The document discusses linear programming, which involves optimizing an objective function subject to constraints. It provides examples of formulating linear programming problems from descriptions of resource allocation scenarios. Linear programming can be used to maximize profits or minimize costs by determining the optimal allocation of limited resources among competing activities. The key components of a linear programming model are decision variables, constraints, and an objective function. Graphical and algebraic (simplex) methods can be used to solve linear programming problems. Special cases like multiple optimal solutions, unbounded solutions, and infeasible solutions are also discussed.
this ppt is helpful for BBA/B.tech//MBA/M.tech students.
the ppt is on simulation topic...its covers -
Meaning
Advantages & Disadvantages
Uses
Process
Monte Carlo SImulation
Advantages & Disadvantages
Its example
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
this ppt is helpful for BBA/B.tech//MBA/M.tech students.
the ppt is on simulation topic...its covers -
Meaning
Advantages & Disadvantages
Uses
Process
Monte Carlo SImulation
Advantages & Disadvantages
Its example
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
Economics is the study of how individuals and societies choose to use the scarce resources that nature and the previous generation have provided. The world‟s resources are limited and scarce. The resources which are not scarce are called free goods. Resources which are scarce are called economic goods.
This ppt will explain you the Defintion ,detailed explanation of phases with necessory diagrams, Applications ,Limitations and scope of Operations Research
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
Economics is the study of how individuals and societies choose to use the scarce resources that nature and the previous generation have provided. The world‟s resources are limited and scarce. The resources which are not scarce are called free goods. Resources which are scarce are called economic goods.
This ppt will explain you the Defintion ,detailed explanation of phases with necessory diagrams, Applications ,Limitations and scope of Operations Research
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
Engineering design is the process of developing a system, component or process to satisfy the desired
requirements. It is a decision making process, in which the basic mathematics and engineering disciplines are
utilized to convert resources optimally to achieve a predetermined objective. It also includes a variety of
realistic constraints such as reliability, safety, economic factors, ethical and social impacts. This work proposes
a methodology and a procedure for the make-or-buy problem. Companies following this methodology are
guided through a structured sequence comprising identification of factors for the make-or-buy decision, and the
comparison of internal sourcing and external sourcing factors against each other. Multi-attribute decisionmaking
is utilized to present an overall make-or-buy decision recommendation.
Exam 3 Extra Problems1 Formulating the Production Planning.docxcravennichole326
Exam 3 Extra Problems
1 Formulating the Production Planning Model
A company manufactures two electric products, air conditioners and fans. The assembly process
is similar in that both of them must pass through two stages of production: wiring and drilling.
Each air conditioner takes 7 hours of wiring and 5 hours of drilling, while each fan must go through
3 hours of wiring and 2 hours of drilling. During the next period, 260 hours of wiring time are
available and 160 hours of drilling time may be used. Each air conditioner sold yields a profit of
$27, and each fan sold yields a profit of $17.
1. Formulate the objective function algebraically.
2. Write the resource constraints. Would demand constraints or non-negativity constraints be
more appropriate for this system of equations?
3. Upon looking over previous years demand, management has decided that to ensure an ad-
equate supply of air conditioners for a contract, at least 24 units should be manufactured.
Additionally, because they incurred an oversupply in the previous period, management has
also requested that no more than 50 fans be produced during this period. Assist the company
by reformulating the necessary constraints from the above question.
1
The corresponding sensitivity report for the original problem is shown below:
Variable cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$6 Air conditioners 0 A 27 15.5 1E+30
$C$6 Fans 80 B 17 1E+30 6.2
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$4 Wiring 240 0 260 1E+30 20
$D$5 Drilling 160 8.5 160 13.3 160
4. Calculate the missing values “A” and “B” in the above report.
5. Suppose management has decided to sell some of it’s drilling capacity for $6 per hour. If 15
hours were sold, what is the impact on profit? What is the new objective function value?
6. The previous page mentions management requiring at least 24 units of air conditioners to be
produced. If this policy were to be implemented, how will this affect contribution?
2
2 Understanding the Sensitivity Report
A manufacturing company produces two products X and Y with contributions to profit per unit
of $10 and $9, respectively. Each product must pass through 4 departments during the manufac-
turing process. The company has formulated a linear programming model to develop a production
strategy to maximize profit, and satisfy the demand of 300 and 500 units for product X and Y ,
respectively. The corresponding sensitivity report for the problem is shown below:
Variable cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$16 Product X 300 0 10 17 1E+30
$B$17 Product Y 500 0 9 1E+30 5.6666667
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$B$21 Department 1 1300 0 6500 1E+30 5200
$B$22 Department 2 2100 0 6000 1E+30 3900
$B$23 Dep.
Question 1.Which of the following is considered to be an advantage.docxIRESH3
Question 1.Which of the following is considered to be an advantage of using both nonfinancial and financial information in the balanced scorecard? (Points : 2)
Nonfinancial information is most helpful in analyzing a company's past performance, while financial information is most useful in evaluating potential future performance.
Nonfinancial information provides the short-term perspective while financial information provides the long-term perspective of performance.
Nonfinancial information reflects the company's current and potential competitive advantage, while financial information tends to focus on a firm's achieved financial performance.
Nonfinancial information should be included with financial information because it is more reliable than financial information.
Question 2.Over the past several years it has become increasingly important for firms to improve achievement towards their social and environmental responsibilities. What is the best way the management accountant can help the firm improve on sustainability? (Points : 2)
Participate in programs of environmental organizations.
Develop and implement a legal staff and public relations staff for dealing with sustainability issues that may affect the firm.
Develop and implement a sustainability scorecard.
Risk management.
Question 3.The range of the cost driver in which the actual value of the driver is expected to fall is the: (Points : 2)
Actual cost range.
Driver range.
Activity range.
Relevant range.
Question 4.The difference between wholesalers and retailers is: (Points : 2)
Wholesalers are merchandisers that sell directly to customers whereas retailers are merchandisers that sell to other merchandisers.
Wholesalers are merchandisers that sell to other merchandisers whereas retailers are merchandisers that sell directly to consumers.
Wholesalers are merchandisers that sell directly to the government whereas retailers are merchandisers that sell to other merchandisers.
Wholesalers are merchandisers that sell directly to customers whereas retailers are merchandisers that sell directly to the government.
Question 5.In a local factory, employees are rewarded for finding new and better ways of changing the way they work. This company is motivating its employees to use what management technique? (Points : 2)
Benchmarking.
Activity-Based Costing.
Theory of Constraints.
Continuous Improvement.
Total Quality Management.
Question 6.Assume the following information pertaining to Moonbeam Company:
Beginning Finished Goods Inventory = $130,000
Ending Finished Goods Inventory = $124,000
Beginning WIP Inventory = $85,000
Ending WIP Inventory = $104,000
Beginning Direct Materials = $117,000
Ending Direct Materials = $130,000
Costs incurred during the period are as follows:
Total Manufacturing Costs = $896,000
Factory Overhead = $199,000
...
[Note: This is a partial preview. To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
Leading companies such as Nike, Toyota, and Siemens are prioritizing sustainable innovation in their business models, setting an example for others to follow. In this Sustainability training presentation, you will learn key concepts, principles, and practices of sustainability applicable across industries. This training aims to create awareness and educate employees, senior executives, consultants, and other key stakeholders, including investors, policymakers, and supply chain partners, on the importance and implementation of sustainability.
LEARNING OBJECTIVES
1. Develop a comprehensive understanding of the fundamental principles and concepts that form the foundation of sustainability within corporate environments.
2. Explore the sustainability implementation model, focusing on effective measures and reporting strategies to track and communicate sustainability efforts.
3. Identify and define best practices and critical success factors essential for achieving sustainability goals within organizations.
CONTENTS
1. Introduction and Key Concepts of Sustainability
2. Principles and Practices of Sustainability
3. Measures and Reporting in Sustainability
4. Sustainability Implementation & Best Practices
To download the complete presentation, visit: https://www.oeconsulting.com.sg/training-presentations
Business Valuation Principles for EntrepreneursBen Wann
This insightful presentation is designed to equip entrepreneurs with the essential knowledge and tools needed to accurately value their businesses. Understanding business valuation is crucial for making informed decisions, whether you're seeking investment, planning to sell, or simply want to gauge your company's worth.
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Implicitly or explicitly all competing businesses employ a strategy to select a mix
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Unveiling the Secrets How Does Generative AI Work.pdfSam H
At its core, generative artificial intelligence relies on the concept of generative models, which serve as engines that churn out entirely new data resembling their training data. It is like a sculptor who has studied so many forms found in nature and then uses this knowledge to create sculptures from his imagination that have never been seen before anywhere else. If taken to cyberspace, gans work almost the same way.
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2. Introduction
Linear programming (LP): model enable
users to find optimal solutions to certain
problems in which the solutions must
satisfy a given set of requirements or
constraints.
A large number of decision problems faced
by a business manager involve allocation of
resources to various activities, with the
objective of increasing profits or decreasing
costs, or both.
Thus, the manager has to take a decision
as to how best the resources be allocated
among the various activities.
11/9/2018
3. Mathematical programming involves
optimization of a certain function, called
the objective function, subject to certain
constraints.
Linear Programming (LP) is a
mathematical modeling technique useful
for economic allocation of “scarce” or
“limited” resources, such as labour,
material, machine, time, warehouse,
space, capital, etc. to several competing
activities such as products, services,
jobs, new equipment, projects etc, on the
basis of a given criterion of optimality.11/9/2018
4. Structure of Linear
Programming Model
The general structure of LP model consists
of three basic elements or
components.These are:
◦ The decision variables, and parameters
◦ The objective function and
◦ The constraints
11/9/2018
5. Decision variables (Activities): The
activities are usually denoted by
decision variables .
The values of these activities represent
the extent to which each of these is
performed.
The values of certain variables may or
may not be under the decision –maker’s
control.
11/9/2018
6. The objective function:
the objective (goal) function of each LP
problem is expressed in terms of
decision variables to optimize the
criterion of optimality (the measure of
performance)
such as profit, cost, revenue, distance,
etc.
In its general form it is expressed as:
Optimize (Maximize or Minimize) Z=
c1x1+c2x2+------cnxn
11/9/2018
7. The constraints:
There are always certain limitations
(or constraints) on the use of
resources, e.g. labor, Machine, Raw
material, space, money, etc.
These limit the degree to which an
objective can be achieved.
11/9/2018
8. Assumptions Underlying Linear
Programming
The basic assumptions of linear programming
include the following:
Proportionality: the measure of effectiveness
(profit or loss) in the objective function and
amount of each resource used must be
proportional to the value of each decision
variable considered individually.
Additivity: It states that the sum of
resources used by different activities must
be equal to the total quantity of resources
used by each activity for all resources
individually and collectively.
11/9/2018
9. Divisibility: This assumption implies
that solutions need not be in whole
numbers (integers). Instead, they are
divisible and may take any fractional
value.
Certainty: the coefficients in the
objective function and constrains are
completely known (deterministic) and do
not change during the period being
studied.
11/9/2018
10. Finiteness: An optimum solution
cannot be computed in the situations
where there are infinite number of
alternative activities and resource
restrictions.
Optimality: In linear programming, the
maximum profit solution or the
minimum cost solution always occur at
the corner point of the set of feasible
solutions.
11/9/2018
11. Formulations of Linear
Programming Problem
formulation is the process of converting
the verbal description and numerical
data into mathematical expressions
which represents the relevant
relationship among decision factors,
objectives and restrictions on the use of
resources
11/9/2018
14. The basic steps in formulating a linear
programming model are as follows-
Step I. Identification of the Decision
Variables
Step II. Identification of the Constraints
Step III. Identification of the Objective
11/9/2018
15. Example 1:-A firm is engaged in
producing two products, A and B. Each
unit of product A requires 2 kg of raw
material and 4 labour hours for
processing, whereas each unit of
product B requires 3 Kg of raw material
and 3 hours of labour, of the same type.
Every week, the firm has an availability
of 60 kg of raw material and 96 labour
hours. One unit of product A sold yields
Birr 40 and one unit of product B sold
gives Birr 35 as profit.
Formulate this problem as a linear11/9/2018
16. Example 2:-
High Quality furniture Ltd. manufactures two
products, tables and chairs. Both the
products have to be processed through two
machines:-Ml andM2.The total machine-
hours available are 200 hours of M1 and 400
hours of M2 respectively.
Time in hours required for producing a chair
and a table on both the machines is as
follows:
Time in hours
Machine TableChair
M1 7 4
M2 5 5
Profit from Sale of table is Br.50 and sale of
a chair is Br. 30
11/9/2018
17. Example 3
A Juice Company has available two kinds of
food Juices: Orange Juice and Grape Juice.
The company produces two types of
punches: Punch A and Punch B. One bottle
of punch A requires 20 liters of Orange Juice
and 5 liters of Grape Juice. One bottle of
punch B requires 10 liters of Orange Juice
and 15 liters of Grape Juice.
From each of bottle of Punch A profit of $4
is made and from each bottle of Punch B a
profit of $3 is made .Suppose that the
company has 230 liters of Orange Juice
and 120 liters of Grape Juice available
11/9/2018
18. Example 4:-
The Agricultural Research institute suggested to a
farmer to spread out at least 4800kg of a special
phosphate fertilizer and not less than 7200 kg of a
special nitrogen fertilizer to raise productivity of crops in
his fields. There are two sources for obtaining these-
mixtures A and B. Both of these are available in bags
weighing 100kg each and they cost Birr 40 and Birr 24
respectively. Mixture A contains phosphate and nitrogen
equivalent of 20 kg and 80 kg respectively, while
mixture B contains these ingredients equivalent of 50 kg
each.
Write this as a linear programming
11/9/2018
19. Example: 5
Suppose that a machine shop has two
different types of machines; machine 1 and
machine 2, which can be used to make a
single product . These machine vary in the
amount of product produced per hr., in the
amount of labor used and in the cost of
operation.
Assume that at least a certain amount of
product must be produced and that we
would like to utilize at least the regular labor
force. How much should we utilize each
machine in order to utilize total costs and still
meets the requirement?11/9/2018
21. Example 6
A company owns two flour mills (A and B) which
have different production capacities for HIGH,
MEDIUM and LOW grade flour. This company
has entered contract supply flour to a firm every
week with 12, 8, and 24 quintals of HIGH,
MEDIUM and LOW grade respectively. It costs
the Co. $1000 and $800 per day to run mill A and
mill B respectively. On a day, mill A produces 6,
2, and 4 quintals of HIGH, MEDIUM and LOW
grade flour respectively. Mill B produces 2, 2 and
12 quintals of HIGH, MEDIUM and LOW grade
flour respectively. How many days per week
should each mill be operated in order to meet the
contract order most economically standardize?
Formulate Linear Programming Model
11/9/2018
22. Methods of Solving Linear Programming
Problems
They can be solved by graphic method or by applying
algebraic method, called the simplex Method.
The graphic approach is restricted in application that it can
be used only when two variables are involved.
11/9/2018
23. Graphical Method
The graphic solution procedure is one
of the methods of solving two variable
linear programming problems. It
consists of the following steps:-
Step I: Defining the problem.
Step II: Plot the constraints graphically.
Step III: Locate the solution space
Use corner point method
11/9/2018
24. Example 1:-A firm is engaged in
producing two products, A and B. Each
unit of product A requires 2 kg of raw
material and 4 labour hours for
processing, whereas each unit of
product B requires 3 Kg of raw material
and 3 hours of labour, of the same type.
Every week, the firm has an availability
of 60 kg of raw material and 96 labour
hours. One unit of product A sold yields
Birr 40 and one unit of product B sold
gives Birr 35 as profit.
Formulate this problem as a linear11/9/2018
25. Example 2.4: A firm that assembles computer and computer equipment is about to start
production of two new microcomputers. Each type of microcomputer will require assembly time,
inspection time and storage space. The amount of each of these resources that can be devoted to
the production of the microcomputers is limited. The manager of the firm would like to
determine the quantity of each microcomputer to produce in order to maximize the profit
generated by the seal of these microcomputers.
Additional information
In order to develop a suitable model of the problem, the manager has met with design and
manufacturing personnel. As the result of those meetings, the manager has obtained the
following information:
Type 1 Type 2
Profit per unit $60 $50
Assembly time per unit 4 hours 10 hours
Inspection time per unit 2 hours 1 hours
Storage space per unit 3 cubic feet 3 cubic feet
11/9/2018
26. The manager has also acquired information on the availability of company resources. These
amounts are:
Resource Amount available
Assembly time 100 hours
Inspection time 22 hours
Storage space 39 cubic feet
The manager also met with the firm’s marketing manager and learned that demand for the
microcomputers was such that whatever combination of these two types of microcomputers is
produced, all of the outputs can be sold.
Required:
a. Formulate the LP model (problem)
b. Find the product mix so that the company can make a maximum profit.
11/9/2018
27. Example2.5
Minimize
Subjectto
Z=40x1 +24x2
20x1 +50x2 ≥4800
80x1 +50x2 ≥7200
x1,x2≥0
TotalCost
PhosphateRequirement
NitrogenRequirement
Herethe decision variables x1 andx2 represent, respectively, the number of bags of mixture A
andofmixtureB,tobebought.
11/9/2018
28. Example 6
A company owns two flour mills (A and B) which
have different production capacities for HIGH,
MEDIUM and LOW grade flour. This company
has entered contract supply flour to a firm every
week with 12, 8, and 24 quintals of HIGH,
MEDIUM and LOW grade respectively. It costs
the Co. $1000 and $800 per day to run mill A and
mill B respectively. On a day, mill A produces 6,
2, and 4 quintals of HIGH, MEDIUM and LOW
grade flour respectively. Mill B produces 2, 2 and
12 quintals of HIGH, MEDIUM and LOW grade
flour respectively. How many days per week
should each mill be operated in order to meet the
contract order most economically standardize?
Formulate Linear Programming Model
11/9/2018
29. Some Special Cases of Linear
Programming
1. Multiple Optimal Solutions: - Situations arise
when there are more than one optimal solution.
2. Unbounded solution: - This is a situation that
occurs whenever a maximization LPP has
unbounded solution space. No finite solution can
be determined.
3. Infeasible Solution: - This indicates that
there may be a solution which can satisfy each
constraint but there is no feasible solution.
11/9/2018
30. Multiple Optimal Solutions
Example: Solve the following linear programming problem.
0,
3
3605020
2
9
110X110Z
21
2
21
1
21
21
XX
x
xx
X
XX
toSubject
XMaximize
11/9/2018
32. Infeasible Solution
Example: - solve the following linear programming problem
using graphic method.
0,
81
5
46Zmax
21
21
21
xx
x
xx
toSubject
xximize
11/9/2018