For this course you may (if compatible with your PC) use a professional software called WinQSB, available at: WinQSB Software. To download it on your own computer, click on the link, save it on your C-drive with a name, say WinQSB. To run, go to you PC C-drive WinQSB click on it, choose "unzip", NOT "unzip and install." Once the window pops up with all of the files, then click the one that says "SETUP." Then register under your own name and University of Baltimore. After completing the above steps, you should find that you will be able to access the various WinQSB software modules on your PC for this and other quantitative-based business courses.
Welcome to: Decision Science: Making Good Strategic Decisions. I look forward to working with you and hope that you will find the course both enjoyable and informative.
This course site is created for you. No one needs to be ashamed of what he or she does not know or how long it takes to master new information. learning on the Web can be nonjudgmental and self-paced. Using advantages of this technology to expand learning opportunities is particularly crucial because we live in a time when learning is becoming a necessity not a luxury.
The letters in OR, or OPRE stand for OPerations RE-search. OPRE is a science assisting you to make decisions (based on some numerical and measurable scales) by searching, and re-searching for a solution. I refer you to What Is OR/MS?, for a deeper understanding of what OPRE is all about. Decision-making process must be based on data neither on personal opinion neither on belief.
In our increasingly complex world, the tasks of the decision-makers are becoming more challenging every day. The decision-maker must respond quickly to events that take place at an ever-increasing speed. A decision-maker must incorporate an often bewildering array of choices and consequences into his or her decisions.
The site for this course was designed and created for you. No one need be ashamed of what he or she does not know or how long it takes to master new information. Learning by the web-enhanced course material can be self-paced and non-judgmental. Using advantages of this technology to expand learning opportunities is especially crucial because we live in a time when learning is a necessity and no longer a luxury.
At one time, it was sufficient for a firm to produce a quality product. As competition grows in today's market, simply producing a quality product is not sufficient. Today, a firm must produce a quality product at less cost than its competitors and simultaneously manage inventory, warehouse space, procurement requirements, etc. In the future, still greater demands will be placed upon decision-makers.
A manager makes many decisions everyday. Some decisions are routine and inconsequential, while others may impact the operations of a firm. Some decisions cause a firm to lose or gain money or determine whether goals are reached. The field of Decision Science (DS), known also as Operations Research (OR), Management Science (MS), and Success Science (SS), has helped managers develop the expertise and tools to understand decision problems, put them into mathematical terms and solve them.
Many tools and techniques help individuals and organization make better decisions. This course provides decision makers and analysts the tools that provide a logical structure to understand the mathematical techniques to solve formulated (i.e. modeled) problems. The primary tools are linear programming and decision analysis, which provide structure and value in helping define and under-stand a problem. In this course you will learn OR/MS/DS/SS methodologies to determine optimal strategic solutions to described problems. Personal Computers allow application of these techniques even in the small business environment. Finally, a clear understanding of a general approach to problem solving enables you to use other applied decision-making and planning techniques in this course.
Since the strategic solution to any problem involves assumptions, it is necessary to determine how much the strategic solution changes when the assumptions change. You learn this by performing "what-if" scenarios or sensitivity analysis.
Preparation for management, whether it is related to technology, business, production, or services, requires knowledge of tools, which aid in determining feasible and optimal policies. In addition to communication and qualitative reasoning skills, enterprises wishing to remain competitively viable in the future, need decision support systems to help them understand the complex interactions between all components of an organization's internal and external system. Such components are found in environmental design, transportation planning and control, facilities management, military mission planning and execution, disaster relief operations, investment management, and manufacturing operations.
An organization, like other organisms, must keep itself in a state of homeostasis--subsystems regulate one another so none of the parts is ahead or behind the system as a whole. This interaction is not trivial; mathematical modeling assists in understanding these fundamental relationships. OR/MS/DS/SS concepts focus on communication of results and recommended action. This helps build a consensus concerning the possible outcomes and recommended action. The decision-maker might incorporate other perspectives of the problem, such as culture, politics, psychology, etc., into the management scientist's recommendations.
The creation of Decision Science software is one of the most important events in decision-making. OR/MS/DS/SS software systems are used to construct examples, to understand existing concepts, and to find new managerial concepts. New developments in decision-making often motivate developments in solution algorithms and revisions of software systems. OR/MS/DS/SS software systems rely on a cooperation of OR/MS/DS/SS practitioners, algorithms designers and software developers.
This course overviews the major quantitative modeling tools successfully used to model the complex interactions described above. Although not exhaustive, this course provides framework for further study. The following tools will be studied: analytically based solutions to math models, linear programming, and decision theory, Decision Science encompasses many disciplines of study because decision-making is a central human activity. Appreciation of decision making is wonderful: it makes what is excellent in this thinking process belongs to you as well.
Just like you, most of your classmates are employed full time. They are engineers, doctors, lawyers, and other professionals. You and your classmates want to learn the business side of your professions. It is important to learn the language of the managers to overcome communication barriers. For example, engineers will learn how to translate "precision" into extra dollars in earning/saving.
In each class I teach, there are some students who find it difficult to rethink and re-evaluate their pre-conceived ideas. In decision-making, one must have an open-mind to be able to think differently and to see from many perspectives. University classrooms provide the environment for debate and the exchange of ideas. Open-mindedness is the main requirement in achieving the ultimate goal of education, which is to be able to think for yourself. Change of opinion is often the progress of sound thought and growing knowledge.
Upon completion of this course, you may find that it "validates" what you think about making good strategic decisions and causes a peace of mind. The contents of this course will help you to systematize what you already know from your own professional experience.
For my teaching philosophy statements, visit the web site On Learning & Teaching.
Feel free to contact me via phone, fax, or email. There is a lot of material to cover, so let's start now!
Many people still remain in the bondage of self-incurred tutelage. Tutelage is a person's inability to make his/her own decisions. Self-incurred is this tutelage when its cause lies not in lack of reason but in lack of resolution and courage to use it without wishing to have been told what to do by something or somebody else. Sapere aude! "Have courage to use your own reason!"- was the motto of the Enlightenment era. During this period, Francisco Goya created his well-known "The sleep of reason produces monsters" masterpiece.
Through the Enlightenment era's struggle and much suffering, "the individual" finally appeared. Eventually human beings gained their natural freedom to think for themselves. However, this has been too heavy a responsibility for many people to carry. There has been an excess of failure. They easily give up their natural freedom to any cult in exchange for an easy life. The difficulty in life is the choice. They do not even have the courage to repeat the very phrases which our founding fathers used in the struggle for independence. What an ironic phenomenon it is that you can get men to die for the liberty of the world who will not make the little sacrifice that it takes to free themselves from their own individual bondage.
Good decision-making brings about a better life. It gives you some control over your life. In fact, many frustrations with oneself are caused by not being able to use one's own mind to understand the decision problem, and the courage to act upon it.
A bad decision may force you to make another one, as Harry Truman said, "Whenever I make a bum decision, I go out and make another one." Remember, if the first button of one's coat is wrongly buttoned, all the rest will be crooked.
A good decision is never an accident; it is always the result of high intention, sincere effort, intelligent direction and skillful execution; it represents the wise choice of many alternatives. One must appreciate the difference between a decision and an objective. A good decision is the process of optimally achieving a given objective.
When decision making is too complex or the interests at stake are too important, quite often we do not know or are not sure what to decide. In many instances, we resort to informal decision support techniques such as tossing a coin, asking an oracle, visiting an astrologer, etc. However formal decision support from an expert has many advantages. This web site focuses on the formal model-driven decision support techniques such as mathematical programs for optimization, and decision tree analysis for risky decisions. Such techniques are now part of our everyday life. For example, when a bank must decide whether a given client will obtain credit or not, a technique, called credit scoring, is often used.
Rational decisions are often made unwillingly, perhaps unconsciously. We may start the process of consideration. It is best to learn the decision-making process for complex, important and critical decisions. Critical decisions are those that cannot and must not be wrong. Ask yourself the objective: What is the most important thing that I am trying to achieve here?
The decision-maker's style and characteristics can be classified as: The thinker, the cowboy (snap and uncompromising), Machiavellian (ends justifies the means), the historian (how others did it), the cautious (even nervous), etc. For example, political thinking consists in deciding upon the conclusion first and then finding good arguments for it.
As the title of this site indicates, it is applied which means it is concrete not abstract or "knowledge for the sake of knowledge". It is axiomatic that if learning occurs, there is change in you. Change might occur in your attitude, thinking, beliefs and/or behavior. Something will have changed or else learning simply did not occur. This course changes your life for the better. The aim of this site is to make you a better decision maker by learning the decision-making process:
The explicit information about the course of actions may also expand your set of alternatives. The more alternatives you develop the better decisions you may make. Creativity in the decision-making process resides in the capacity for evaluating uncertain, hazardous, and conflicting information. You must become a creative person to expand your set of alternatives. Creativity, arises out of thinking hard (i.e., becoming of a thinker) rather than working hard (i.e., becoming of a workaholic). A bulldozer must work hard, a human being must think hard.
A deep immersion in your decision-making process makes you more creative. The roots of creativity lie in consciousness incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness, by the usage of mental images, symbols, words, and logic. Saturation or too Narrow thinking; Inability to incubate (this, one must learn from cows); and the Fear of standing alone doing something new; block creativity . Most people treat knowledge as a liquid to be swallowed easily rather than as a solid to be chewed, and then wonder why it provides so little nourishment. Aristotle noted, "We call in others to aid us in deliberation on important questions, distrusting ourselves as not being equal to deciding."
Be objective about yourself and your business. More than half of my students, semester after semester, raise their hands when I ask, "Is your judgment better than that of the average person?" It is important to identify your weaknesses as well as your strengths.
There is no such thing as a creative/non-creative person. It is the creative process which make you more creative. Pablo Picasso realized this fact and said about himself: "All human beings are born with the same creative potential. Most people squander theirs away on a million superfluous things. I expend mine on one thing and one thing only: my art." Creative decision alternatives are original, relevant, and practical.
Any careful strategizing and policy-making cannot be easy tasks; however the methodologies and techniques presented here can be used for improving procedural rationality during the process of strategizing. The efficiency and effectiveness of such applications depends on the selection of strategizing process.
Many people treat goal setting this way -- they dream about where they want to go, but they do not have a map to get there. What is a map? In essence, the written words and careful planning. Decision-making is a complicated process. This complication arises from the fact that your present goal (including wants, resources, and abilities) dictates your choices, however, your choices will change your goals. This influential-cycle keeps the decision-maker busy all the time. Selecting your goals and your criteria for success is a dynamic process and changes over time. This is true in almost all cases dealing with personal growth or organizational growth
The logic of worldly success rests on a fallacy: the strange error that our perfection depends on the thoughts and opinions and applause of other men! A weird life it is, indeed, to be living always in somebody else's imagination, as if that were the only place in which one could at last become real!
On a daily basis a manager has to make many decisions. Some of these decisions are routine and inconsequential, while others have drastic impacts on the operations of the firm for which he/she works. Some of these decisions could involve large sums of money being gained or lost, or could involve whether or not the firm accomplishes its mission and its goals. In our increasingly complex world, the tasks of decision-makers are becoming more challenging with each passing day. The decision-maker (i.e., the responsible manager) must respond quickly to events that seem to take place at an ever-increasing pace. In addition, a decision-maker must incorporate a sometimes-bewildering array of choices and consequences into his or her decision. Routine decisions are often made quickly, perhaps unconsciously without the need for a detailed process of consideration. However, for complex, critical or important managerial decisions it is necessary to take time to decide systematically. Being a manager means making critical decisions that cannot and must not be wrong or fail. One must trust one's judgment and accept responsibility. There is a tendency to look for scapegoats or to shift responsibility.
Decisions are at the heart of any organization. At times there are critical moments when these decisions can be difficult, perplexing and nerve-wracking. Making decisions can be hard for a variety of structural, emotional, and organizational reasons. Doubling the difficulties are factors such as uncertainties, having multiple objectives, interactive complexity, and anxiety.
Strategic decisions are purposeful actions. The future of your organization and the progress of your career might be profoundly affected by what you decide.
Good decisions are made with less stress, and it is easier to explain the reasons for the decision that was made. Decisions should be made strategically. That is, one should make decisions skillfully in a way that is adapted to the end one wishes to achieve. To make strategic decisions requires that one takes a structured approach following a formal decision making process. Otherwise, it will be difficult to be sure that one has considered all the key aspects of the decision.
Making good strategic decisions is learnable and teachable through an effective, efficient, and systematic process known as the decision-making process. This structured and well-focused approach to decision-making is achieved by the modeling process, which helps in reflecting on the decisions before taking any actions. Remember that: one must not only be conscious of his/her purposeful decisions, one must also find out the causes for which they are made. There is no such thing as "free-will". Those who believe in their free wills are in fact ignorant to the causes that impel them to their decisions. There is no such thing as arbitrary in any activity of man, least of all in his decision-making. Just as he has learned to be guided by objective criteria in making his physical tools, so he is guided by unconscious objective criteria in forming his decision in most cases.
The simplest decision model with only two alternatives, is known as Manicheanism, which was adapted by Zarathustra (B.C. 628-551), and then taken by all other organized religions. Manicheanism is the duality concept, which divides everything in the world into discrete either/or and opposite polar, such as good and evil, black and white, night and day, mind (or soul) and body, etc. This duality concept was a sufficient model of reality for those old days in order to make their world manageable and calculable. However, nowadays we very well know that everything is becoming and has a wide continuous spectrum. There are no real opposites in nature. We have to see the world through our colorful mind's eyes; otherwise we do not understand complex ideas well.
The Industrial Revolution of the 19th century probably did more to shape life in the modern industrialized world than any event in history. Large factories with mass production created a need for managing them effectively and efficiently. The field of Decision Science (DS) also known as Management Science (MS), Operations Research (OR) in a more general sense, started with the publication of The Principles of Scientific Management in 1911 by Frederick W. Taylor. His approach relied on the measurement of industrial productivity and on time /movement studies in the factories. The goal of his scientific management was to determine the best method for performing tasks in the least amount of time, while unfortunately using the stopwatch in an inhumane manner.
A basic education in OR/MS/DS/SS for managers is essential. They are responsible for leading the business system and the lives in that system. The business system is dynamic in nature and will respond as such to disturbances internally and externally.
The OR/MS/DS/SS approach to decision making includes the diagnosis of current decision making and the specification of changes in the decision process. Diagnosis is the identification of problems (or opportunities for improvement) in current decision behavior; it involves determining how decisions are currently made, specifying how decisions should be made, and understanding why decisions are not made as they should be. Specification of changes in decision process involves choosing what specific improvements in decision behavior are to be achieved and thus defining the objectives.
Nowadays, the OR/MS/DS/SS approach has been providing assistance to managers in developing the expertise and tools necessary to understand the decision problems, put them in analytical terms and then solve them. The OR/MS/DS/SS analysts are, e.g., "chiefs of staff for the president", "advisors", "R&D modelers" "systems analysts", etc. Applied Management Science is the science of solving business problems. The major reason that MS/OR has evolved as quickly as it has is due to the evolution in computing power.
Foundations of Good Decision-Making Process: When one talks of "foundations", usually it includes historical, psychological, and logical aspects of the subject. The foundation of OR/MS/DS/SS is built on the philosophy of knowledge, science, logic, and above all creativity. In this course the decision "problem", does not refer to prefabricated exercises or puzzles with which most educators continually confront students, such as the problem of finding a solution to a system of equations, without giving any motivation for its need-to-know.
Since some decision problems are so complicated and so important, the individuals who analyze the problem are not the same as the individuals who are responsible for making the final decision. Therefore, this site distingushes between a management scientist, someone who studies what decision to make, and a decision maker, someone responsible for making the decision.
This site is about how to make good decisions when confronted with decision problems. It means real problems, the effective handling of which can make a significant difference. Almost all decision problems have environments with similar components as follows:
Deterministic versus Probabilistic Models: Before going further, we distinguish between deterministic and probabilistic decision-making problems. All the decision models can be classified as either deterministic or probabilistic models. In deterministic models your good decisions bring about good outcomes. You get that which you expect, therefore the outcome is deterministic (i.e., risk-free). However, in probabilistic decision models, the outcome is uncertain, therefore making good decisions may not produce good outcomes. Unlike deterministic models where good decisions are judged by the outcome alone, in probabilistic models, the decision maker is concerned with both the outcome value and the amount of risk each decision carries. When the outcome of your decision is rather certain and all the important consequences occur within a single period, then your decision problem is classified as a deterministic decision. However, in many instances, these types of models are encumbered with the two most difficult factors -- uncertainty and delayed effects. Both difficulties can be overcome by probabilistic modeling which includes the time discounting factor. We will cover both deterministic and probabilistic decision-making models.
After recognizing this no-nonsense classification of decision-making components, the OR/MS/DS/SS analyst performs the following sequence with some possible feedback loops between its steps:
Problem understanding encompasses a problem structure, and a diagnostic process to assist us in problem formulation (i.e., giving a Form to a complex situation) and representation. This stage is the most important aspect of the decision-making process. Problem understanding is an interactive process between the decision maker and the OR/MS/DS/SS analyst. The decision maker may be unfamiliar with the analytic details of the problem formulation such as what elements to include in the model, and how to include them as variables, constraints, indexes, etc.
Since the strategic solution to any problem involves making certain assumptions, it is necessary to determine the extent to which the strategic solution changes when the assumptions change. You will learn this by performing the "what-if" scenarios and the necessary sensitivity analysis. Ensure that both plan and dispositions are flexible, adaptable to circumstances. Your plan should foresee and provide for a next step in case of success or failure.
Gathering reliable information at the right time is a component of good decisions. It is helpful to understand the nature of the problem by asking "who?", "what?", "why?", "when", "where" and "how?". Finally, break them into three input groups, namely: Parameters, Controllable, and Uncontrollable inputs. Uncontrollable factors are the main components of decision-making which must be dealt with, by, e.g., forecasting. In making conscious decisions, we all make forecasts. We may not think that we are forecasting, but our choices will be directed by our anticipation of results of our actions or omissions.
One must evaluate the various courses of actions within the controllable inputs, consider various scenarios for uncontrollable inputs, and then decide the best course of action. As you know, the whole process of managerial decision-making is synonymous with the practice of management. Decision-making is at the core of all managerial functions. Planning, for example, involves the following decisions: What should be done? When? How? Where? By whom?
As indicated in the above diagram, perceiving the need to face the decision problem is a point of departure and no more. As soon as you elaborate, it becomes transformed by thought process to a mental model. The decision-making process contains a few well-defined stages, including describing, prescribing, and controlling the problem, each of these stages requires a set of relevant questions to be asked. Moreover, this process is never ending since the problem keeps changing, therefore there is a always need for feedback to measure the effect of your decision. Moreover, each decision problem that you make successfully became a rule, which served afterward to make other decisions. This happens when your are facing a sequential decision-making problem.
At the "what-if" analysis stage of modeling, the modeler and the owner of the problem must concentrate on what can happen rather that what would happen. Most of the management activity is a "rear view." That is, no manager can ever have any information other than what has happened in the past, hence managing is done by looking in the rear view mirror. The "what-if" analysis provides "look ahead" management. The management can use a dynamic model to experiment with future consequences of new policies. It provides information on what is likely to happen, not what necessarily will happen.
You may ask, how do we differentiate between "what can happen" and "what would happen?" Here are two concrete examples:
Preparation for management, whether it is related to technology, business, production, or services, requires knowledge of tools, which can aid in the determination of feasible, optimal policies. In addition to skills related to communication and qualitative reasoning, enterprises wishing to remain competitively viable in the future, need model-driven decision support systems to help them understand the complex interactions between all components of a given organization's system, both internal, and external situations. The strategic assessment at this stage must recognize both the internal analysis such as the strengths and weaknesses, and the external analysis such as threats and opportunities.
There are also situations where some may feel that the decision-maker should rely on simply "do the right thing" and damn the analytical strategic thinking . Whereas many agree that for defensible and responsible decisions one should at least know the balance of the analytical approach as well as the human-side of the decision which includes the ethical elements.
All OR/MS/DS/SS concepts focus on communication of the results and recommended courses of actions (strategies). This helps all involved to build a consensus concerning the possible outcomes and recommended course of action. The decision-maker might incorporate some other perspectives of the problem, such as cultural, political, psychological, etc., into the management scientist's recommendations.
Successful OR/MS/DS/SS modeling approach to decision-making demands a proper attitude as well as an understanding of more technical matters. Although both the OR/MS/DS/SS analyst and decision-maker should understand problem identification, model building, and solution techniques, the attitudes of both are probably the most important elements of successful application. Although proper attitude is not sufficient for successful application, it is necessary. An analyst who focuses more on techniques for solution than on model formulation will not be successful. The analyst's main interest should be in providing assistance in decision-making and not in finding methods of solution that are more elegant or marginally faster than existing methods. A decision maker who thinks that she or he can turn the analyst loose without guidance and expect to get relevant information back that can be applied directly to the problem and then forgotten will not make the best use of quantitative inputs. Instead, the interaction between the decision-maker and OR/MS/DS/SS analyst must be open, interactive, and focused on the ultimate goal of the effort: to develop and make the best use of the quantitative input to a decision problem.
Today's business decisions are driven by data. In all aspects of our lives, and importantly in the business context, an amazing diversity of data is available for inspection and given insights. Moreover, business managers and decision makers are increasingly encouraged to justify decisions on the basis of data. Taking this course gives you an edge. Graduates with strong quantitative skills are in demand. This phenomenon will grow as the impetus for data-based decisions strengthens and the amount and availability of data increases. The quantitative toolkit can be developed and enhanced at all stages of your career.
You may ask what are the quantitative modeling tools? Quantitative modeling tools are statistics and mathematics you used to apply to solve the word-problems in your high school days. These tools are being used to solve (real) business (decision) problems, such as optimizing profit you will learn shortly.
Good Questions with Useful Answers (Word.Doc)
A note on the Required Reading: In addition to reading the weekly lecture notes your textbook, the required reading provides deeper understanding of weekly topics. In some cases it enhances your understanding by looking at the same topic from different perspectives including relevancy to other topics, and as a bridge to the future topics. The aim is create a coherent statistical learning system with its wholeness and manifoldness
Modeling of Linear Programs (LP)
If you've only got limited resources at your disposal, then it's helpful to calculate how best to maximize those resources '" whether that's time, money, or space.
Let's say, for example, that you have 50 square feet of office space to use for storage. Your budget is $200, and there are a variety of cabinet types and sizes from which to choose. How do you optimize the space you have available, and stay within the allotted budget?
Or suppose you have three delivery trucks, and 10 drop-off points. How do you plan the most efficient route and schedule for these trucks?
Or consider that you manufacture three products using the same basic raw materials. However, as each product uses different amounts of material, some are more expensive to produce than others. A few of the materials are perishable, and need to be used quickly. How much of each product should you manufacture to minimize your cost? And which combination produces the least waste?
Questions like these may seem very complex. With so many variables and constraints to take into consideration, how do you decide what to do? The answer is to use linear programming.
Linear programming is a mathematical technique that determines the best way to use available resources. Managers use the process to help make decisions about the most efficient use of limited resources '" like money, time, materials, and machinery.
Linear programming (LP) is often a favorite topic for both professors and students. The ability to introduce LP using a graphical approach, the relative ease of the solution method, the widespread availability of LP software packages, and the wide range of applications make LP accessible even to students with relatively weak mathematical backgrounds. Additionally, LP provides an excellent opportunity to introduce the idea of "what-if" analysis, due to the powerful tools for post-optimality analysis developed for the LP model.
Linear Programming is a mathematical procedure for determining optimal allocation of scarce resources. LP is a procedure that has found practical application in almost all facets of business, from advertising to production planning. Transportation, distribution, and aggregate production planning problems are the most typical objects of LP analysis. In the petroleum industry, for example a data processing manager at a large oil company recently estimated that from 5 to 10 percent of the firm's computer time was devoted to the processing of LP and LP-like models.
Linear programming deals with a class of programming problems where both the objective function to be optimized is linear and all relations among the variables corresponding to resources are linear. This problem was first formulated and solved in the late 1940's. Rarely has a new mathematical technique found such a wide range of practical business, commerce, and industrial applications and simultaneously received so thorough a theoretical development, in such a short period of time. Today, this theory is being successfully applied to problems of capital budgeting, design of diets, conservation of resources, games of strategy, economic growth prediction, and transportation systems. In very recent times, linear programming theory has also helped resolve and unify many outstanding applications.
It is important for the reader to appreciate, at the outset, that the "programming" in Linear Programming is of a different flavor than the "programming" in Computer Programming. In the former case, it means to plan and organize as in "Get with the program!", it programs you by its solution. While in the latter case, it means to write codes for performing calculations. Training in one kind of programming has very little direct relevance to the other. In fact, the term "linear programming" was coined before the word "programming" became closely associated with computer software. This confusion is sometimes avoided by using the term linear optimization as a synonym for linear programming.
Any LP problem consists of an objective function and a set of constraints. In most cases, constraints come from the environment in which you work to achieve your objective. When you want to achieve the desirable objective, you will realize that the environment is setting some constraints (i.e., the difficulties, restrictions) in fulfilling your desire or objective. This is why religions such as Buddhism, among others, prescribe living an abstemious life. No desire, no pain. Can you take this advice with respect to your business objective?
What is a function: A function is a thing that does something. For example, a coffee grinding machine is a function that transform the coffee beans into powder. The (objective) function maps and translates the input domain (called the feasible region) into output range, with the two end-values called the maximum and the minimum values.
When you formulate a decision-making problem as a linear program, you must check the following conditions:
For example, the following problem is not an LP: Max X, subject to X < 1. This very simple problem has no solution.
As always, one must be careful in categorizing an optimization problem as an LP problem. Here is a question for you. Is the following problem an LP problem?
Max X2
subject to:
X1 + X2 < = 0
X12 - 4 < = 0
Although the second constraint looks "as if" it is a nonlinear constraint, this constraint can equivalently be written as:
X1 > =, -2, and X2 < = 2.
Therefore, the above problem is indeed an LP problem.
For most LP problems one can think of two important classes of objects: The first is limited resources such as land, plant capacity, or sales force size; the second, is activities such as "produce low carbon steel", "produce stainless steel", and "produce high carbon steel". Each activity consumes or possibly contributes additional amounts of the resources. There must be an objective function, i.e. a way to tell bad from good, from an even better decision. The problem is to determine the best combination of activity levels, which do not use more resources than are actually available. Many managers are faced with this task everyday. Fortunately, when a well-formulated model is input, linear programming software helps to determine the best combination.
The Simplex method is a widely used solution algorithm for solving linear programs. An algorithm is a series of steps that will accomplish a certain task.
To formulate an LP problem, I recommend using the following guidelines after reading the problem statement carefully a few times.
Any linear program consists of four parts: a set of decision variables, the parameters, the objective function, and a set of constraints. In formulating a given decision problem in mathematical form, you should practice understanding the problem (i.e., formulating a mental model) by carefully reading and re-reading the problem statement. While trying to understand the problem, ask yourself the following general questions:
Learn that the feasible region has nothing or little to do with the objective function (min or max). These two parts in any LP formulation come mostly from two distinct and different sources. The objective function is set up to fulfill the decision-maker's desire (objective), whereas the constraints which shape the feasible region usually comes from the decision-maker's environment putting some restrictions/conditions on achieving his/her objective.
The following is a very simple illustrative problem. However, the way we approach the problem is the same for a wide variety of decision-making problems, and the size and complexity may differ. The first example is a product-mix problem.
During a couple of brain-storming sessions with a carpenter (our client), he told us that he, solely, makes tables and chairs, sells all tables and chairs he makes at a market place, however, does not have a stable income, and wishes to do his best.
The objective is to find out how many tables and chairs he should make to maximize net income. We begin by focusing on a time frame, i.e., planning time-horizon, to revise our solution weekly if needed. To learn more about his problem, we must go to his shop and observe what is going on and measure what we need to formulate (i.e., to give a Form, to make a model) of his problem. We must confirm that his objective is to maximize net income. We must communicate with the client.
The carpenter's problem deals with finding out how many tables and chairs to make per week; but first an objective function must be established:
Since the total cost is the sum of the fixed cost (F) and the variable cost per unit multiplied by the number of units produced. Therefore, the decision problem is to find X1 and X2 such that:
Maximize 9X1 + 6X2 '" [(1.5X1 + X2) + (2.5X1 + 2X2) + F1 + F2],
where X1 and X2 stand for the number of tables and chairs; the cost terms in the brackets are the raw material, and labor costs respectively. F1 and F2 are the fixed costs for the two products respectively. Without loss of generality, and any impact on optimal solution, let us set F1 = 0, and F2 = 0. The objective function reduces to the following net profit function:
Maximize 5X1 + 3X2
That is, the net incomes (say, in dollars, or tens of dollars) from selling X1 tables and X2 chairs.
The constraining factors which, usually come from outside, are the limitations on labors (this limitation comes from his family) and raw material resources (this limitation comes from scheduled delivery). Production times required for a table and a chair are measured at different times of day, and estimated to be 2 hours and 1 hour, respectively. Total labor hours per week are only 40 hrs. Raw materials required for a table and a chair are 1, and 2 units respectively. Total supply of raw material is 50 units per week. Therefore, the LP formulation is:
Maximize 5 X1 + 3 X2
Subject to:
2 X1 + X2 < = 40 labor constraint
X1 + 2 X2 < = 50 material constraint
and both X1, X2 are non-negative.
This is a mathematical model for the carpenter's problem. The decision variables, i.e., controllable inputs are X1, and X2. The output for this model is the total net income 5 X1 + 3 X2. All functions used in this model are linear (the decision variable have power equal to 1). The coefficients of these constraints are called Technological Factors (matrix). The review period is one week, an appropriate period within which the uncontrollable inputs (all parameters such as 5, 50, 2,..) are less likely to change (fluctuate). Even for such a short planning time-horizon, we must perform the what-if analysis to react to any changes in these inputs in order to control the problem, i.e., to update the prescribed solution.
Notice that since the carpenter is not going out of business at the end of the planning horizon, we added the conditions that both X1, X2 must be non-negative instead of the requirements that X1, and X2 must be positive integers. The non-negativity conditions are also known as "implied constraints." Again, a Linear Program would be fine for this problem if the carpenter were going to continue to manufacture these products. The partial items would simply be counted as work in progress and would eventually become finished goods say, in the next week.
We may try to solve for X1 and X2 by listing possible solutions for each and selecting the pair (X1, X2) that maximize 5X1 + 3X2 (the net income). However, it is too time consuming to list all possible alternatives and if the alternatives are not exhaustively listed, we cannot be sure that the pair we select (as a solution) is the best of all alternatives. This way of solving a problem is known as "sequential thinking" versus "simultaneous thinking". More efficient and effective methodologies, known as the Linear Programming Solution Techniques are based on simultaneous thinking are commercially available in over 400 different software packages from all over the world.
The optimal solution, i.e., optimal strategy, is to make X1 = 10 tables, and X2 = 20 chairs. We may program the carpenter's weekly activities to make 10 tables and 20 chairs. With this (optimal) strategy, the net income is $110. This prescribed solution was a surprise for the carpenter since, because of more net income of selling a table ($5), he used to make more tables than chairs!
Hire or Not? Suppose the carpenter can hire someone to help at a cost of $2 per hour. This is, in addition, hourly-based wage he/she is currently paying; otherwise $2 is much lower than the current minimum wage in US. Should the carpenter hire and if yes then for how may hours?
Let X3 be the number of extra hours, then the modified problem is:
Maximize 5 X1 + 3 X2 - 2 X3
Subject to:
2 X1 + X2 < = 40 + X3 labor constraint with unknown additional hours
X1 + 2 X2 < = 50 material constraint
Under this new condition, we will see that the optimal solution is X1 = 50, X2 = 0, X3 = 60, with optimal net income of $130. Therefore, the carpenter should be hired for 60 hours. What about only hiring 40 hours? The answer to this and other types of what-if questions are treated under sensitivity analysis in this Web site.
As an exercise, use your LP software to find the largest range for X values satisfying the following inequality with two absolute value terms:
| 3X '" 4 | - | 2X '" 1 | < = 2
Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a net income of $1.00 for each regular pizza and $1.50 for each deluxe pizza produced. The firm currently has 150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 pound of dough mix and 4 ounces (16 ounces= 1 pound) of topping mix. Each deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix. Based on the past demand per week, Bryant can sell at least 50 regular pizzas and at least 25 deluxe pizzas. The problem is to determine the number of regular and deluxe pizzas the company should make to maximize net income. Formulate this problem as an LP problem.
Let X1 and X2 be the number of regular and deluxe pizza, then the LP formulation is:
Maximize X1 + 1.5 X2
Subject to:
X1 + X2 < = 150
0.25 X1 + 0.5 X2 < = 50
X1 > = 50
X2 > = 25
X1 > = 0, X2 > = 0
Linear programming is a powerful tool for selecting alternatives in a decision problem and, consequently, has been applied in a wide variety of problem settings. We will indicate a few applications covering the major functional areas of a business organization.
Finance: The problem of the investor could be a portfolio-mix selection problem. In general, the number of different portfolios can be much larger than the example indicates, more and different kinds of constraints can be added. Another decision problem involves determining the mix of funding for a number of products when more than one method of financing is available. The objective may be to maximize total profits, where the profit for a given product depends on the method of financing. For example, funding may be done with internal funds, short-term debt, or intermediate financing (amortized loans). There may be limits on the availability of each of the funding options as well as financial constraints requiring certain relationships between the funding options so as to satisfy the terms of bank loans or intermediate financing. There may also be limits on the production capacity for the products. The decision variables would be the number of units of each product to be financed by each funding option.
Optimal Portfolio: Determining an Optimal Portfolio An investment club has a set of clearly defined goals regarding the liquidity and risk of their investment. Given the best available projections for the expected annual return on each of a set of possible investments, the club can use a linear model to select the appropriate set of stocks and determine the investment amounts in order to maximize its overall rate of return. (See Ch. 3, Problem 4.)
Production and Operations Management: Quite often in the process industries a given raw material can be made into a wide variety of products. For example, in the oil industry, crude oil is refined into gasoline, kerosene, home-heating oil, and various grades of engine oil. Given the present profit margin on each product, the problem is to determine the quantities of each product that should be produced. The decision is subject to numerous restrictions such as limits on the capacities of various refining operations, raw-material availability, demands for each product, and any government-imposed policies on the output of certain products. Similar problems also exist in the chemical and food-processing industries.
Human Resources: Personnel planning problems can also be analyzed with linear programming. For example, in the telephone industry, demands for the services of installer-repair personnel are seasonal. The problem is to determine the number of installer-repair personnel and line-repair personnel to have on the work force each month where the total costs of hiring, layoff, overtime, and regular-time wages are minimized. The constraints set includes restrictions on the service demands that must be satisfied, overtime usage, union agreements, and the availability of skilled people for hire. This example runs contrary to the assumption of divisibility; however, the work-force levels for each month would normally be large enough that rounding to the closest integer in each case would not be detrimental, provided the constraints are not violated.
Marketing: Linear programming can be used to determine the proper mix of media to use in an advertising campaign. Suppose that the available media are radio, television, and newspapers. The problem is to determine how many advertisements to place in each medium. Of course, the cost of placing an advertisement depends on the medium chosen. We wish to minimize the total cost of the advertising campaign, subject to a series of constraints. Since each medium may provide a different degree of exposure of the target population, there may be a lower bound on the total exposure from the campaign. Also, each medium may have a different efficiency rating in producing desirable results; there may thus be a lower bound on efficiency. In addition, there may be limits on the availability of each medium for advertising.
Distribution: Another application of linear programming is in the area of distribution. Consider a case in which there are m factories that must ship goods to n warehouses. A given factory could make shipments to any number of warehouses. Given the cost to ship one unit of product from each factory to each warehouse, the problem is to determine the shipping pattern (number of units that each factory ships to each warehouse) that minimizes total costs. This decision is subject to the restrictions that demand at each factory cannot ship more products than it has the capacity to produce.
Mathematical Modeling: LP Formulation
Good Questions with Useful Answers (Word.Doc)
I do recommend refreshing your knowledge about solving systems of equations by visiting the Web site Solving System of Equations.
Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others. Here is a problem formulation submitted by one of your classmates. Click Here (Word.Doc) to view How things can go wrong in your LP problem formulation? Lecture Notes and Question: Learn To Graph Re-do what is in the lecture notes by hand computation, answer the questions therein. Read section 2.7 of your textbook, then as another part of your homework classify the feasible regions given at: Read about LP solution: Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others: The following are submitted by your classmates: Here is a solution set on different types of feasible region. Here is solutions to graphical representation of linear
. Here is an Excel feasible region with computed four corner points coordinates (A-E) of the Wilson feasible region. Click Here (Word.Doc) to view step-by-step for graphing problem 2.7 feasible region. Click Here (Word.Doc) to view a complete set. Here (Word.Doc) is another one. Procedure for Graphical Method of Solving LP Problems: All variables have power of 1, and they are added or subtracted (not divided or multiplied). The constraint must be of the following forms ( < =, > =, or =, that is, the LP-constraints are always closed), and the objective must be either maximization or minimization. For example, the following problem is not an LP: Max X, subject to X < 1. This very simple problem has no solution. After all the constraints are graphed, you should have a non-empty (convex) feasible region, unless the problem is infeasible. Unfortunately, some of the boundaries of the feasible regions described in your textbook are wrong. See, e.g., the figures depicted on page 56. Almost all inequalities must be changed to equality. Right? In general, if the feasible region is within the first quadrant of the coordinate system (i.e., if X1 and X2 ³ 0), then, for the maximization problems you are moving the iso-value objective function parallel to itself far away from the origin point (0, 0), while having at least a common point with the feasible region. However, for minimization problems the opposite is true, that is, you are moving the iso-value objective parallel to itself closer to the origin point, while having at least a common point with the feasible region. The common point provides the optimal solution. If you do not have access to any software use: Graphical Method for LP Linear Programming (LP): Graphical Solution Algorithms, Read Ch. 2 and the course lecture notes. Do all parts of problem 2.7, with a detailed step-by-step description of the graphical method, using graph paper (Word.Doc , graph paper (PDF). This part of assignment, makes one conscious about what one does. Unfortunately, in some browsers the Graphical Methods of WinQSB may not be available. However, use e.g. the following JavaScript as an alternative: Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others. Click Here (Word.Doc) to view step-by-step solution set for problem 2.7 submitted by one of your classmates. Click Here (Word.Doc) to view a submitted complete solution set for problem 2.7. Another one (Word.Doc). Click Here (Word.Doc) to view How things can go wrong in your graphical LP solution? You are certainly welcome to use the discussion board to post your questions, responses to any parts of the above files' contents. I do thank everyone for active-learning participation. While you were doing your last week homework, you may have wondered, why there is no good programs or software that do [the] graphical method easily? The graphical method is limited to its usefulness by aiding the conceptualization of LP problems having two variables. Most real-world problems involve numerous variables and constraints. As the number of variables and constraints increase, the problems must be solved algebraically using techniques such as the Simplex Method, which is based on an efficient computerized implementation (in e.g. WinQsb, www.lindo.com, http://www.maximal-usa.com, http://www.dashoptimization.com, etc., there are well over 400 LP sofware packages indicating its wide range of applications, visit e.g., http://lionhrtpub.com/software-surveys.shtml) of the Algebraic Method. The question is how can you utilize a software package to learn the business concepts better? The Value of Performing Experiment: If the learning environment is focused on background information, knowledge of terms and new concepts, the learner is likely to learn that basic information successfully. However, this basic knowledge may not be sufficient to enable the learner to carry out successfully the on-the-job tasks that require more than basic knowledge. Thus, the probability of making real errors in the business environment is high. On the other hand, if the learning environment allows the learner to experience and learn from failures within a variety of situations similar to what they would experience in the "real world" of their job, the probability of having similar failures in their business environment is low. This is the realm of simulations-a safe place to fail. The appearance of management science software is one of the most important events in decision making process. OR/MS software systems are used to construct examples, to understand the existing concepts, and to discover useful managerial concepts. On the other hand, new developments in decision making process often motivate developments of new solution algorithms and revision of the existing software systems. OR/MS software systems rely on a cooperation of OR/MS practitioners, designers of algorithms, and software developers. The major change in learning this course over the last few years is to have less emphasis on strategic solution algorithms and more on the modeling process, applications, and use of software. This trend will continue as more students with diverse backgrounds seek MBA degrees without too much theory and mathematics. Our approach is middle-of-the-road. It does not have an excess of mathematics nor too much of software orientation. For example, we lean how to formulate problems prior to software usage. What you need to know is how to model a decision problem, first by hand and then using the software to solve it. The software should be used for two different purposes. Personal computers, spreadsheets, professional decision making packages and other information technologies are now ubiquitous in management. Most management decision-making now involves some form of computer output. Moreover, we need caveats to question our thinking and show why we must learn by instrument. In this course, the instrument is your computer software package. Every student taking courses in Physics and Chemistry does experimentation in the labs to have a good feeling of the topics in these fields of study. You must also perform managerial experimentation to understand the Management Science concepts and techniques. Learning Objects: My teaching style deprecates the 'plug the numbers into the software and let the magic box work it out' approach. Computer-assisted learning is similar to the experiential model of learning. The adherents of experiential learning are fairly adamant about how we learn. Learning seldom takes place by rote. Learning occurs because we immerse ourselves in a situation in which we are forced to perform. You get feedback from the computer output and then adjust your thinking-process if needed. Unfortunately, most classroom courses are not learning systems. The way the instructors attempt to help their students acquire skills and knowledge has absolutely nothing to do with the way students actually learn. Many instructors rely on lectures and tests, and memorization. All too often, they rely on "telling." No one remembers much that's taught by telling, and what's told doesn't translate into usable skills. Certainly, we learn by doing, failing, and practicing until we do it right. The computer assisted learning serve this purpose. Unfortunately, the first objective is missing in all management science/operations research textbooks. What is critical and challenges for you is to lean the new technology, mainly the use of software within a reasonable portion of your time. The learning curve of the software we will be using is very sharp. We need caveats to question our thinking and show why we must learn-to by instrument that in this course is your computer package. Every student taking courses in Physics and Chemistry does experimentation in the labs to have a good feeling of the topics in these fields of study. You must also perform experimentation to understand the Management Science concepts. For example, you must use your computer packages to perform the "what if" analysis. Your computer software allows you observe the effects of varying the "givens". You will be engaged in thinking process of building models rather than how to use the software to model some specific problems. Software is a tool, it cannot substitute for your the thinking process. We will not put too much focus on the software at the expense of leaning the concepts. We will lean step-by-step problem formulation, and managerial interpretation of the software output. Managerial Interpretations: The decision problem is stated by the decision maker often in some non-technical terms. When you think over the problem, and finding out what module of the software to use, you will use the software to get the solution. The strategic solution should also be presented to the decision maker in the same style of language which is understandable by the decision maker. Therefore, just do not give me the printout of the software. You must also provide managerial interpretation of the strategic solution in some non-technical terms. Learn by Doing: We will use WinQSB package as a computer-assisted learning tool to gain a good "hands-on" experience on concepts and techniques used in this course. These labs experimentation will enhance your understanding of the technical concepts covered in this course. The QSB (Quantitative Systems for Business) is developed and maintains by Yih-Long Chang. This software package contains the most widely used problem-solving algorithms in Operations Research and Management Science (OR/MS). The WinQSB is the Windows version of the QSB software package runs under the CD-ROM Windows. There is no learning-curve for this package, you just need a few minutes to master its useful features. The WinQSB Decision Support Software for MS/OM is available from the John Wiley & Sons publisher, ISBN 0-471-40672-4, 2003. Further Reading: Use the LP/ILP module in your WinQSB package for two purposes: to solve large problems, and to perform numerical experimentation for understanding concepts we have covered in the LP and ILP sections. Variable Type: Select the variable type from the "Problem Specification" screen (the first screen you see when introducing a new problem); for linear programming use the default "Continuous" option. Input Data Format: Select the input data format from the "Problem Specification" screen. Usually, it is preferred to use the Matrix format to input the data. In the Normal format the model appears typed in. This format may be found more convenient when solving a large problem with many variables. You can go back and forth between the formats, selecting the "Switch to the... " from the Format menu. Variable/Constraint Identification: It is a good idea to rename variables and constraints to help identify the context they represent. Changing the names of variables and constraints is done in the Edit menu. Best Fit: Using the best fit from the Format menu lets each column have its own width. Solving for the Optimal Solution (if it exists): Select Solve the problem from the Solve and analyze menu, or use the "solve" icon at the top of the screen. The run returns a "Combined Report" that gives the solution and additional output results (reduced costs, ranges of optimality, slack/surplus, ranges of feasibility, and shadow prices). Solving by the Graphic Method: Select the Graphic method from the Solve and Analyze menu (can only be used for a two-variable problem.) You can also click the graph icon at the top the screen. You can re-scale the X-Y ranges after the problem has been solved and the graph is shown. Choose the Option menu and select the new ranges from the drop down list. Alternate Optimal Solutions (if they exist): After solving the problem, if you are notified that "Alternate solution exists!!", you can see all the extreme point optimal solutions by choosing the Results menu and then select Obtain alternate optimal. Also visit Multiple Solutions section on this Web site for some warnings. Notes: Use the "Help" file in WinQSB package to learn how to work with it. For entering problem into the QSB software; for a constraint such as X1 + X2 < = 50, the coefficient is 1 and should be entered that way in the software. For any variables that are not used in that particular constraint (for example if there was X3 in the problem but it was not part of the above constraint), just leave the cell blank for that constraint. You can change the direction of a constraint easily by clicking on < = (or > =) cell. To construct the dual of a given problem click on Format, then select Switch to the Dual Form. If you are not careful you may have difficulty with entering LP problem in WinQSB. For example, in a given problem a few of your constraints may have variable on the right hand side (RHS). You cannot enter variable name on the RHS cell otherwise you keep getting an in-feasibility response. Only numbers can be entered on the RHS. For example, for the constraint X2 + X4 < = .5X5 one must write it first in the form of X2 + X4 - .5X5 < = 0, then using any LP package including your QSB. The LP/ILP module in WinQSB, like any other popular Linear programming software pac
kages solves large linear models. Most of the software packages use the modified Algebraic Method called the Simplex algorithm. The input to any package includes: The combine report is a solution report consists of both the solution to the primal (original) problem and it Dual. The typical output generated from linear programming software includes: The following are detailed descriptions and the meaning of each box in the WinQSB output beginning in the upper left-hand corner, proceeding one row at a time. The first box contains the decision variables. This symbol (often denoted by X1, X2, etc.) represents the object being produced. The next box entitled "solution value" represents the optimal value for the decision variables, that is, the number of units to be produced when the optimal solution is used. The next box entitled "unit costs" represents the profit per unit and is the cost coefficient of the objective function variables. The next box "total contribution", is the dollar amount that will be contributed to the profit of the project, when the total number of units in the optimal solution is followed. This will produce the optimal value. The next bow is the "Reduced Cost", which is really the increase in profit that would need to happen if one were to begin producing that item, in other words the product, which is currently is not produce becomes profitable to produce. The next box over is the "allowable minimum" and "allowable maximum", which shows the allowable change in the cost coefficients of that particular item that can happen and still the current the optimal solution remains optimal. However, the optimal value may change if any cost coefficient is changed but the optimal solution will stay the same if the change is within this range. Remember that these results are valid for one-change-at-a-time only and may not be valid for simultaneous changes in cost coefficients. The next line is the optimal value, i.e., and the value of objective function evaluated at optimal solution strategy. This line shows the maximum (or minimum) value that can be derived under the given the optimal strategy. The next line down contains the constraints; often C1, C2, etc. denote the constraints. Starting on the left-hand side the first box contains the symbol C1 that represents the first constraint. The next box is the constraint value. That is, the left-hand-side (LHS) of each C1 evaluated ate the optimal solution. The next box over is the "direction box", which is either greater than or equal to / less than or equal to, which are the direction of the each constraint. The next box is the right hand side value, which states the value that is on the right hand side of each constraint. The next box is the difference between RHS and LHS numerical values called the slack or surplus box. If it is slack, it will have a less than or equal to sign associated with it, which means there is leftover of resources/raw material. If there is a surplus it will have a greater than or equal to sign associated with it, which means that there are over production. Next box over is the shadow price. If any slack or surplus is not zero then its shadow price is zero, however the opposite statement may not be correct. A shadow price is the additional dollar amount that will be earned if the right hand side constraint is increased by one unit while remaining within the sensitivity limits for that RHS. The next two boxes show the minimum and maximum allowable for the right hand side constraints. The first box (minimum box) shows the minimum value that the RHS constraint can be moved to and still have the same current shadow price. The second box shows the maximum number that the constraint can be moved to and still have the same current shadow price. Recall that the shadow prices are the solution to the dual problem. Therefore, the allowable change in the RHS suggest how far each RHS can go-up or down while maintaining the same solution to the dual problem. In both cases the optimal solution to the primal problem and the optimal value may change. Doing Integer LP by Excel Solver: Using the Constraint menu, for the LP Problem, select Normal Constraint and then the icon for (< = or > = or = , choose INT for Integer, or bin fir Binary) to obtain the Add Constraint window. Then we can designate any variable (e.g. , inter the variable, say B4 in the Cell Reference) as Integer in the Add Constraint window. For LP you may use LP by Excel to complete the necessary computations and graphing in Excel, as your textbook does. Lecture Notes: You may try getting WinQSB from WinQSB Free Download on your C-drive computer. If you do not have access to WinQsb, use the following on: Linear Program Solver Computer Implementation: Solve problem 2.7 by the WinQSB and compare the results with your graphical solution. As part of your homework implement WinQsb on all the problems at: As another part of your homework, Here there are two applications of LP, read them carefully to understand then use WinQSB to verify the solution given therein. Send me a short report of your findings. If it is possible and easy for you, then send me any copy of the printout or send me the output electronically (fax, email-attach, etc.). However, what I really need to receive form you is your written managerial interpret and description of all the computational results in the computer output.
Graphical Method for Linear Programs
- Doing your homework by Excel implementation alone is not complete. You must do and show your hand computations too. In your Exam you are not allowed to use Excel or any computer software.
Read to understand: Graphical Method for Linear ProgramsPractice More to Enhance Your Understanding
- Doing your homework by e.g., Excel implementation alone is not complete. You must do and show your hand computations too. In your Exam you are not allowed to use Excel or any computer software.
Here is a solution set. Wilson Solution. One more is Here.
Computer Package: WinQSB
Chang Y-L., QSB+: Quantitative Systems for Business Plus, Prentice Hall, 1994.WinQSB Installation Instructions
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Computer Assisted Learning
Managerial Interpretation of the WinQSB Combined Report
For example, in the Wilson problem if you change cell F7 t0 3299 sq ft. Baseball dozens(B4) and softball dozens(C4) come up to 209.5 and 375.25 respectively. Then when you change B4 and C4 to int, the answer comes up to 211 and 374. When you change to Binary Cells C4 and B4 go to 1 if any anser is greater than 0. They go to 0 if the answer is 0.
The Dark Side of LP
MultipleUnbounded
Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others.
Click for Excel: Wilson's Problem Formulation
and click for the Excel Solution for Wilson's Decisions: Step-by-Step
Click Here (Word.Doc) to view step-by-step solution set for problem 2.7 submitted by one of your classmates.
Here is a solution set. Wilson Solution. One more is Here.
Yet a comprehensive WinQsb implementation is Here.
Click Here (Word.Doc) to view a submitted complete solution set for problem 2.7. Another one (Word.Doc).
Click Here (Word.Doc) to view How things can go wrong in your graphical LP solution?
What Combined Report Says?
Managerial What-if Analysis
The LP/ILP module in WinQSB, like any other popular Linear programming software packages solves large linear models. Most of the software packages use the modified Algebraic Method called the Simplex algorithm. The input to any package includes:
The combine report is a solution report consists of both the solution to the primal (original) problem and it Dual.
The typical output generated from linear programming software includes:
The following are detailed descriptions and the meaning of each box in the WinQSB output beginning in the upper left-hand corner, proceeding one row at a time. The first box contains the decision variables. This symbol (often denoted by X1, X2, etc.) represents the object being produced. The next box entitled "solution value" represents the optimal value for the decision variables, that is, the number of units to be produced when the optimal solution is used. The next box entitled "unit costs" represents the profit per unit and is the cost coefficient of the objective function variables.
The next box "total contribution", is the dollar amount that will be contributed to the profit of the project, when the total number of units in the optimal solution is followed. This will produce the optimal value. The next bow is the "Reduced Cost", which is really the increase in profit that would need to happen if one were to begin producing that item, in other words the product, which is currently is not produce becomes profitable to produce.
The next box over is the "allowable minimum" and "allowable maximum", which shows the allowable change in the cost coefficients of that particular item that can happen and still the current the optimal solution remains optimal. However, the optimal value may change if any cost coefficient is changed but the optimal solution will stay the same if the change is within this range. Remember that these results are valid for one-change-at-a-time only and may not be valid for simultaneous changes in cost coefficients.
The next line is the optimal value, i.e., and the value of objective function evaluated at optimal solution strategy. This line shows the maximum (or minimum) value that can be derived under the given the optimal strategy.
The next line down contains the constraints; often C1, C2, etc. denote the constraints. Starting on the left-hand side the first box contains the symbol C1 that represents the first constraint. The next box is the constraint value. That is, the left-hand-side (LHS) of each C1 evaluated ate the optimal solution. The next box over is the "direction box", which is either greater than or equal to / less than or equal to, which are the direction of the each constraint. The next box is the right hand side value, which states the value that is on the right hand side of each constraint.
The next box is the difference between RHS and LHS numerical values called the slack or surplus box. If it is slack, it will have a less than or equal to sign associated with it, which means there is leftover of resources/raw material. If there is a surplus it will have a greater than or equal to sign associated with it, which means that there are over production. Next box over is the shadow price. If any slack or surplus is not zero then its shadow price is zero, however the opposite statement may not be correct. A shadow price is the additional dollar amount that will be earned if the right hand side constraint is increased by one unit while remaining within the sensitivity limits for that RHS.
The next two boxes show the minimum and maximum allowable for the right hand side constraints. The first box (minimum box) shows the minimum value that the RHS constraint can be moved to and still have the same current shadow price. The second box shows the maximum number that the constraint can be moved to and still have the same current shadow price. Recall that the shadow prices are the solution to the dual problem. Therefore, the allowable change in the RHS suggest how far each RHS can go-up or down while maintaining the same solution to the dual problem. In both cases the optimal solution to the primal problem and the optimal value may change.
Primal and Dual problems and Their Solutions
Sensitivity Analysis: Review the Sensitivity Analysis section of the course lecture notes. Apply the right-hand-side (RHS) value and coefficients of the objective function (known as the cost coefficients, because historically during World War II, the first LP problem was a cost minimization problem) sensitivity range to problem 2.7 in Ch. 2, computer implementation together with managerial interpretations of the computer solution. Construct the dual problem, solve it and then provide economical interpretations for the dual and its solution. To construct the dual of a given problem by using WinQSB, click on Format, then select "Switch to the Dual Form".
As you know by now, this course has three ingredients:
A set of Keywords and Phrases,
A Collection of Problem-Solving Algorithms, and Managerial Interpretations,
and the most important of all their Implications and Applications to Business Decision-Making.
As I pointed out this course is not about say, linear programming (LP), we are using LP as an application and as a tool. Since you have mastered, the Keywords & Phrase, and Techniques, now we are able to concentrate on the Managerial Business Decision-Making Process. The lecture note section on Managerial Interpretation of the WinQSB Combined Report deals with how to interpret and describe the computational results in computer output such as, the optimal strategic solution, sensitivity ranges, shadow prices, and other useful information for the decision-maker.
Homework: Perform some "what-if" scenarios analysis on Problem 2.7. That is, use your computer software package to do some numerical experimentation on variations of Problem 2.7. Again, this computer-assisted learning assignment provides a "hands-on" experience, which will enhance your understanding of the technical concepts, involved in various topics of controlling the problems, which we have covered. This computer-assisted learning concepts provides a "hands-on" experience which will enhance your understanding of the technical concepts involved in various topics of sensitivity analysis that we have covered. .
For example:
A Review of Linear Programming (power Point).
WinQSB Computational Tools that can be applied to the current topics and then perform some numerical experiment for deeper understanding of the concepts. For example, you may like checking your hand-computations for the homework problem(s).
Warnings:
- Doing your homework by Excel implementation alone is not complete. You must do and show your hand computations too. In your Exam you are not allowed to use Excel or any computer software.
Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others.
Click Here (Word.Doc) to view the sensitivity ranges, the dual construction and its solution, and a set of what-if analysis.
Click Here (Word.Doc) to view the Dual of Problem 2.7 and its managerial meanings submitted by one of your classmates.
Click Here (Word.Doc) to view Managerial Interpretation of the Computer Implementation submitted by one of your classmates.
Click Here (Word.Doc) for a submitted Sensitivity Analysis.
Click Here (Word.Doc) to view a set of What-if Analysis performed on Problem 2.7 by one of your classmates.
Click Here (Word.Doc) to view a set of Post-optimality Activities performed on Problem 2.7 by one of your classmates.
Click Here (Word.Doc) to view a set of Sensitivity Analysis Activities.
You are certainly welcome to use the discussion board to post your questions, responses to any parts of the above files' contents. I do thank everyone for active-learning participation.
Decision AnalysisMotivations
Payoff Table Analysis: A campus bookstore must determine how many copies of a particular text to order for the upcoming semester. The bookstore's profit is a function of the number of sections of the course that will be offered as well as the number of textbooks the store orders. To assist the store in determining the correct number of books to order, the manager has constructed a payoff table. (See Ch.6, Problem 1.)
Decision-Making Criteria: An operator of a multiplex movie theater is trying to determine how many screens a new movie release should be booked into. This decision must be made before the theater operator has access to critics' reviews of the movie. (See Ch. 6, Problem 25.)
Value of Information: A candy maker is trying to decide whether to introduce a new line of lower-calorie candy. To help make this decision, the candy maker has hired a market research firm which, based on data from customer focus groups, will give its expert opinion as to whether customer attitudes are favorable or unfavorable toward the new product. Using this information, the candy firm will make a determination regarding the product introduction. (See Ch.6, Problems 27 and 28.)
Introduction:
The mathematical models and techniques considered in decision analysis are concerned with prescriptive theories of choice (action). This answers the question of exactly how a decision maker should behave when faced with a choice between those actions which have outcomes governed by chance, or the actions of competitors.
Decision analysis is a process that allows the decision maker to select at least and at most one option from a set of possible decision alternatives. There must be uncertainty regarding the future along with the objective of optimizing the resulting payoff (return) in terms of some numerical decision criterion.
The elements of decision analysis problems are as follow:
Read Ch. 6 and the course lecture notes. Do problems 6.1, and 6.2 by hand and WinQSB applications. How stable is your decision? The computer packages such as your WinSQB, are necessary a very helpful tools for the decision maker in performing stability and sensitivity aspects of the decision whenever there is uncertainty in the payoffs and or in assigning probabilities in any decision analysis.
WinQSB Computational Tool:
Select Da.exe "Decision Analysis" module then perform some numerical experiment for deeper understanding of the concepts. For example, you may like checking your hand-computations for the homework problem(s), or checking the numerical examples from your textbook. Submit along with the rest of your homework, a short report entitle "Computer Assignments" describing your findings. As a stating point (worth 20%)
The Da.exe "Decision Analysis" module in your WinQSB package is used for two distinct purposes: to solve large problems, and to perform numerical experimentation. Numerical experimentation including what-if analysis of the payoff matrix and the subjective probability assignments to the states of nature.
The following functions are available in the Da.exe module:
Bayesian Analysis: Select this option from the Problem Specification screen to input prior probabilities and conditional probabilities (probability of an indicator value given a state of nature). Then, press the 'solve' icon to obtain the posterior probabilities.
Decision Trees: You must draw the decision tree first to number all nodes, including the terminal nodes. These numbers become the node IDs, when building the decision tree within the program.
When you are ready to enter the data, select the option 'Decision Tree Analysis' from the 'Problem Specification' screen.
For each node, you will indicate the number of nodes immediately connected to it (type ,... ,
Mistakes may be corrected by directly typing the changes into the proper cells.
WinQSB Numerical Example: An Illustrative Solution Set,
Warnings:
- You have to send me your own solution. DO NOT submit the solution done by your classmate(s). Submitting any posted solution has zero value.
- Doing your homework by Excel implementation alone is not complete. You must do and show your hand computations too. In your Exam you are not allowed to use Excel or any computer software.
Collaborative Learning: It is a fact that we learn from each other, and it is good to rub and polish our mind against that of others.
Sample of Solutions: A solution set (Word.Doc), submitted by one of your clasmates.
Here (Word.Doc), is another one.
You are certainly welcome to use the discussion board to post your questions, responses to any parts of the above files' contents. I do thank everyone for active-learning participation.
Decision Analysis Part II
Decision Trees: Sequential Decision Making
A Motivation: A construction company specializing in restoring historical homes has several different options for doing the restoration work. The least expensive plan would require approval of the town's historic commission. The firm can opt for this plan or a more expensive plan that does not require the historic commission's approval. If it goes for the least expensive plan, the firm can wait for the commission's approval or it can begin work immediately, hoping the commission will give its approval. To determine its best course of action, the firm will undertake a decision tree analysis. (See Ch. 6, Problem 33.)
There are a few satisfactory description of uncertainty, one of which is the concept and the algebra of probability.
To make serious business decisions one is to face a future in which ignorance and uncertainty increasingly overpower knowledge, as ones planning horizon recedes into the distance. The deficiencies about our knowledge of the future may be divided into three domains, each with rather murky boundaries:
In fact, all highly man-made systems, such as, large communications networks, nuclear-powered electric-generating stations and spacecraft are full of hidden "paths to failure", so numerous that we cannot think of all of them, or not able to afford the time and money required to test for and eliminate them. Individually each of these paths is a black swan, but there are so many of them that the probability of one of them being activated is quite significant.
While making business decisions, we are largely concerned with the domain of risk and usually assume that the probabilities follow normal distributions. However, we must be concerned with all three domains and have an open mind about the shape of the distributions.
Continuum of pure uncertainty and certainty: The domain of decision analysis models falls between two extreme cases. This depends upon the degree of knowledge we have about the outcome of our actions, as shown below:
| Ignorance | Risky Situation | Complete Knowledge |
| _______________________________________________________________ | ||
| Pure Uncertainty | Probabilistic | Deterministic |
| Model | Model | Model |
One "pole" on this scale is deterministic, such as the carpenter's problem. The opposite "pole" is pure uncertainty. Between these two extremes are problems under risk. The main idea here is that for any given problem, the degree of certainty varies among managers depending upon how much knowledge each one has about the same problem. This reflects the recommendation of a different solution by each person.
Probability is an instrument used to measure the likelihood of occurrence for an event. When you use probability to express your uncertainty, the deterministic side has a probability of 1 (or zero), while the other end has a flat (all equally probable) probability. For example, if you are certain of the occurrence (or non-occurrence) of an event, you use the probability of one (or zero). If you are uncertain, and would use the expression "I really don't know," the event may or may not occur with a probability of 50%. This is the Bayesian notion that probability assessment is always subjective. That is, the probability always depends upon how much the decision maker knows. If someone knows all there is to know, then the probability will diverge either to 1 or 0.
The decision situations with flat uncertainty have the largest risk. For simplicity, consider a case where there are only two outcomes, with one having a probability of p. Thus, the variation in the states of nature is p(1-p). The largest variation occurs if we set p = 50%, given each outcome an equal chance. In such a case, the quality of information is at its lowest level. Remember from your Statistics course that the quality of information and variation are inversely related. That is, larger variation in data implies lower quality data (i.e. information).
Relevant information and knowledge used to solve a decision problem sharpens our flat probability. Useful information moves the location of a problem from the pure uncertain "pole" towards the deterministic "pole".
Probability assessment is nothing more than the quantification of uncertainty. In other words, quantification of uncertainty allows for the communication of uncertainty between persons. There can be uncertainties regarding events, states of the world, beliefs, and so on. Probability is the tool for both communicating uncertainty and managing it (taming chance).
Good Questions with Useful Answers (Word.Doc)
As a cautious note, you may experience some difficulties in comprehending the decision analysis problems, this is true for everyone while translating the way the problems are worded and the type of questions that are asked. Therefore, the most difficult part of decision analysis is the translation of the problem. Here are my suggestions: Read the problem may time, slowly. I suggest also drawing a decision tree to start with, then read the problem few time to modify the tree. Remember that, the mathematical representation of a decision analysis problem is the decision tree.
I do suggest that you prepare a few pages of your own Summary Sheets. Your mind is what your brain does. Self-consciousness is self-knowledge. The process of becoming conscious distributes what you know throughout your brain via the brain neural network branches, unlike memorizing, which connects only two nodes of the network. The availability and expansion of what you know throughout your neural network branches make the information processing of your brain accurate. Thus, you possess a reflective, brilliant knowledgeable mind. The process of making your own summary-sheet is the idea of contemplating the topics you have learned. By definition of aesthetics, the longer you contemplate on what you have learned the more beautiful the subject matter becomes. Beauty and contemplation is distinguished from other mental manifestations; contemplation is the result of the perfect apprehension of relations and topics. Use the following guide to prepare your Summary Sheets: The above process helps to crystallize your mind to be reflective and responsive to questions posed about topics you've learned in this course and reinforces the topics in your mind. I Am Confused: How to know when to apply what formulas and calculations in word problems? I have received the following honest concern and difficulty from one of your classmates. It will be beneficial to read and apply my recommendations: ...The muddiest points still remaining are how to know when to apply what formulas and calculations in word problems? You are not alone on this. Since you are learning little-by-little every week, it is very natural desire to see the wholeness and manifoldness of topics. Therefore, it is natural to feel confused because of accumulation of different topics. However we must cross over to the other side of confusion where by thinking clearly and distinctively you will feel comfortable. As you know by now, the ingredient components of what you should master are: To master what you a
Review, and Examination
Preparations:
Your final is similar is designed to know how reflective is your mind on decision making. Therefore, there is a two-hour time limit. You should expect about 15 questions both familiar applications and conceptual type questions. Your test has a total of 100 points for correct answers. You may need a couple of graph paper (Word.Doc) , graph paper (PDF). Print a few copies before taking your test.
Your test has a total of 100 points for correct answers.
You may need a couple of graph paper (Word.Doc) , graph paper (PDF). Print a few copies before taking your test.
Preparation
Review your past assignments. Ask yourself: What have I learned up to now? Your preparation is a very important undertaking in terms of integrating what you have learned each week in order to see the whole picture and inter-connectivity of the topics. Since you are allowed ONLY to use your own Pre-Prepared Summary-Sheets for Exam. Read carefully the Summary-Sheets for the Exam on this page, while preparing one for the test.
You may ask: "Will you be posting the solutions to Sample test?" No, however, I will be glad to check your submitted solutions, at least one week prior to the examination dates.
Examination
You are allowed to use your own Pre-Prepared Summary-Sheets for Exam. Read carefully the Summary-Sheets for the Exam on this page, while preparing one for the test. The exam is not in any particular format so expect both standard numerical problem solving and conceptual type questions. The exams will test your understanding of the material covered in this course.
Samples of Past Exams are available for inspection.
Please read the following information carefully.
Read carefully the Summary-Sheets for the Exam on this page, while preparing one for the test. You can use a calculator. Instead of a regular calculator, you are not allowed to use WinQSB, Excel, JavaScript, cell phone, computer, or any other computational tools for the exam.
The main purpose of taking the examinations is to find out how reflective your mind is in answering a set of questions correctly. The objective is to maximize the number of correct solutions, subject to a limited time constraint.
Your examination has the similar format as the sample test, consisting of two kinds of questions.
Sample of Questions with Useful Answers (Word.Doc)
Sample of LP Conceptual Questions (Word.Doc)
Review your past assignments. Ask yourself: What have I learned up to now? Your preparation is a very important undertaking in terms of integrating what you have learned each week in order to see the whole picture and inter-connectivity of the topics.
Exercise your knowledge on this Sample Final Exam.
You may ask: "Will you be posting the solutions to Sample test?" No, however, I will be glad to check your submitted solutions, at least one week prior to the examination dates.
The conceptual part contains a few questions that come from your weekly readings assignments. These questions test how careful and reflective your readings have been by coming up with correct, exclusive, and inclusive answers.I do suggest that you prepare a few pages of your own Summary Sheets.
Your mind is what your brain does. Self-consciousness is self-knowledge. The process of becoming conscious distributes what you know throughout your brain via the brain neural network branches, unlike memorizing, which connects only two nodes of the network. The availability and expansion of what you know throughout your neural network branches make the information processing of your brain accurate. Thus, you possess a reflective, brilliant knowledgeable mind.