Jon Curwin is Principal Lecturer in Business Analysis and Senior Learning and Teaching Fellow, Birmingham City Business School
Title | Quantitative Methods for Business Decisions |
Author | |
Edition | 7, illustrated |
Publisher | Cengage Learning, 2013 |
ISBN | 1408064987, 9781408064986 |
Length | 606 pages |
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Learn the complex mathematical models and problem-solving techniques encountered in later courses in economics, accounting, finance and production and operations management..
The course covers the complex mathematical models and problem-solving techniques encountered in later courses in economics, accounting, finance and production and operations management. Topics include descriptive statistics, probability and probability distributions, hypothesis testing, populations and sampling, analysis of variance, simple and multiple regression analysis, time series forecasting and modeling and introductory chi-square techniques.
Kaizen is about changing the way things are. If you assume that things are all right the way they are, you can’t do kaizen. So change something! —Taiichi Ohno
Inspect & adapt: overview.
The Inspect and Adapt (I&A) is a significant event held at the end of each PI, where the current state of the Solution is demonstrated and evaluated. Teams then reflect and identify improvement backlog items via a structured problem-solving workshop.
The Agile Manifesto emphasizes the importance of continuous improvement through the following principle: “At regular intervals, the team reflects on how to become more effective, then tunes and adjusts its behavior accordingly.”
In addition, SAFe includes ‘relentless improvement’ as one of the four SAFe Core Values as well as a dimension of the Continuous Learning Culture core competency. While opportunities to improve can and should occur continuously throughout the PI (e.g., Iteration Retrospectives ), applying some structure, cadence, and synchronization helps ensure that there is also time set aside to identify improvements across multiple teams and Agile Release Trains .
All ART stakeholders participate along with the Agile Teams in the I&A event. The result is a set of improvement backlog items that go into the ART Backlog for the next PI Planning event. In this way, every ART improves every PI. A similar I&A event is held by Solution Trains .
The I&A event consists of three parts:
Participants in the I&A should be, wherever possible, all the people involved in building the solution. For an ART, this includes:
Additionally, Solution Train stakeholders may also attend this event.
The PI System Demo is the first part of the I&A, and it’s a little different from the regular system demos after every iteration. This demo shows all the Features the ART has developed during the PI. Typically the audience is broader; for example, Customers or Portfolio representatives are more likely to attend this demo. Therefore, the PI system demo tends to be a little more formal, and extra preparation and setup are usually required. But like any other system demo, it should be timeboxed to an hour or less, with the level of abstraction high enough to keep stakeholders actively engaged and providing feedback.
Before or as part of the PI system demo, Business Owners collaborate with each Agile Team to score the actual business value achieved for each of their Team PI Objectives , as illustrated in Figure 1.
The achievement score is calculated by separately totaling the business value for the plan and actual columns. The uncommitted objectives are not included in the total plan. However, they are part of the total actual. Then divide the actual total by the planned total to calculate the achievement score illustrated in Figure 1.
In the second part of the I&A event, teams collectively review any quantitative and qualitative metrics they have agreed to collect, then discuss the data and trends. In preparation for this, the RTE and the Solution Train Engineer are often responsible for gathering the information, analyzing it to identify potential issues, and facilitating the presentation of the findings to the ART.
Each team’s planned vs. actual business value is rolled up to create the ART predictability measure, as shown in Figure 2.
Reliable trains should operate in the 80–100 percent range; this allows the business and its external stakeholders to plan effectively. (Note: Uncommitted objectives are excluded from the planned commitment. However, they are included in the actual business value achievement, as can also be seen in Figure 1.)
The teams then run a brief (30 minutes or less) retrospective to identify a few significant issues they would like to address during the problem-solving workshop . There is no one way to do this; several different Agile retrospective formats can be used [3].
Based on the retrospective and the nature of the problems identified, the facilitator helps the group decide which issues they want to tackle. Each team may work on a problem, or, more typically, new groups are formed from individuals across different teams who wish to work on the same issue. This self-selection helps provide cross-functional and differing views of the problem and brings together those impacted and those best motivated to address the issue.
Key ART stakeholders—including Business Owners, customers, and management—join the retrospective and problem-solving workshop teams. The Business Owners can often unblock the impediments outside the team’s control.
The ART holds a structured, root-cause problem-solving workshop to address systemic problems. Root cause analysis provides a set of problem-solving tools used to identify the actual causes of a problem rather than just fixing the symptoms. The RTE typically facilitates the session in a timebox of two hours or less.
Figure 3 illustrates the steps in the problem-solving workshop.
The following sections describe each step of the process.
American inventor Charles Kettering is credited with saying that “a problem well stated is a problem half solved.” At this point, the teams have self-selected the problem they want to address. But do they agree on the details of the problem, or is it more likely that they have differing perspectives? To this end, the teams should spend a few minutes clearly stating the problem, highlighting the ‘what,’ ‘where,’ ‘when,’ and ‘impact’ as concisely as possible. Figure 4 illustrates a well-written problem statement.
Effective problem-solving tools include the fishbone diagram and the ‘5 Whys.’ Also known as an Ishikawa Diagram , a fishbone diagram is a visual tool to explore the causes of specific events or sources of variation in a process. Figure 5 illustrates the fishbone diagram with a summary of the previous problem statement written at the head of the ‘fish.’
For our problem-solving workshop, the main bones often start with the default categories of people, processes, tools, program, and environment. However, these categories should be adapted as appropriate.
Team members then brainstorm causes that they think contribute to solving the problem and group them into these categories. Once a potential cause is identified, its root cause is explored with the 5 Whys technique. By asking ‘why’ five times, the cause of the previous cause is uncovered and added to the diagram. The process stops once a suitable root cause has been identified, and the same process is then applied to the next cause.
Pareto Analysis, also known as the 80/20 rule, is used to narrow down the number of actions that produce the most significant overall effect. It uses the principle that 20 percent of the causes are responsible for 80 percent of the problem. It’s beneficial when many possible courses of action compete for attention, which is almost always the case with complex, systemic issues.
Once all the possible causes-of-causes are identified, team members then cumulatively vote on the item they think is the most significant factor contributing to the original problem. They can do this by dot voting. For example, each person gets five votes to choose one or more causes they think are most problematic. The team then summarizes the votes in a Pareto chart, such as the example in Figure 6, which illustrates their collective consensus on the most significant root cause.
The next step is to pick the cause with the most votes and restate it clearly as a problem. Restating it should take only a few minutes, as the teams clearly understand the root cause.
At this point, the restated problem will start to imply some potential solutions. The team brainstorms as many possible corrective actions as possible within a fixed timebox (about 15–30 minutes). The rules of brainstorming apply here:
The team then cumulatively votes on up to three most viable solutions. These potential solutions are written as improvement stories and features, planned in the following PI Planning event. During that event, the RTE helps ensure that the relevant work needed to deliver the identified improvements is planned. This approach closes the loop, thus ensuring that action will be taken and that people and resources are dedicated as necessary to improve the current state.
Following this practice, problem-solving becomes routine and systematic, and team members and ART stakeholders can ensure that the train is solidly on its journey of relentless improvement.
The above describes a rigorous approach to problem-solving in the context of a single ART. If the ART is part of a Solution Train, the I&A event will often include key stakeholders from the Solution Train. In larger value streams, however, an additional Solution Train I&A event may be required, following the same format.
Due to the number of people in a Solution Train, attendees at the large solution I&A event cannot include everyone, so stakeholders are selected that are best suited to address the problems. This subset of people consists of the Solution Train’s primary stakeholders and representatives from the various ARTs and Suppliers .
Last update: 22 January 2023
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Mr. Robinson has a B.S. in Math, University of Pittsburgh, And M.A. in Secondary Education, University of Phoenix. He is certified to teach in secondary schools. He has several years experience in the classroom. and as a computer programmer at NASA's Goddard Spaceflight Center..
Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
An example of quantitative reasoning would be one of George Polya 's steps to problem solving, developing a plan. This means after understanding the problem, then determining how to solve it.
In quantitative reasoning, we study how to understand the problem, how to develop a plan to solve the problem, and how to execute the plan and check the results.
What is quantitative reasoning, what are quantitative skills, types of quantitative reasoning, quantitative reasoning process, quantitative reasoning examples, lesson summary.
Quantitative Reasoning is the ability to use mathematics and information to solve real world problems. Learning about quantitative reasoning may also help in solving non-mathematical problems. For this lesson, the focus will be on translating real world problems into solvable mathematical problems. In 1945 George Polya published a book, How To Solve It . It quickly became his most prized publication. It sold over 1,000,000 copies and has been translated into 17 languages. In his book he identifies the four basic principles of quantitative reasoning in solving problems.
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Quantitative skills or quantitative thinking are skills that enable one to solve problems with numbers. The quantitative skills definition includes having skills in number operations, algebraic, geometric, trigonometric and any other numerical calculations. Quantitative skills are needed especially in the areas of developing and carrying out a plan to solve a problem.
Quantitative skills include the following:
1. Reading and identifying mathematical Information
2. Interpreting and analyzing mathematical information
3. Finding appropriate methods of solving problems
4. Evaluating the validity of results
5. Effective communication of quantitative concepts
Cognitive skills include some of the following:
1. Attention span
3. Logic and reasoning
4. Auditory and visual ability
Quantitative skills help in developing cognitive skills, interpreting, and analyzing mathematical information, finding appropriate methods of solving problems and evaluating the validity of results. These quantitative skills enhance the logic and reasoning skills. Reading and identifying mathematical information increase the attention span and memory skills. Effective communication of quantitative concepts improves auditory and visual ability.
Problem solving is the primary focus of quantitative reasoning. Understanding the problem, setting up a plan and then carrying out the plan are the major steps.
Using quantitative reasoning determines if enough information (data sufficiency) is available to solve the problem
This is the 4 step process of problem solving made famous by George Poyla:
In order to understand the problem, what is unknown needs to be determined. What data is available? These are the first challenges.
In order to develop a plan, the data must be connected with the unknown. Can previous solved problems be used to solve this problem? What formulas and equations are needed to set up a plan? Have all the aspects of the problem been looked at?
Execute the plan.
Check the work. Are the results consistent?
What will be the perimeter of a room given the width and area? The problem is to find the perimeter. Given the width is 8 feet. The area is 80 square feet. There are two ways to solve this problem. The first solution is to use the equations that compute the area and perimeter. The second solution is to draw a diagram of the problem.
Understand the problem: The first way to solve the problem will be to use equations that compute the area and perimeter of a rectangle .
Devise a plan:
Let W represent the width of the room, and A the area. then
W = 8 ft., A = 80 sq. ft., L = length, P = perimeter
Equation to find the area is W x L = A
Equation to find the perimeter is 2W + 2L = P
The plan will be to use these equations in solving the problem.
Execute the plan: Find the length L.
W x L = 80 sq. ft.
8 x L = 80 sq. ft.
(8 x L)/ 8 = (80 sq. ft.)/8
L = 10 sq. ft.
Find the perimeter P.
P = 2W + 2L
P = 2(8) + 2(10)
P = 16 + 20
Look back: Check.
8 x 10 = 80 sq. ft.
Understand the problem:
Find the length L.
Devise a plan: Use the formula to find the perimeter
2W + 2L = P
Use this formula to find the length L.
Execute the plan: W = 8 ft.
(8 x L)/8 = 80/8
2(8) + 2(10) = P
16 + 20 = P
John is a short order cook at Big Boys. His salary is $18 per hour. He is given a 12% raise. John works 35 hours per week. How much more money will he receive per week because of his increase in pay?
In order to find a solution, find the difference between what he made before the raise and what he will make after the raise.
(Weekly Pay After Increase) minus (Weekly Pay Before Increase) = Difference
Oldpay = $18/hr, Newpay = Oldpay x 1.12 = $20.16
Oldweek = old weekly pay,
New-week = new weekly pay after increase
Difference = New-week minus Oldweek
Execute the plan:
Newpay = $18/hr x 1.12 = $20.16/hr
Oldweek = Oldpay x 35 hrs
Oldweek = $18/hr x 35 hrs
New-week = Newpay x 35
= $20.16 x 35
= $705.60 minus $630
John makes an extra $75.60 a week.
$630 + $75.60 = $705.60
_Quantitative Reasoning is the ability to use mathematics and information to solve real world problems. In this lesson, the George Polya quantitative reasoning method was used to solve problems. The 4 steps from his method are as follows:
Quantitative Reasoning uses Quantitative skills or quantitative thinking in solving problems with numbers. The quantitative skills definition includes having skills in number operations, algebraic, geometric, trigonometric and any other numerical calculations.
Quantitative skills help in developing cognitive skills , interpreting and analyzing mathematical information, finding appropriate methods of solving problems and evaluating the validity of results. Cognitive skills enhance logic and reasoning skills. Reading and identifying mathematical information enhances the attention span and memory skills. Effective communication of quantitative concepts improves auditory and visual ability.
Quantitative reasoning.
In an effort to develop a program to decrease the amount of sugar the people in the city of Stoneville are eating, the mayor is gathering facts about the town's residents. He finds that the city's population is 12,322, and that the city's stores sell a total of 19,820 pounds of candy each year. He wants to figure out how much candy the average resident buys in a year.
He first notes that if he could divide the number of pounds out evenly between the residents, then the amount assigned to each resident would equal the average number of pounds that resident bought. Ah-ha! He simply needs to divide the number of pounds of candy by the number of residents, so this is the plan he devises to solve the problem.
Now, he just needs to carry out the plan.
19,820 / 12,322 ≈ 1.6
It looks like, on average, each resident of the city buys 1.6 pounds of candy per year. The mayor decides that this makes sense based on the facts of the problem, so he has his answer.
The reasoning that the mayor used in this scenario is an example of using quantitative reasoning to solve a real-world problem. Quantitative reasoning is the act of understanding mathematical facts and concepts and being able to apply them to real-world scenarios.
Many standardized tests have a quantitative reasoning section. Tackling these types of problems can be done using a number of strategies. First and foremost, when dealing with any type of quantitative reasoning problem, it's a good idea to have a plan.
Thankfully, there's a nice four-step process that George Polya, a Hungarian mathematician, developed to solve problems in general, and it can often be used to organize your thoughts and develop a plan to solve a given problem.
This process can aid in quantitative reasoning in that it gives a nice strategy on ways to think about the problem in an organized manner. If we look back at how the mayor solved his problem, we see that he used this process.
This process can be used for any type of problem, but the quantitative reasoning comes in at steps two and three when we devise a plan and carry it out. Knowing how to identify the relationships among the quantities in the problem and connect those relationships to appropriate operations is quantitative reasoning at its finest. Consider another example.
Suppose that the distance around a rectangular flower garden is 22 feet. The length of the garden is 7 feet, but we don't know the width, and we'd like to figure it out. Let's take it through our steps!
First, we make sure that we understand the problem. Well, let's see. The garden is a rectangle, and we know that its length is 7 feet. This tells us that two of the sides of the garden that are opposite one another have lengths of 7 feet. We also know that the distance around the garden, or its perimeter, is 22 feet. Okay, I think we can move on to Step 2 and devise a plan.
There's a couple different strategies we can use here. If we're familiar with the perimeter formula of a rectangle, we can plug our known values into the formula and use that to solve the problem. If we're not familiar with the formula, we could draw a picture of the garden and look for ways to solve the problem that way. Great! We have two strategies we can use. On to Step 3.
The first option is to use the perimeter formula. The perimeter formula for a rectangle is as follows:
We know that P = 22 and l = 7. We can plug these into the formula and solve for w to get our width.
As you can see, after we simplify, we get 22 = 14 + 2 w , and after we subtract 14 from both sides, we see that 8 is 2 w , making w = 4. So, in other words, we get that the width of the garden is 4 feet. Looking back, this makes sense with the problem. Let's see if we get the same with our other strategy for solving. In this strategy, we use a picture.
We know that the distance around the garden is 22 feet, and from the picture, we can see that the sides have lengths 7, 7, w , and w . Therefore, if we add up the lengths of the sides, they should equal 22.
7 + 7 + w + w = 22
Once again, we have an equation we can use to solve for w to find our width.
Like before, after we subtract 14 from both sides, we get 2 w equaling 8, which makes w - yes - 4! Simple! So once again, we get that the width of the garden is 4 feet, solidifying our confidence that this is the correct answer.
Let's take a moment or two to review the important information that we've learned.
Quantitative reasoning is the act of understanding mathematical facts and concepts and being able to apply them to real-world scenarios. A nice four-step process to tackle these types of problems is Polya's problem solving process:
The quantitative reasoning strategies come in at steps two and three. These can include drawing a picture, translating the problem into numerical expressions, working backwards, or the like. Basically, it just involves turning a real-world scenario into a mathematical problem that we can solve using various math strategies. Though this process may seem difficult at first, with some practice it gets much easier, and your quantitative reasoning becomes much stronger - so keep practicing!
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