
Assembly department labour hours >= 0
Chemical processing department labour hours >= 0
Integer restriction:
All decision variables must be integers, since we are dealing with quantities of products and resources.
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Solving the model:
This model can be solved using any integer programming solver, such as CPLEX or Gurobi. The solver will
find the optimal product mix that maximizes total profit, subject to all of the constraints.
Example solution:
Suppose the following values are given for the input parameters
:
The optimal solution to the model is as follows:
The total profit for this solution is
1,825,000.
Conclusion:
This integer programming model can be used to determine the optimal product mix for a company, in
order to maximize total profit, subject to all of the constraints.
(b)
Step-by-Step Solution to the Product Mix Problem Excel Model
Step 1: Enter the input data:
Step 2: Formulate the objective function
The objective function is to maximize total profit. Total profit is calculated as follows:
Total Profit = Total Revenue - Raw Material Cost - Labour Cost
Total revenue is calculated as follows:
Total Revenue = (Product A price * Product A quantity) + (Product B price * Product B quantity) +
(Product C price * Product C quantity)
Raw material cost is calculated as follows:
Raw Material Cost = (Stainless Steel cost per kilogram * Stainless Steel quantity) + (Rustproofing cost
per litre * Rustproofing quantity)
Labour cost is calculated as follows:
Labour Cost = (Assembly department labour cost per hour * Assembly department labour hours) +
(Chemical processing department labour cost per hour * Chemical processing department labour
hours)
The objective function can be formulated in Excel using the following formula:
=MAX(B8-B9-B10)
Step 3: Formulate the constraints
The constraints for the model are as follows:
Stainless Steel constraint
:
Stainless Steel quantity >= Product A quantity * Stainless Steel requirement per Product A + Product B
quantity * Stainless Steel requirement per Product B + Product C quantity * Stainless Steel requirement
per Product C
Rustproofing constraint:
Rustproofing quantity >= Product A quantity * Rustproofing requirement per Product A + Product B
quantity * Rustproofing requirement per Product B + Product C quantity * Rustproofing requirement per
Product C
Assembly department labour hours constraint:
Assembly department labour hours >= Product A quantity * Assembly department labour hours per
Product A + Product B quantity * Assembly department labour hours per Product B + Product C quantity
* Assembly department labour hours per Product C
Chemical processing department labour hours constraint:
Chemical processing department labour hours >= Product A quantity * Chemical processing department
labour hours per Product A + Product B quantity * Chemical processing department labour hours per
Product B + Product C quantity * Chemical processing department labour hours per Product C
Non-negativity constraints:
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Product A quantity >= 0
Product B quantity >= 0
Product C quantity >= 0
Stainless Steel quantity >= 0
Rustproofing quantity >= 0
Assembly department labour hours >= 0
Chemical processing department labour hours >= 0
The constraints can be formulated in Excel using the following formulas:
=B11>=B12*B2+B13*C2+B14*D2
=C11>=B15*B2+B16*C2+B17*D2
=D11>=B18*B2+B19*C2+B20*D2
=E11>=B21*B2+B22*C2+B23*D2
=B12>=0
=C12>=0
=D12>=0
=B13>=0
=C13>=0
=D13>=0
=E13>=0
The total profit for this solution is 1,825,000.
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 a linear programming issue, the problem specifies
that we remove the integer requirement from the decision variables and then create the Answer
Report or Sensitivity Analysis 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.
Create the Report on Answers and the Report on Sensitivity Analysis:Using a linear planning
solver like CPLEX or Gurobi, we may create the Answer Report and the Sensitivity Analysis
Report.In addition to providing details on the best way to solve the issue, the solver will also tell
us how the best answer varies depending on the factors we input.
3.
Please identify the resource that has the fewest available options:Through examining shadow
prices all the limitations in the Answer Report, we may ascertain which resource is the most
restrictive.If a constraint were to be loosened by one unit, the function with objectives would
improve by that amount; this is called the shadow cost of the constraint.The resource with the
greatest shadow price is also the most limited.
4.
Analyze what happens if the availability of the most limiting resource changes:
To analyze what
happens if the availability of the most limiting resource changes, we can use the Sensitivity
Analysis Report. The Sensitivity Analysis Report shows how the optimal solution to the problem
changes as the problem parameters change. We can use this information to predict how the
company's profit would change if the availability of the most limiting resource changed.
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
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 availability of the most limiting resource changes:
The best way to solve the issue will shift if the scarcest resource becomes more or less available. As you
tweak the problem's parameters, the Sensitivity Assessment Report reveals how the ideal solution
evolves. We may see that c1 has an objective function coefficient of 10 in this instance. So, a one-unit
increase in c1 availability will result in a ten-unit rise in the objective function value. Also, a one-unit
drop in c1 availability will result in a ten-unit drop in the objective function value.
Conclusion:
The resource that is limited by c1 is the most important one for the firm. The company's bottom line will
improve when c1 becomes more widely available. The company's bottom line will take a hit if c1
becomes less readily available.
(d)
Changes in the Most Limiting Resource's Availability: A Parametric Analysis
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The following procedures may be used to conduct a parametric study of the change in the availability of
the smallest resource:
1.
Determine which resource is the most constrained.To accomplish this, open the Answer Report
and examine the shadow pricing of the restrictions.If a constraint were to be loosened by one
unit, the function's objective would improve by that amount; this is called the shadow cost of
the constraint.The resource with the greatest shadow price is also the most limited.
2.
Pick a range of numbers representing the accessibility of the scarcest resource.
3.
Find the optimal solution to the linear programs problem for all values in the range.
4.
Make a note of the values of the objective functions and choice variables that provide the best
answer.
5.
Draw a graph showing the value of the goal function and the decision variables against the
availability for the resource with the highest constraints.
6.
Find patterns and trends by analyzing the data.
Graph and Table
For a basic linear programming issue, the following table and graph reflect the outcomes of a parametric
study of the change in availability of most limiting resource:
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
To make more of the scarcest resource available, two options are:
Solution 1: Increase the production capacity of c1.
Solution 2: Purchase c1 from an external supplier.
Advantages and Disadvantages of the Two Solutions:
The company's unique situation will dictate the optimal course of action. The optimal course of action
would be to increase c1's production capacity if the firm had the capital to invest in new machinery and
buildings. It may be more cost-effective for the business to buy c1 from a third party if internal resources
are limited.
Conclusion:
When solving a linear programming issue, parametric evaluation is a potent technique for seeing how
different values for the problem parameters affect the best possible solution. Companies may find the
most profitable approach to enhance profits by doing a parametric examination of the change in access
to the most restrictive resource.
(e)
Assumptions:
1.
The model presupposes a linear connection between the goal function and the choice factors.
Because it is oversimplified, this may not apply in other situations. The items' manufacturing
costs can, for instance, not scale linearly with respect to the amounts produced.
2.
Confidence: The model operates on the assumption that each input parameter is known with
absolute certainty. Once again, this oversimplification may not hold water when put into reality.
Things like raw material price fluctuations and unpredictable product demand are two examples.
3.
Objectivity: The model presupposes that the choice factors are unrelated to one another.
Occasionally, this does not hold true. For instance, there can be limitations on the raw materials'
availability or the methods for producing the goods.
4.
A single goal: The model presupposes that the company's only goal is in order to maximize
profits. Occasionally, this does not hold true. Other than increasing profits, companies may have
goals like reducing their negative influence on the environment or making their employees
happy.
Limitations:
1.
The precision of the model depends on how well the input parameters are defined. The accuracy
of the model's output is dependent on the accuracy of the input parameters.
2.
The model's applicability is restricted to a certain set of circumstances. In cases where there are
limitations on product manufacturing methods or when the objective function and choice
variables do not form a linear connection, it will not be useful.
3.
When it comes to generalizability, the model falls short. Other businesses or sectors may not be
able to use the model's output.
4.
Adding more choice variables and restrictions to the model might make it more difficult to solve.
Because of this, putting the approach into reality might be challenging.
In spite of these caveats and assumptions, the product mix issue model is nevertheless a powerful
resource for product development decision-makers. Under all circumstances, the model may be used to
determine the best combination of products to maximize profitability.
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