The document discusses linear programming models for solving business optimization problems. It provides an overview of linear programming and the steps to formulate a linear programming model, which are to define decision variables, construct the objective function, and formulate constraints. The document also discusses graphical solutions to linear programming problems using examples of maximizing profit from two products given resource constraints and minimizing fertilizer costs given nutrient requirements.
A typical concern in government agencies: the status of funds. In a standardized government accounting system, most managers are not well versed in fund management. Some agencies resort to packaged accounting system or a customized suit of application. Given these options, the perennial issue is: how effective and how efficient? The other issue is the cost. It is not an understatement to say that most, if not all, government agencies are equipped with the simple spreadsheet, the more common is the Excel. With simple worksheet formatting with the use of standard accounting codes, simple task assignment and coordinated schedules of update and transmission, the Excel offers the least cost, yet effective and efficient method of reporting fund status for the provincial, regional, central and national levels.
This is the objective of “Simple Excel in Fund Monitoring”, to facilitate the monitoring of fund status, particularly disbursements. It is significantly accurate because of its transaction-based data capture, right from the disbursement voucher as it is being processed.
,
capital budgeting
,
concept of capital budgeting
,
the capital budgeting process
,
significance of capital budgeting
,
classification of investment project proposals
,
techniques of capital budgeting
,
types of project
A typical concern in government agencies: the status of funds. In a standardized government accounting system, most managers are not well versed in fund management. Some agencies resort to packaged accounting system or a customized suit of application. Given these options, the perennial issue is: how effective and how efficient? The other issue is the cost. It is not an understatement to say that most, if not all, government agencies are equipped with the simple spreadsheet, the more common is the Excel. With simple worksheet formatting with the use of standard accounting codes, simple task assignment and coordinated schedules of update and transmission, the Excel offers the least cost, yet effective and efficient method of reporting fund status for the provincial, regional, central and national levels.
This is the objective of “Simple Excel in Fund Monitoring”, to facilitate the monitoring of fund status, particularly disbursements. It is significantly accurate because of its transaction-based data capture, right from the disbursement voucher as it is being processed.
,
capital budgeting
,
concept of capital budgeting
,
the capital budgeting process
,
significance of capital budgeting
,
classification of investment project proposals
,
techniques of capital budgeting
,
types of project
Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. It may be positive, zero or negative.
NPV is used in capital budgeting and investment planning to analyze the profitability of a projected investment or project.
Also known as sophisticated technique for capital budgeting exercise.
It accounts for time value of money by using discounted cash flows in the calculation.
Accounting plays a vital role in running a business because it helps you track income and expenditures, ensure statutory compliance, and provide investors, management, and government with quantitative financial information.
Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. It may be positive, zero or negative.
NPV is used in capital budgeting and investment planning to analyze the profitability of a projected investment or project.
Also known as sophisticated technique for capital budgeting exercise.
It accounts for time value of money by using discounted cash flows in the calculation.
Accounting plays a vital role in running a business because it helps you track income and expenditures, ensure statutory compliance, and provide investors, management, and government with quantitative financial information.
Crystallization, or crystallisation, is the process of atoms or molecules arr...Mayurkumarpatil1
Crystallization, or crystallisation, is the process of atoms or molecules arranging into a well-defined, rigid crystal lattice in order to minimize their energetic state
Micro RNA genes and their likely influence in rice (Oryza sativa L.) dynamic ...Open Access Research Paper
Micro RNAs (miRNAs) are small non-coding RNAs molecules having approximately 18-25 nucleotides, they are present in both plants and animals genomes. MiRNAs have diverse spatial expression patterns and regulate various developmental metabolisms, stress responses and other physiological processes. The dynamic gene expression playing major roles in phenotypic differences in organisms are believed to be controlled by miRNAs. Mutations in regions of regulatory factors, such as miRNA genes or transcription factors (TF) necessitated by dynamic environmental factors or pathogen infections, have tremendous effects on structure and expression of genes. The resultant novel gene products presents potential explanations for constant evolving desirable traits that have long been bred using conventional means, biotechnology or genetic engineering. Rice grain quality, yield, disease tolerance, climate-resilience and palatability properties are not exceptional to miRN Asmutations effects. There are new insights courtesy of high-throughput sequencing and improved proteomic techniques that organisms’ complexity and adaptations are highly contributed by miRNAs containing regulatory networks. This article aims to expound on how rice miRNAs could be driving evolution of traits and highlight the latest miRNA research progress. Moreover, the review accentuates miRNAs grey areas to be addressed and gives recommendations for further studies.
"Understanding the Carbon Cycle: Processes, Human Impacts, and Strategies for...MMariSelvam4
The carbon cycle is a critical component of Earth's environmental system, governing the movement and transformation of carbon through various reservoirs, including the atmosphere, oceans, soil, and living organisms. This complex cycle involves several key processes such as photosynthesis, respiration, decomposition, and carbon sequestration, each contributing to the regulation of carbon levels on the planet.
Human activities, particularly fossil fuel combustion and deforestation, have significantly altered the natural carbon cycle, leading to increased atmospheric carbon dioxide concentrations and driving climate change. Understanding the intricacies of the carbon cycle is essential for assessing the impacts of these changes and developing effective mitigation strategies.
By studying the carbon cycle, scientists can identify carbon sources and sinks, measure carbon fluxes, and predict future trends. This knowledge is crucial for crafting policies aimed at reducing carbon emissions, enhancing carbon storage, and promoting sustainable practices. The carbon cycle's interplay with climate systems, ecosystems, and human activities underscores its importance in maintaining a stable and healthy planet.
In-depth exploration of the carbon cycle reveals the delicate balance required to sustain life and the urgent need to address anthropogenic influences. Through research, education, and policy, we can work towards restoring equilibrium in the carbon cycle and ensuring a sustainable future for generations to come.
Artificial Reefs by Kuddle Life Foundation - May 2024punit537210
Situated in Pondicherry, India, Kuddle Life Foundation is a charitable, non-profit and non-governmental organization (NGO) dedicated to improving the living standards of coastal communities and simultaneously placing a strong emphasis on the protection of marine ecosystems.
One of the key areas we work in is Artificial Reefs. This presentation captures our journey so far and our learnings. We hope you get as excited about marine conservation and artificial reefs as we are.
Please visit our website: https://kuddlelife.org
Our Instagram channel:
@kuddlelifefoundation
Our Linkedin Page:
https://www.linkedin.com/company/kuddlelifefoundation/
and write to us if you have any questions:
info@kuddlelife.org
WRI’s brand new “Food Service Playbook for Promoting Sustainable Food Choices” gives food service operators the very latest strategies for creating dining environments that empower consumers to choose sustainable, plant-rich dishes. This research builds off our first guide for food service, now with industry experience and insights from nearly 350 academic trials.
Diabetes is a rapidly and serious health problem in Pakistan. This chronic condition is associated with serious long-term complications, including higher risk of heart disease and stroke. Aggressive treatment of hypertension and hyperlipideamia can result in a substantial reduction in cardiovascular events in patients with diabetes 1. Consequently pharmacist-led diabetes cardiovascular risk (DCVR) clinics have been established in both primary and secondary care sites in NHS Lothian during the past five years. An audit of the pharmaceutical care delivery at the clinics was conducted in order to evaluate practice and to standardize the pharmacists’ documentation of outcomes. Pharmaceutical care issues (PCI) and patient details were collected both prospectively and retrospectively from three DCVR clinics. The PCI`s were categorized according to a triangularised system consisting of multiple categories. These were ‘checks’, ‘changes’ (‘change in drug therapy process’ and ‘change in drug therapy’), ‘drug therapy problems’ and ‘quality assurance descriptors’ (‘timer perspective’ and ‘degree of change’). A verified medication assessment tool (MAT) for patients with chronic cardiovascular disease was applied to the patients from one of the clinics. The tool was used to quantify PCI`s and pharmacist actions that were centered on implementing or enforcing clinical guideline standards. A database was developed to be used as an assessment tool and to standardize the documentation of achievement of outcomes. Feedback on the audit of the pharmaceutical care delivery and the database was received from the DCVR clinic pharmacist at a focus group meeting.
Willie Nelson Net Worth: A Journey Through Music, Movies, and Business Venturesgreendigital
Willie Nelson is a name that resonates within the world of music and entertainment. Known for his unique voice, and masterful guitar skills. and an extraordinary career spanning several decades. Nelson has become a legend in the country music scene. But, his influence extends far beyond the realm of music. with ventures in acting, writing, activism, and business. This comprehensive article delves into Willie Nelson net worth. exploring the various facets of his career that have contributed to his large fortune.
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Introduction
Willie Nelson net worth is a testament to his enduring influence and success in many fields. Born on April 29, 1933, in Abbott, Texas. Nelson's journey from a humble beginning to becoming one of the most iconic figures in American music is nothing short of inspirational. His net worth, which estimated to be around $25 million as of 2024. reflects a career that is as diverse as it is prolific.
Early Life and Musical Beginnings
Humble Origins
Willie Hugh Nelson was born during the Great Depression. a time of significant economic hardship in the United States. Raised by his grandparents. Nelson found solace and inspiration in music from an early age. His grandmother taught him to play the guitar. setting the stage for what would become an illustrious career.
First Steps in Music
Nelson's initial foray into the music industry was fraught with challenges. He moved to Nashville, Tennessee, to pursue his dreams, but success did not come . Working as a songwriter, Nelson penned hits for other artists. which helped him gain a foothold in the competitive music scene. His songwriting skills contributed to his early earnings. laying the foundation for his net worth.
Rise to Stardom
Breakthrough Albums
The 1970s marked a turning point in Willie Nelson's career. His albums "Shotgun Willie" (1973), "Red Headed Stranger" (1975). and "Stardust" (1978) received critical acclaim and commercial success. These albums not only solidified his position in the country music genre. but also introduced his music to a broader audience. The success of these albums played a crucial role in boosting Willie Nelson net worth.
Iconic Songs
Willie Nelson net worth is also attributed to his extensive catalog of hit songs. Tracks like "Blue Eyes Crying in the Rain," "On the Road Again," and "Always on My Mind" have become timeless classics. These songs have not only earned Nelson large royalties but have also ensured his continued relevance in the music industry.
Acting and Film Career
Hollywood Ventures
In addition to his music career, Willie Nelson has also made a mark in Hollywood. His distinctive personality and on-screen presence have landed him roles in several films and television shows. Notable appearances include roles in "The Electric Horseman" (1979), "Honeysuckle Rose" (1980), and "Barbarosa" (1982). These acting gigs have added a significant amount to Willie Nelson net worth.
Television Appearances
Nelson's char
2. 2-2
Topics
Linear Programming – An overview
Model Formulation
Characteristics of Linear Programming Problems
Assumptions of a Linear Programming Model
Advantages and Limitations of a Linear Programming.
A Maximization Model Example
Graphical Solutions of Linear Programming Models
A Minimization Model Example
Irregular Types of Linear Programming Models
3. 2-3
Objectives of business decisions frequently involve
maximizing profit or minimizing costs.
Linear programming uses linear algebraic relationships
to represent a firm’s decisions, given a business objective,
and resource constraints.
Steps in application:
1. Identify problem as solvable by linear programming.
2. Formulate a mathematical model of the unstructured
problem.
3. Solve the model.
4. Implementation
Linear Programming: An Overview
4. 2-4
Decision variables - mathematical symbols representing levels
of activity of a firm.
Objective function - a linear mathematical relationship
describing an objective of the firm, in terms of decision variables
- this function is to be maximized or minimized.
Constraints – requirements or restrictions placed on the firm by
the operating environment, stated in linear relationships of the
decision variables.
Parameters - numerical coefficients and constants used in the
objective function and constraints.
Model Components
5. 2-5
Summary of Model Formulation Steps
Step 1 : Clearly define the decision variables
Step 2 : Construct the objective function
Step 3 : Formulate the constraints
6. 2-6
Characteristics of Linear Programming Problems
A decision amongst alternative courses of action is required.
The decision is represented in the model by decision variables.
The problem encompasses a goal, expressed as an objective
function, that the decision maker wants to achieve.
Restrictions (represented by constraints) exist that limit the
extent of achievement of the objective.
The objective and constraints must be definable by linear
mathematical functional relationships.
7. 2-7
Proportionality - The rate of change (slope) of the objective
function and constraint equations is constant.
Additivity - Terms in the objective function and constraint
equations must be additive.
Divisibility -Decision variables can take on any fractional value
and are therefore continuous as opposed to integer in nature.
Certainty - Values of all the model parameters are assumed to
be known with certainty (non-probabilistic).
Assumptions of Linear Programming Model
8. 2-8
It helps decision - makers to use their productive resource
effectively.
The decision-making approach of the user becomes more
objective and less subjective.
In a production process, bottle necks may occur. For example, in
a factory some machines may be in great demand while others
may lie idle for some time. A significant advantage of linear
programming is highlighting of such bottle necks.
Advantages of Linear Programming Model
9. 2-9
Linear programming is applicable only to problems where the
constraints and objective function are linear i.e., where they can
be expressed as equations which represent straight lines. In real
life situations, when constraints or objective functions are not
linear, this technique cannot be used.
Factors such as uncertainty, and time are not taken into
consideration.
Parameters in the model are assumed to be constant but in real
life situations they are not constants.
Linear programming deals with only single objective , whereas in
real life situations may have multiple and conflicting objectives.
In solving a LPP there is no guarantee that we get an integer
value. In some cases of no of men/machine a non-integer value
is meaningless.
Limitations of Linear Programming Model
10. 2-10
LP Model Formulation
A Maximization Example (1 of 4)
Product mix problem - Beaver Creek Pottery Company
How many bowls and mugs should be produced to maximize
profits given labor and materials constraints?
Product resource requirements and unit profit:
Resource Requirements
Product
Labor
(Hr./Unit)
Clay
(Lb./Unit)
Profit
($/Unit)
Bowl 1 4 40
Mug 2 3 50
12. 2-12
LP Model Formulation
A Maximization Example (3 of 4)
Resource 40 hrs of labor per day
Availability: 120 lbs of clay
Decision x1 = number of bowls to produce per day
Variables: x2 = number of mugs to produce per day
Objective Maximize Z = $40x1 + $50x2
Function: Where Z = profit per day
Resource 1x1 + 2x2 40 hours of labor
Constraints: 4x1 + 3x2 120 pounds of clay
Non-Negativity x1 0; x2 0
Constraints:
13. 2-13
LP Model Formulation
A Maximization Example (4 of 4)
Complete Linear Programming Model:
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
14. 2-14
A feasible solution does not violate any of the constraints:
Example: x1 = 5 bowls
x2 = 10 mugs
Z = $40x1 + $50x2 = $700
Labor constraint check: 1(5) + 2(10) = 25 < 40 hours
Clay constraint check: 4(5) + 3(10) = 70 < 120 pounds
Feasible Solutions
15. 2-15
An infeasible solution violates at least one of the
constraints:
Example: x1 = 10 bowls
x2 = 20 mugs
Z = $40x1 + $50x2 = $1400
Labor constraint check: 1(10) + 2(20) = 50 > 40 hours
Infeasible Solutions
16. 2-16
Graphical solution is limited to linear programming models
containing only two decision variables (can be used with three
variables but only with great difficulty).
Graphical methods provide visualization of how a solution for
a linear programming problem is obtained.
Graphical methods can be classified under two categories:
1. Iso-Profit(Cost) Line Method
2. Extreme-point evaluation Method.
Graphical Solution of LP Models
17. 2-17
Coordinate Axes
Graphical Solution of Maximization Model (1 of 12)
Figure 2.2 Coordinates for Graphical Analysis
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
X1 is bowls
X2 is mugs
18. 2-18
Labor Constraint
Graphical Solution of Maximization Model (2 of 12)
Figure 2.3 Graph of Labor Constraint
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
19. 2-19
Labor Constraint Area
Graphical Solution of Maximization Model (3 of 12)
Figure 2.4 Labor Constraint Area
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
20. 2-20
Clay Constraint Area
Graphical Solution of Maximization Model (4 of 12)
Figure 2.5 Clay Constraint Area
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
21. 2-21
Both Constraints
Graphical Solution of Maximization Model (5 of 12)
Figure 2.6 Graph of Both Model Constraints
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
22. 2-22
Feasible Solution Area
Graphical Solution of Maximization Model (6 of 12)
Figure 2.7 Feasible Solution Area
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
23. 2-23
Objective Function Solution = $800
Graphical Solution of Maximization Model (7 of 12)
Figure 2.8 Objection Function Line for Z = $800
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
24. 2-24
Alternative Objective Function Solution Lines
Graphical Solution of Maximization Model (8 of 12)
Figure 2.9 Alternative Objective Function Lines
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
25. 2-25
Optimal Solution
Graphical Solution of Maximization Model (9 of 12)
Figure 2.10 Identification of Optimal Solution Point
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
26. 2-26
Optimal Solution Coordinates
Graphical Solution of Maximization Model (10 of 12)
Figure 2.11 Optimal Solution Coordinates
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
27. 2-27
Extreme (Corner) Point Solutions
Graphical Solution of Maximization Model (11 of 12)
Figure 2.12 Solutions at All Corner Points
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
28. 2-28
Optimal Solution for New Objective Function
Graphical Solution of Maximization Model (12 of 12)
Maximize Z = $70x1 + $20x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
Figure 2.13 Optimal Solution with Z = 70x1 + 20x2
29. 2-29
Standard form requires that all constraints be in the form
of equations (equalities).
A slack variable is added to a constraint (weak
inequality) to convert it to an equation (=).
A slack variable typically represents an unused resource.
A slack variable contributes nothing to the objective
function value.
Slack Variables
30. 2-30
Linear Programming Model: Standard Form
Max Z = 40x1 + 50x2 + s1 + s2
subject to:1x1 + 2x2 + s1 = 40
4x2 + 3x2 + s2 = 120
x1, x2, s1, s2 0
Where:
x1 = number of bowls
x2 = number of mugs
s1, s2 are slack variables
Figure 2.14 Solution Points A, B, and C with Slack
31. 2-31
LP Model Formulation – Minimization (1 of 8)
Chemical Contribution
Brand
Nitrogen
(lb/bag)
Phosphate
(lb/bag)
Super-gro 2 4
Crop-quick 4 3
Two brands of fertilizer available - Super-gro, Crop-quick.
Field requires at least 16 pounds of nitrogen and 24 pounds of
phosphate.
Super-gro costs $6 per bag, Crop-quick $3 per bag.
Problem: How much of each brand to purchase to minimize total
cost of fertilizer given following data ?
33. 2-33
Decision Variables:
x1 = bags of Super-gro
x2 = bags of Crop-quick
The Objective Function:
Minimize Z = $6x1 + 3x2
Where: $6x1 = cost of bags of Super-Gro
$3x2 = cost of bags of Crop-Quick
Model Constraints:
2x1 + 4x2 16 lb (nitrogen constraint)
4x1 + 3x2 24 lb (phosphate constraint)
x1, x2 0 (non-negativity constraint)
LP Model Formulation – Minimization (3 of 8)
34. 2-34
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2 16
4x2 + 3x2 24
x1, x2 0
Figure 2.16 Graph of Both Model Constraints
Constraint Graph – Minimization (4 of 8)
35. 2-35
Figure 2.17 Feasible Solution Area
Feasible Region– Minimization (5 of 8)
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2 16
4x2 + 3x2 24
x1, x2 0
36. 2-36
Figure 2.18 Optimum Solution Point
Optimal Solution Point – Minimization (6 of 8)
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2 16
4x2 + 3x2 24
x1, x2 0
37. 2-37
A surplus variable is subtracted from a constraint to
convert it to an equation (=).
A surplus variable represents an excess above a
constraint requirement level.
A surplus variable contributes nothing to the calculated
value of the objective function.
Subtracting surplus variables in the farmer problem
constraints:
2x1 + 4x2 - s1 = 16 (nitrogen)
4x1 + 3x2 - s2 = 24 (phosphate)
Surplus Variables – Minimization (7 of 8)
38. 2-38
Figure 2.19 Graph of Fertilizer Example
Graphical Solutions – Minimization (8 of 8)
Minimize Z = $6x1 + $3x2 + 0s1 + 0s2
subject to: 2x1 + 4x2 – s1 = 16
4x2 + 3x2 – s2 = 24
x1, x2, s1, s2 0
39. 2-39
For some linear programming models, the general rules
do not apply.
Special types of problems include those with:
Multiple optimal solutions
Infeasible solutions
Unbounded solutions
Irregular Types of Linear Programming Problems
40. 2-40
Figure 2.20 Example with Multiple Optimal Solutions
Multiple Optimal Solutions Beaver Creek Pottery
The objective function is
parallel to a constraint line.
Maximize Z=$40x1 + 30x2
subject to: 1x1 + 2x2 40
4x2 + 3x2 120
x1, x2 0
Where:
x1 = number of bowls
x2 = number of mugs
41. 2-41
An Infeasible Problem
Figure 2.21 Graph of an Infeasible Problem
Every possible solution
violates at least one constraint:
Maximize Z = 5x1 + 3x2
subject to: 4x1 + 2x2 8
x1 4
x2 6
x1, x2 0
42. 2-42
An Unbounded Problem
Figure 2.22 Graph of an Unbounded Problem
Value of the objective
function increases indefinitely:
Maximize Z = 4x1 + 2x2
subject to: x1 4
x2 2
x1, x2 0
43. 2-43
Problem Statement
Example Problem No. 1 (1 of 3)
■ Hot dog mixture in 1000-pound batches.
■ Two ingredients, chicken ($3/lb) and beef ($5/lb).
■ Recipe requirements:
at least 500 pounds of “chicken”
at least 200 pounds of “beef”
■ Ratio of chicken to beef must be at least 2 to 1.
■ Determine optimal mixture of ingredients that will
minimize costs.
44. 2-44
Step 1:
Identify decision variables.
x1 = lb of chicken in mixture
x2 = lb of beef in mixture
Step 2:
Formulate the objective function.
Minimize Z = $3x1 + $5x2
where Z = cost per 1,000-lb batch
$3x1 = cost of chicken
$5x2 = cost of beef
Solution
Example Problem No. 1 (2 of 3)
45. 2-45
Step 3:
Establish Model Constraints
x1 + x2 = 1,000 lb
x1 500 lb of chicken
x2 200 lb of beef
x1/x2 2/1 or x1 - 2x2 0
x1, x2 0
The Model: Minimize Z = $3x1 + 5x2
subject to: x1 + x2 = 1,000 lb
x1 50
x2 200
x1 - 2x2 0
x1,x2 0
Solution
Example Problem No. 1 (3 of 3)
46. 2-46
Solve the following model
graphically:
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2 10
6x1 + 6x2 36
x1 4
x1, x2 0
Step 1: Plot the constraints
as equations
Example Problem No. 2 (1 of 3)
Figure 2.23 Constraint Equations
47. 2-47
Example Problem No. 2 (2 of 3)
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2 10
6x1 + 6x2 36
x1 4
x1, x2 0
Step 2: Determine the feasible
solution space
Figure 2.24 Feasible Solution Space and Extreme Points
48. 2-48
Example Problem No. 2 (3 of 3)
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2 10
6x1 + 6x2 36
x1 4
x1, x2 0
Step 3 and 4: Determine the
solution points and optimal
solution
Figure 2.25 Optimal Solution Point