Choose a Strategy

Choose a Strategy

Choose a Strategy
Seeking the ‘best choice’ is the most obvious strategy in decision-making situations.
However, other strategies often play a role in decision-making.
Eliminate the worst:
Sometimes it is more important to avoid making a poor choice than it is to make the best choice.
This strategy is often used during the first visit to a new restaurant. When there are many menu selections and you are not sure what you might enjoy most, a common strategy is to narrow the range of choices by eliminating the choices that definitely do not appeal. A vegetarian might first mentally delete all the meat dishes. Someone on a first date might first delete all the dishes with garlic. Someone on a diet might first delete all the dishes with rich creams and sauces.
During World War II, the psychologist J. P. Guildford devised aptitude tests that were used by the US Army Air Force to help in screening candidates for pilot school . After implementing these tests the failure rate in pilot training was reduced by over 60%.
Guilford’s psychological tests were not designed to select the best pilots, but to help identify and eliminate the candidates who were unlikely to become pilots.
Some employers use a similar strategy when hiring and promoting – especially when union rules and/or civil laws make it very difficult to dismiss an employee for incompetence. In such a situation, an employer’s first concern is to ensure that they do not hire an unsuitable candidate. After unsuitable candidates have been eliminated from the process, the employer cannot go too far wrong in hiring any of the remaining applicants.
Pick the lowest bid
In the lowest bid strategy, you create a set of specifications describing what you are looking for and then allow qualified suppliers to submit their offers. Many contracts for public works are awarded this way.
The lowest bid process has two main objectives. First, the process is intended to be open and fair so that all suppliers have an equal opportunity to make their bids.
Second, the process is intended to reduce costs. If a bid is inflated then it probably will not be lowest and thus will not be accepted.
It might be fun to buy a new car using a lowest bid process. You would describe the car that you want and then all the auto dealerships in town would have until the end of the week to submit their bids. All you would have to do is open the envelopes and call the winning dealer to get the car you want at the lowest price in town.
Rank the choices
Another decision-making strategy is to rank all your choices, from highest to lowest, and then select the top ranked choice. In order to rank choices, you have to first establish a set of criteria to rate various aspects of each choice. Then you apply those criteria and construct a final ranking for all your choices.
Suppose that you are trying to decide among ten possible destinations for your next vacation. Using a scale of one-to-five, you could rate cost, accommodation, travel time, local attractions, and safety for each possible destination. Then all you would have to do is add the scores to determine where you are going for your vacation.

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Clarify Your Choices

Clarify Your Choices

Clarify Your Choices
The first step in the decision-making process is to clarify what your choices are. This involves gathering the pertinent information, arranging that information so that similar features in the choices can be easily compared, and then summarizing the options that are available with each choice.
The next step is to analyze each choice to consider all the potential consequences. Once the costs and benefits of each available choice are clearly defined, the decisionmaking process is greatly simplified.
Benefits
What is a benefit? A benefit could be a gain in physical resources, financial resources, social status, or future opportunities. A benefit could represent enhanced chances of survival for yourself, your family, or your nation. A benefit could be enhanced comfort, increased personal power, or decreased workload.
In most situations, each of the possible choices that you can make has some associated benefits; otherwise there is no point in making it a choice. Some benefits associated with a choice are obvious; some are more obscure; some may require reasonable predictions of events in the future. Some benefits occur almost immediately; some may not occur unless other favourable events also occur. There can be major benefits, secondary benefits, and deferred benefits associated with any choice
The benefit of a choice may be that it has the least negative consequences.
Costs
In most situations, each of your possible choices also has some associated costs. Some costs are obvious; some are hidden; some may require reasonable predictions of events in the future. Some costs may occur almost immediately; some may never occur unless other unfavourable events occur as well. There can be major costs, secondary costs, and deferred costs associated with any choice.
The failure to recognize all the costs, especially ‘hidden’ costs is one of the commonest sources of difficulty in decision-making. Sometimes hidden costs are simply overlooked; sometimes they are ignored or deliberately hidden to bias the decision-making process.
Money, time, and physical energy are obvious costs. Lost purchasing power, psychic wear and tear, damage to the environment, and social repercussions are some of the costs that tend to be ignored.
Other factors
There are a few other factors that should also be considered before making a choice.
Can a decision be reversed? If it could be reversed, what would be the cost? What would be the cost of not making a particular decision? What is the cost of not making any decision? What is the cost of delaying a decision? Is a particular choice guaranteed? Are any of the choices ‘win-win’? Are there other choices that have yet to be considered?
Suppose you plan to purchase a new car this year. There are literally thousands of makes, models, and options to choose from. It is a challenging task to collect and compare all the available information. The Canadian Automobile Association (www.caa.ca) provides some information on costs that allows you to begin comparing the expenses associated with different vehicles. The safety ratings of automobiles can be checked at www.safecar.gov. As you gather more specific information for side-byside comparisons, it becomes easier to make a reasoned decision.

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Decision-Making

Decision-Making

Decision-Making
Every action we take is the result of a decision. Do we take our coffee with milk or cream? Do we wear the red outfit or the blue outfit? Do we turn left or right to go around that tree? Do we save or spend? Do we apply for that job or not? Which of the political candidates should we vote for?
Decision-making can involve a choice between two alternatives, the ranking of alternatives, the elimination of the worst alternatives, or the selection of the best alternatives.
The simple ‘if – then –jump’ construct is one of most useful processes controlling all the computers in the world. This simple line of code provides a computer program with the ability to compare two numbers, decide if they are equal or not, and then jump to other corresponding lines in the program. This very basic decision-making process underlies the ability of computer programs to solve complex problems and appear to mimic some aspects of human thinking.
The following example illustrates this important concept. Just about any quality can be expressed as a number, for example the colour green could be ‘6’ and red could be ‘7’. At some point in a computer program, suppose the variable A corresponds to the colour of a sweater that you want to order over the Internet, and B corresponds to the colour of the sweaters in stock. Then a segment of a computer program might look something like this:
300 If A = B then jump to line 400.
400 Print “We have your sweater in stock.”
600 Else If A <> B then jump to line 700
700 Print “Sorry we do not have that colour at the moment.
(The symbol <> is often used in computer programs to mean ‘not equal to’.)
The rest of this is section is devoted to examining strategies for making the best decisions in challenging situations.

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Dealing with Employee Theft

Dealing with Employee Theft

Dealing with Employee Theft
Suppose that you own a small retail store with a dozen employees. Your sales have remained steady, but your profits have dropped and you suspect that the cause may be employee theft. What can you do to deal with this problem?
First you should remind yourself that reliable and efficient employees are a business’s greatest asset. You want to take all reasonable steps to encourage and reward your good employees. Any action taken against an employee needs be judicious and planned to cause minimum disruption to your business.
Second, you should conduct some background research and reassess the general situation. ‘Shrinkage’, as it is often called, is a common problem in any business.
Somehow, the revenues at the end of the week are less than they should be. There are a number of potential causes, including:
1. Poor inventory control – sometimes you do not receive all the inventory that you have paid for, some inventory may be damaged and has to be written off, some sold items are returned, some items are discounted, or some items are given away for advertising.
2. Poor accounting – through poor bookkeeping you may not really know how much inventory was sold at what prices, you may not know the true total of your expenses, or you may not know how much you owe others, or others owe you.
3. Theft – items may be stolen by shoppers, by employees, or by third parties after your business is closed for the day. Theft may be described as pilfering – a little here, a little there – or theft may be described as a money-making operation for a professional thief.
Prevention is a major component in controlling shrinkage. You may need to update and develop better procedures for inventory control and accounting. You may need to do a better job of selecting employees in the first place.
You may need to hire a security expert to review your procedures for dealing with customers, handling cash, and securing inventory. You may need to organize a professional development program for your employees outlining any new procedures and emphasizing the benefits of being part of a prosperous business.
If an employee is caught with incriminating evidence, you may call the police and press charges, you may fire the employee, or if the offence was minor you might be able to work out an ad hoc rehabilitation program. In all such cases it is essential to carefully document your evidence and your actions. In general, problems involving people need to be dealt with carefully and according to established procedures.

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The Monty Hall Paradox

The Monty Hall Paradox

The Monty Hall Paradox
There was a popular television game show in the 1970’s called ‘Let’s Make a Deal’ with host Monty Hall. In a regular segment of the show, a contestant was shown three doors and was given the details of a really good prize that was behind one door, and poor consolation prizes behind the other two. With a lot of showmanship, the typical game then proceeded as follows:
1. The contestant selected Door #1, Door #2, or Door #3, hoping the prize was behind that door. The door was not opened at this stage of the game.
2. Monty then opened one of the other two doors to reveal a consolation prize.
3. The contestant was then given the opportunity to switch from the original selection to the third and last door.
The problem is to determine whether or not switching doors would have been a good strategy. This is a problem in deduction. The key to its solution is to find a way to express the components of the problem so that the implications are more obvious. 

At a first glance, the following statements seem to be correct:
1. Since there are three doors and your selection is essentially random, the probability of winning the grand prize after your first selection should be one in three.
2. After Monty reveals a consolation prize, there are only two doors left, so your chances of winning should increase to one in two.
3. There appears to be no obvious advantage to switching your choice; there are still two doors with a big prize behind only one door.
In reality, participants who switched doors won about two times out of three. That is the paradox. How could switching doors be an advantage? If you search the Internet using the key words ‘Monty Hall paradox’ you will find hundreds of sites that explore this problem.
To simplify the discussion, let’s assume that the prize is behind Door #1, but as a contestant you do not know that. You still have a random choice of one of the three doors. As Monty takes his turn he will always open one of the two remaining doors to reveal a consolation prize, never the door to the grand prize.
Working through the possibilities:
1. If you picked Door #1 – then Monty will open either Door #2 or Door #3. In this case, if you switch from Door #1 you lose.
2. If you picked Door #2 – since Monty knows the prize is behind Door #1, he has to open Door #3 revealing a consolation prize. In this case if you switch to Door #1 you win.
3. If you picked Door #3 – then similarly Monty has to open Door #2 revealing another consolation prize. Again, in this case if you switch to Door #1 you win.
The net result is that if the prize is behind Door #1 and your first choice is random, then two times out of three, you will win the grand prize if you switch doors after Monty reveals a consolation prize. The same pattern applies if the prize is behind Door #2 or Door #3.
The key to the paradox is the phrase Monty ‘has to open’ in the second and third possibilities. For example, if the prize is behind Door #1 and you have selected Door #2, then Door #3 is the only choice available for Monty to open without spoiling the game. In two cases out of three, your initial choice plus Monty’s forced choice direct you to the winning door.

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Two Applications

Two Applications

Two Applications
As you read through the following descriptions of problems, try to identify the various problem-solving strategies that are being used. You can probably devise alternate strategies that would work just as well.

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Recreational Problems

Recreational Problems

Recreational Problems
Many of us enjoy working on problems as a form of recreation.
Jigsaw puzzles, Sudoku puzzles, crosswords, table games, and video games are all designed to provide invigorating challenges for the human mind. The only reward in these activities is the satisfaction of solving the puzzle.
Many of our creative hobbies involve significant problem solving. What are the best materials to use? How can certain effects be achieved? Is there a more interesting way to achieve a similar result? A partial list of creative hobbies includes amateur astronomy, carpentry, collecting, painting, photography, sewing, sculpting, and writing. The products of these hobbies can often be purchased elsewhere readymade, but there is a great source of satisfaction in creating something yourself and being able to solve all the attendant problems.

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Professional Problem Solvers

Professional Problem Solvers
Experts and professionals in any field have specialized knowledge and experience that allow them to deal efficiently with problems that a lay person would find very challenging. Experts and professionals like to deal with these problems; their livelihoods depend on being presented with problems that require their knowledge and expertise.
Professional problem solvers include medical doctors, dentists, lawyers, architects, engineers, computer programmers, auto mechanics, and police detectives.
There are others, including university professors, government and industrial researchers, and inventors who deliberately seek out problems that no one else has been able to solve. Their professional success is based on their ability to find and solve unique problems. Their work contributes to humanity’s library of knowledge and
problem-solving techniques.

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Problems for Students

Problems for Students
Problems constructed specifically for students provide both learning challenges and a means for evaluating student performances.
In most student assignments, tests, exams, and contests – problems are provided in written form along with rules restricting time and the resources that can be used.
Most student problems are intended to help students learn and consolidate the material being studied. Students can usually assume that any assigned problems can actually be solved. Lots of practice in solving problems in a given subject area is the very best strategy for developing problem-solving skills in that subject area. Relativestudent- skills in problem solving are often used to grade and rank students.

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Looking for Problems

Looking for Problems

Looking for Problems
Rather than avoiding problems, some individuals, professions, and industries thrive on problems. Many professions are based on dealing with problems that are beyond the capacities of lay people. Many recreational activities are based on finding and working with entertaining problems.
Humans like challenges. Detectives need crimes to solve. Surgeons need damaged bodies to repair. Mathematicians need equations to solve. Plumbers need leaks to fix.
Teachers need children to teach.

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Shaping the Environment

Shaping the Environment

Shaping the Environment
One last approach to avoiding problems is to shape your environment so that events and activities lead naturally to the intended result with minimal intervention.
As a very simple example, the drain in any bathtub is located at the lowest point so that water will drain completely when the plug is pulled. If the drain were located anywhere else in the tub, there would always be extra work required to remove all the water after a bath.
The Montessori method of education makes frequent use of equipment that is selfevaluating.
If children can place sets of geometric shapes in the corresponding recessed places on a board, then they can see for themselves that the task has been completed correctly. If a child can prepare fruit and pour juice without an accident, then they have obviously succeeded in serving a snack.
A proper fence around a yard can provide a safe place for children to play with minimal supervision.
In industry it is easier to operate in a neat control room with clearly labelled dials and colour coded switches than it is if all the switches and dials look alike and are distributed across a control panel at random.
Building a canal can simplify the transport of goods by water. Building a bridge across a valley can simplify travel. Reducing air pollution can ease suffering due to asthma and allergies.
Shaping your environment using elements of design, engineering, and creativity can eliminate a host of potential problems before they occur.

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Advanced Planning and Preparation

Advanced Planning and Preparation

Advanced Planning and Preparation
When dealing with large and complex systems, advanced planning and preparation can also help you avoid, or at least reduce the impact of potential problems.
Dealing with complexity
When you are dealing with any project that is large, complex, and/or expensive then careful planning is the best way to avoid significant problems. Careful planning should include a detailed written plan, consultation with experts, and the creation of stepby- step procedures. It is far easier to deal with a potential problem at the beginning of a project than it is to make major modifications after the project has been partially completed. It is far easier to conduct a careful survey and then build a wall in its correct location, than it is to try to move the wall after it has been built.
Timing and scheduling often play a role in major projects. Projects grind to a halt if supplies, equipment, and/or workers are not available when needed. Costs mount rapidly if supplies, equipment, and/or workers are sitting around before they are needed.
On a personal level, two examples of endeavours that can benefit from careful planning are your educational career and your vacations.
In education you should look ahead several years to find a direction that will best satisfy your needs, ambitions, interests, and abilities. Once you have decided on an educational goal, you should work backwards from that goal to your present situation and make lists of courses you need and standards that you have to meet, since most advanced programs require prerequisite courses with minimum academic standards.
Without adequate planning, you can end up spending an extra year or two at school making up for missing prerequisites.
When going on vacation, you often have to plan several months in advance in order to get the flights and accommodations that you want. Tourism is one of the world’s largest industries, and chances are that thousands of other travellers want to visit the same places that you do. You can also enhance your vacation experience by learning more about your travel destination.
Carpenters have a saying that sums up the essence of this section, ‘measure twice, and cut once’. In other words, plan carefully and you will not have to repeat your efforts or waste material.
Contingency plans
Contingency plans are plans that you make for situations that are unlikely to occur, but are very serious if they do. When making a contingency plan you can carefully consider alternatives and identify the essential steps of the best plans.
Fire drills at home, at school, and in industry; life-boat drills on cruise ships; and the pre-flight review of emergency procedures on airplanes are common examples of contingency planning. Note that contingency planning requires the development of procedures, the design and provision of equipment, and step-by-step rehearsals. Then
if an emergency situation does arise, people can quickly react by following the recommended procedures.
A clear thinker should anticipate, make contingency plans, and be ready to act in response to situations with high stakes and limited response times.

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Anticipate

Anticipate

Anticipate
Knowledge of specific situations and the expectations, skills, and limitations of yourself and others often allows you to manage events in order to reduce the potential for encountering related problems.
Anticipate social needs
If you know that a person is liable to be cranky on Monday morning, why not arrange for low demand activities on Monday and wait until Tuesday to announce a new project? If you can predict that a teenager will ask to use the family car on the weekend, why not ask him/her to take it in for a tune-up and a safety check earlier in the week? That way you have some assurance that the car is in good working order, and the teenager has a vested interest in keeping it that way. If your spouse likes to go out for dinner on special occasions and an anniversary is approaching, why not make reservations at a favourite restaurant?
Anticipate physical requirements
The last time you went for a hike you ended up with feet that were wet and sore; why not shop for better hiking boots and socks before you venture forth again? There is always a possibility that your computer will crash with the loss of important computer files; why not get into the habit of creating back-up files? The last time the power went out you had difficulty finding a working flashlight, why not purchase a handcranked flashlight and store it in a convenient location in your home?
In engineering, it is standard practice to calculate the required strength of elements in a structure and then to build components with double or quadruple those strengths. The extra strength provides a ‘factor of safety’ to allow for unexpected stress and loading. Most major systems on the American space shuttle are built in triplicate so that the ship can continue to function even if a main system and its first backup have failed.
Anticipate the reactions of others
Driving a car provides lots of opportunities to avoid problems through clear thinking. Even though other drivers are supposed to follow the rules of the road and make reasonable decisions, they often do not. They may be forgetful, distracted, unskilled, or simply discourteous – but in many situations, after observing the behaviour of the drivers around you for a few seconds, you can predict that they are going to change lanes, turn, stop, or make some unusual manoeuvre. This type of anticipation and planning can help you avoid problems on the road before they occur.

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Avoiding Problems

Avoiding Problems

Avoiding Problems
Your knowledge of any working system can allow you to make reasonable predictions about how it will react to various inputs. At work and at play, your ability to think clearly and anticipate what might happen, can help you avoid a range of problems before they occur.
The basic strategies for avoiding problems are to anticipate, plan ahead, and shape the appropriate environment.

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Review and Consolidate Your Work

Review and Consolidate Your Work

Review and Consolidate Your Work
Finally, it is important to document the methods you used, and the solution that you obtained.
When your problem solving efforts are well documented, you are prepared to apply what you have learned to future efforts. This is particularly true for students. It is not enough to solve a problem and obtain the ‘right answer’. You also need to think about how you solved that problem and how you might use those same methods on similar problems in the future. Without this type of consolidation, little learning takes place and every new problem has to be solved again from scratch.
Often a solution by itself is not sufficient. You may also have to convince others that your solution actually works and that it is the best, fastest, most practical, most economical, or most environmentally friendly approach. You may have to defend your solution and demonstrate that any funds were spent properly, that it represents your own work, that safety rules were followed, or that the laws of the land were not broken.
You may also want to examine your solution to determine if it provides any new insights, or a better understanding of the general situation.
You should also consider the implications and ramifications of your solution. Sometimes a solution to one problem can create other problems that are more serious than the original. You can buy the biggest, most comfortable sofa in the store but if you do not have enough space for it in your current home, you have created another problem. You can kill all the weeds and then find that nothing else will grow either.
You can build an expressway to relieve traffic and end up facilitating the abandonment of the city core as more people move to the suburbs.
Starting about 1980 and continuing for most of a decade, the Central Intelligence Agency (CIA) of the United States supplied funding, training, and weapons to Muslim extremists in Afghanistan and Pakistan. The objective was assist in the overthrow of the Soviet-backed government in Afghanistan. The strategy worked – but it also led to the creation of the Taliban and Al Queda movements, which soon became enemies of the United States.

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Verify Your Solution

Verify Your Solution

Verify Your Solution
No solution is complete until you have checked that it contains no obvious errors; it satisfies the parameters of the original problem; and field-testing has shown that it works under realistic conditions.
First organize, rethink and rewrite your final solution. The process of organizing and simplifying often uncovers a variety of minor errors. Second, apply your solution to a few very simple cases for which you already know the results. If your solution does not work for simple cases, it probably will not work in more realistic situations.
Always double-check your solution against the requirements of the original problem. You may have found a magnificent solution, but for a different problem. It is always reassuring to confirm that the original requirements have been met. Field-testing is recommended whenever others are expected to apply your solution to a variety of new cases. Field-testing a solution serves three vital roles:
1. It demonstrates that others can understand your solution. You may have produced a brilliant solution, but if no one else can understand it, it is of little value.
2. It demonstrates that your solution is workable and can be effectively used by those for whom it was intended.
3. It provides an opportunity to receive comments and suggestions for making further improvements to your solution.

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Still More Strategies

Still More Strategies

Still More Strategies
Despite your best efforts you will always encounter problems that seem unsolvable.
What do you do then?
Re-evaluation
Sometimes it is wise to stop and re-evaluate your problem solving efforts. Is this problem that important? Is it worth more of your time, energy, and resources? Are there more productive ventures you could work on?
You may want to upgrade the significance of a problem and bring in more resources; you may want to downgrade the significance of a problem and withdraw some of your resources. You may want to abandon a problem altogether. While re-evaluating the need to solve a difficult problem you should clarify your own motives and prepare a plan of action that will maximize your chances of success.
Luck
We are all ready to accept a quick solution to a problem that occurs by happenstance. Why work hard when you can get lucky? That is why gambling and the sale of lottery tickets are so popular. Some problems, such as solving a jigsaw puzzle, do involve an element of chance. The first piece you pick up may complete a challenging part of the puzzle, or you may have to search through pieces for several hours to accomplish the same task. More often, experience and a systematic method lead to a simple solution that makes it appear that luck played a role.
Luck can play a role in situations that require just one positive event to greatly simplify your problem. Most real world problems involve many steps and require a systematic approach in which luck has no potential role. Luck may play a role in your taking a different road home and avoiding a traffic jam, but luck will not help you paint the garage.
Hard work
Most challenging problems are ultimately solved with hard work over an extended period of time. If you face a difficult problem and are motivated to solve it, then be prepared to spend a lot of time and effort.
Suppose that one of your children is misbehaving on a regular basis. This is a major problem that can require years of effort, much patience, consultation with experts, trial-and-error, and lots of luck.
If you want to start your own business, you face a host of problems that have to be solved. It is not enough to have a good idea and a grand opening ceremony. Hard work over an extended period of time will be required to make your business a success.
Novel point of view
There is a tendency to stick with your initial approach and assumptions when working on a given problem. This tendency can lock you into a limited range of thought patterns that can, in turn, limit your ability to solve that problem.
We are creatures of habit and tend to reuse recently activated neural pathways when we repeat similar actions. If you sit in the same position, with the same information spread in front of you, as you repeatedly go over aspects of a difficult problem, there is a tendency to repeat exactly the same line of reasoning. When you get stuck on a problem and seem to be repeating the same steps over and over, a new approach can lead to a wider range of thought and action that may lead to better results.
Finding a novel approach may be as simple as sitting in a different chair, asking a different question, or changing an assumption.
In a procedure called ‘brainstorming’ you are encouraged to speculate, without restriction, and to write down all the ideas that come to mind. While a list of unrestricted speculations typically contains mostly irrelevant material, it can also include an idea that stimulates further thought in a new and useful direction.
Brainstorming can work with an individual, but it is more effective with a small group. Brainstorming with a group adds the possibility that one person’s wild speculation will stimulate novel thoughts in another person, and those novel thoughts may lead to a more practical problem solving approach.
Closest similar problem
A particular problem may prove very difficult to solve, but has features similar to other problems that have already been solved. It is possible to gain insights into your current problem by studying the solutions to these related problems. If your problem follows the pattern of the most closely related cases, then perhaps your problem will have a similar solution.
A Viking, lost at sea, could utilize sailing lore and conclude that a stationary smudge of clouds on the horizon might be surrounding the mountains of a distant island. His knowledge of similar situations could thus provide him with the ability to find land when none was in sight.
Medical diagnosis often employs similar-case analysis. When you visit a doctor with your own personal set of flu symptoms, your doctor may have already seen several patients with similar symptoms. Those other similar cases could then form the basis for an initial diagnosis of your particular situation.
Change a problem to make it simpler
Sometimes a problem, as you find it, is simply too difficult to solve. However, if you make one or two simplifying assumptions, then solving the problem may become a realistic task. This technique is often used in mathematics, science, and engineering.
The key to this technique is the judicious choice of simplifying assumptions. There is little point in solving your financial problems by assuming you are a millionaire, or solving a problem in rocket science by assuming you have invented an anti-gravity device. A good assumption eliminates awkward details while having minimal effect on the overall situation.
When estimating the amount of material required for a job, carpenters typically calculate the amount of wood in the finished product, and then increase that amount by ten percent to allow for waste (splits, knots, and short pieces that cannot be used). Experience has shown that expecting ten percent waste is a reasonable assumption in carpentry. It is not necessary to examine every piece of wood and preplan every cut before ordering material.
Einstein’s equations of general relativity are very difficult to solve for the real universe, so cosmologists usually begin by making a fundamental assumption.
Although the universe obviously contains planets, stars, and galaxies, on a very large scale it is useful to assume that the mass of the universe is evenly distributed in all directions. This basic assumption allows the equations of general relativity to be solved for a number of interesting cases and allows cosmologists to make predictions about the behaviour of the universe for billions of years into the future, and backward into the past.
Ask for assistance
When you are having serious difficulty with an important problem, ask for assistance. Almost always, there are others with expertise and experience who can help you with particular aspects of a problem. Often there are co-workers or friends who are pleased to be asked for help. In helping you they are able to make good use of their own skills, are able to feel a sense of satisfaction in providing assistance, and are helping to create a problem-solving network that they may someday need to access themselves.
Time for reflection
Compared to the speed of a computer, the human brain works at a snail’s pace. If possible give yourself time to mull over the different aspects of a problem. Your conscious mind has to work with the limitations of immediate-term memory, but your subconscious mind can sift through thousands of your memories. Give your subconscious mind time to do its work.
We have all had the experience of reflecting on a past conversation and thinking of more effective things that we could have said, if only we had thought of them at the time. Given time, your subconscious mind can often come up with better ideas.
Time for reflection is most beneficial when you have already organized, analyzed, and attempted to solve a problem. After that, your subconscious has all the pertinent information laid out before it and can peacefully chug away through your stored memories looking for items that might provide useful clues the next time you consciously consider the problem. Given more time, there is a better chance that you will find a better solution.

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Some Advanced Strategies

Some Advanced Strategies

Some Advanced Strategies
If the basic strategies are ineffective, there are a variety of other approaches that can be applied to problem solving. These strategies involve models and analogies, logical analysis, and statistical analysis.
Models and analogies
The use of models and analogies allows you to create and examine a similar, but simpler problem. Analyzing the simpler problem often provides insights into the behaviour of the original and more complex system. 

For example, predicting the motion of a tornado in a turbulent atmosphere may be a daunting task beyond the power of a supercomputer. However, analyzing the motion of a rotating cone moving through a smoothly flowing fluid is a much simpler task and might provide useful insights and clues for predicting the motions real-world tornados.
Logical analysis
In a situation with clear definitions and rules governing relationships, it is possible to make use of logical analysis in problem solving. Algebra, geometry, calculus – in fact all branches of mathematics – are subject to logical analysis.
Situations that combine human interactions with logical analysis are ‘fuzzy’ in nature and require approaches that are more flexible. ‘Game theory’ has evolved as a branch of applied mathematics that deals with strategic interactions between humans, when the optimal choice of one person depends on the choices made by other participants.
The underlying assumption is that all participants will make rational decisions based on their own preferences.
Statistical analysis
In some situations there are no precise rules to govern interactions, but many observations have been made and lots of data is available. In those situations, a statistical analysis can reveal patterns in the data, and the patterns can be used for making predictions.
Weather forecasts are based on statistical analyzes. No one knows exactly how wind, sun and rain are going to interact over hours and days, but powerful computer analyzes of data collected from thousands of weather stations and weather satellites can be used to make reasonable forecasts of national and local weather patterns for the near future.
In modern cryptography, the breaking of secret codes involves statistical analyzes and powerful computers to examine millions of possible coding schemes. Alan Turing pioneered this approach at Bletchley Park, in Britain, during World War II. With the help of first-generation electric computers designed by Turing, the code breaking team at Bletchley Park was able to decipher many of the intercepted German communications. That information played a significant role in shortening the duration of the war.
Most studies of the safety and effectiveness of new medicines are based on statistical analysis. Biological systems are so complex that no one knows exactly what is happening at the molecular level in your body when you take a medicine. So, large groups of volunteers are required to test a new drug. Statistical analyzes are used to establish that most of the time when you take a particular drug, you can expect these benefits and occasionally those side effects.

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Basic Strategies

Basic Strategies

Basic Strategies
There are a number of basic problem-solving strategies that can be applied to most problems.
Top-down
With a top-down strategy, you begin with general principles and work towards specific examples that ultimately include the problem situation. The direction of general tospecific gives the strategy its name.
If you planned a meal using a top-down strategy, you might start with the nutritional requirements of an average adult and the nutritional values for a variety of foods, and eventually arrive at an appropriate shopping list, and finally a recipe for tonight’s supper.
The creation of legislation and governmental institutions in the United States of America has followed a top-down process. The Declaration of Independence was issued in June 1776. The new Constitution was written in 1787 and approved by congress in March 1789. This was followed in the same year by the passage of the Bill of Rights containing the first ten amendments to the constitution. Since that time, all other laws and governmental bodies of the United States have had to fit within the framework of the amended Constitution.
Most large computer programs are designed top-down, beginning with a detailed set of performance specifications. Then the primary and secondary features of the program are designed and specifications are created for exchanging information between parts of the program. The actual writing of computer code does not begin until the top-down design has been completed.
Bottom-up
As the name suggests, the bottom-up strategy is the opposite of the top-down. When using a bottom-up strategy for problem solving, you start with a specific aspect of a problem and then work towards a general solution.
If you want to improve your physical fitness, a top-down approach might include buying a new jogging suit and a book on weight lifting. A bottom-up approach might begin with skipping dessert and walking to work.
A bottom-up approach to home-building would begin with digging foundations and hammering nails. A top-down approach to home-building would begin with a site plan and an architectural design.
A trip to your family doctor for a yearly check-up would be part of a top-down approach to your health. A trip to the emergency department at your local hospital is usually a bottom-up strategy.
Left-right
The top-down and bottom-up approaches both imply that solving a problem involves an ordered sequence of events. The term ‘left-right’ is used here to imply that a problem can be approached from a number of different directions at the same time.
The left-right approach is commonly used in problem solving when you have a definite starting point and a definite goal. For instance, if you want to unravel a tangled ball of string, you often have to work alternately with one loose end, and then the other.
If you want to build a bridge across a river, it is often feasible to start construction from both banks with the goal of meeting in the middle.
Problems in deductive geometry often require a left-right approach. In these problems you typically begin with a given set of information and are seeking a particular conclusion. A left-right approach involves starting with the given information to see what other conclusions can be reached. Then you contemplate the ‘answer’ and work backwards; ‘If this statement is true, then what else must also be true?’ You work alternately from the ‘beginning’ and then the ‘end’ with the goal of finding a sequence of logical statements that will meet somewhere in the middle.
(Mathematicians, and wise students, then rewrite their solutions to make it seem as if there was a smooth and obvious deductive path from the given information to the desired conclusion.)
Trial-and-error
The trial-and-error approach involves making a reasonable estimate of what is required and then attempting to solve the problem using that estimate. The trial anderror approach is most effective when your problem is part of a well-behaved system.
Then the results tend to improve systematically as your estimates get closer to some optimum value. The key to success with a trial-and-error approach is to pause and analyze the results after each attempt at a solution. Then you can make systematic improvements in your estimates. Random guessing is generally a waste of time.
For equation solving, there are a variety of numerical techniques that make the trialand- error process as efficient as possible and ensure that no potential solutions are missed. The essential feature of ‘numerical methods’ is that successive trials are not random, but are based on computations that systematically lead towards a solution.
The ability of computers to rapidly carry out extensive and repeated computations makes them an ideal aid when using numerical methods for problem solving.
Trail-and-error methods should be avoided when dealing with delicate situations such as brain surgery, and when dealing with systems with critical values such as explosive devices.

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Allocate Your Resources

Allocate Your Resources

Allocate Your Resources
Resource allocation is a significant aspect of problem solving. While the problems in life are without end, your resources are limited so choices have to be made. Your limited resources include your time, money, physical resources, skills, expert knowledge, and good will.
Consider the classic case of writing a university exam. How much time should you allocate to answering any given question, and which questions should you attempt first? It is always reasonable to take a few minutes to read over all the questions at the beginning of an exam and rate their degree of difficulty. Then you should always answer the easy questions first. And you should allocate your time for each question according to a minutes-per-mark ratio. Suppose that you have 120 minutes to earn 100 marks; the implication is that a question worth 10 marks should be answered in approximately 12 minutes. Rating questions and establishing a marks-per-minute ratio provide useful guides for allocating your resources during an academic exam.

Home renovations provide another example of resource allocation and problem solving. While you may wish to totally renovate your home, your financial resources are probably limited and your patience in tolerating disruptions may also be limited.
You have to choose with care the extent of the changes that you can afford and the scheduling of renovation activities.
Every attempt at problem solving involves making choices and allocating resources. The allocation of your resources should be based on obtaining a reasonable chance of success while still holding some resources in reserve for dealing with any potential complications and unexpected emergency situations.

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