This is a Web-text companion site for the Web-enhanced textbooks:
Success Science    Leadership Decision Making     Linear Programs (LP)     Integer Optimization and the Network Models    Decision Analysis     Decision Analysis JavaScript
Excel for Linear Programming     Systems Simulation and
Compendium of Web Site Reviews    Queueing and Simulation    Required Readings    Course E-Labs     LINDO: Download for week 2 the student version (free-of-charge) from Lindo.Com




The MENU:

0. Final Exam is scheduled for this Friday 13, BU.520.601.K2 at 9 AM and BU.520.601.K1 at 1PM.

1. Any Question?
Whenever you have any question on what are the HW assignments, please ask me via email (harsham@jhu.edu), Thank you.

2. Bring-in your scientific calculator and textbook in every class meeting

3. Please Email me (harsham@jhu.edu) of any Broken Link(s). Thank you for keeping this site up to date.


I am looking forward to working with you and hope that you will find the course both enjoyable and informative.
Professor Hossein Arsham   

          Contents

  1. Welcome Message
  2. Review of the Math You Need
  3. Solving Linear Programs by Excel
  4. Homework Assignment to Do Before
    Each Class Meeting and Sample Tests
  5. LINDO: Install it in your PC via Lindo.Com


Dear Student and Decision-Maker

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.

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 Web 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, decision theory, integer programming and network models 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 their 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. For more watch the video Do you know what Decision Science is?

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, faxes, or email. There is a lot of material to cover, so let's start now!

Course Ingredients: The Course Ingredient Components Include:

  1. A set of Technical Keywords and Phrases,
  2. A Collection of Problem-Solving Methodologies, and
  3. Managerial Interpretations, Their Implications and Applications.

I am sure that your careful readings and effective use of the above learning objects, provide various perspectives, create a deep understanding of the topic, together with the wholeness and manifoldness of this course.


General Instructions

  1. Write everything you know about the topics, one by one.

  2. When you can't think of anything more, give yourself time to look for topics and details you may have missed.

  3. Ask yourself, is there anything else I may have missed? Be as inclusive as possible.

  4. Summarize your writing to create fewer pages.

  5. Re-organize to make even fewer pages.

  6. Ask, How do the topics fit together? What elements are related and how?

  7. Ask, What is the significance for me? What can I do with it? What are the implications?

  8. Go back to step 3, until you have as few pages of summary as possible.

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.

To view a Summary-Sheet for the first test, prepared by one of your classmates, click here. If you think you have prepared a better one, kindly send it to me via an attached email. Thank you.


What Math Do I Need for this Course?

Don't Panic, high school math will suffice! There will be some refreshers. The following sites may help:

High school math refresher.


Homework Assignment to Do Before
Each Class Meeting and Sample Tests

Please read and follow the Instructions for your homework assignment. Thank you.

We will proceed in the following sequence (Not a weekly-schedule of topics).

  1. Reading Assignment and Essay: Read the Preface (Introduction section), Chapter 1 (all sections), and Chapter 2 (Sections 2.1, and 2.2) of your textbook Management Science. Read also the introductory sections of all the chapters (i.e., sections 1 of all chapters) in your textbook. This is a good way to get a grip on your textbook and the concepts contained therein.
    Visit the main Web site of this course and go over the title of the topics therein. Visit the following Web sites:
    INFORMS
    Decision Sciences Institute
    OR Page
    Operational Research Society

    After you did your reading assignment, then write a 2-page essay (format-free) entitled: "What is Applied Management Science?" Your essay should, among others, address some of the following questions:

    • What is Applied, what is Management, what is Science?
    • What benefits would you expect from using management science?
    • Why do you think the topic of decision-making process has not received mainstream attention?
    • Business, education and psychology communities embrace decision aides. Do you think the general public will? (why or why not?)
    • What is your perception of the problem people have with indecisiveness as it relates to major life decisions (i.e. career, family, etc.)?
    • It seems to me that the topic of decision-making as problems-solving should be taught in high schools, too, because of its implications on individuals and society. Would you agree or disagree and why?
    • Is management science a form of Applied Decision-Making in Organizations, and Business Problem Solving?
    • Does management science make decisions?
    • Why has quantitative analysis for management problems become more widely used in the past years?

    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 an essay submitted by one of your classmates. Here is another one.
    Comments on The Two 2Essays by one of your classmates.

    Click Here (Word.Doc) to view my general comments on your essays.

  2. Analytical Geometry Review: Read Chapter 11 (sections 11.1 - 11.6), the course lecture notes and then Formulate (do not solve) The Wilson Problem as a linear program (LP).
    I do recommend refreshing your knowledge about solving systems of equations by e.g., visiting the Web site Solving System of Equations. Do Lego Formulation, and this LP Formulation

    Visit the following Web site:
    The Zero Saga & Confusions With Numbers

    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.
    Wilson, and Other Problems are formulations submitted by one of your classmates.

    Click Here (Word.Doc) to view How things can go wrong in your LP problem formulation?

    More Optimization Models

  3. Linear Programming (LP) Graphical Solution Algorithms, Read the course lecture notes. Do all parts of The Wilson Problem, 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. Another LP Graphical Solution. Walking Through Another LP Graphical Method.

    The Gape Between the Modeler and the Manager: Not all managers are aware of modeling concepts for decision-making or the practice of modeling for decisions. As a Modeler give a managerial explanations for the Decision Maker for the following Three Models:
    Advertising Problem Pages 96-99
    Allocation Golf Problem: Similar to Veerman Furniture Company model on page 242.
    Nutrition Problem page 247

    Lego Solution,    Managerial I,    Managerial II,    Managerial III submitted by your classmates.

    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 The Wilson Problem submitted by one of your classmates.
    Click Here (Word.Doc) to view a complete solution set for The Wilson Problem submitted by another classmates.

    Click Here (Word.Doc) to view How things can go wrong in your graphical LP solution?

  4. Computer Implementation and Computer-assisted Learning: Solve The Wilson Problem

    Modeling Validation: The Dark Side of LP: Graph each one of the model therein (homework)
    Stability Problem and the MFL Situation Verify each one of the model therein (homework)

    Unfortunately, in some browsers the Graphical Methods of Excel may not be available. However, one may, e.g., use the following JavaScript instead.

    1. Interactive Graphical LP
    2. The LP Grapher

    Use any above solver and then compare the results with your hand graphical solution.

    You may use: LINDO: Download the student version-free from Lindo.Com

  5. 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 The Wilson Problem 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.

    As you know by now, this course has three ingredients: A set of Technical 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 LP 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.

    Perform some "what-if" scenarios analysis on The Wilson Problem. That is, use your computer software package to do some numerical experimentation on variations of The Wilson Problem. 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.

    Computer Managerial Interpretation: Item-by-Item

    Quantitative risk analysis is the practice of creating a mathematical model of a project or process that explicitly includes uncertain parameters that we cannot control, and also decision variables or parameters that we can control.  A quantitative risk model calculates the impact of the uncertain parameters and the decisions we make on outcomes that we care about -- such as profit and loss, investment returns, environmental consequences, and the like.  Such a model can help business decision makers and public policy makers understand the impact of uncertainty and the consequences of different decisions.

    Walking Through LP Sensitivity Analysis

    The Dual Problem: The Insurance Problem and the Economics Meaning of the Shadow Prices (homework)

    1. Graph with Sensitivity Analysis
    2. Interactive Graphical with Sensitivity Analysis
    3. Solving Equations with Parametric RHS

    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 Dual of The Wilson Problem 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.

    Computer Implementations of Carpenter and the Wilson Problems
    An Almost Complete Solution with Sensitivity Analysis to the Wilson Problem Submitted by One of Your Classmates.

    Carpenter Structural and What-If Analysis

    How to Compute Sensitivity Ranges: A Solution-Based Approach

    How to Deal with Unrestricted Variables: LP in Non-Standard Form The Use of Dummy Objective Function: Goal-seeking Problem and Solving System of Equations.

  6. The Network Models and Integer LP Read Chapter 12 (Sections 12.1-12.5) Solve at least any two of the problems therein. Provide your managerial interpretation of the optimal solution for each problem.
    You may ask what are the Managerial Interpretations?

    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 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 solution in some non-technical terms. Walking Through Network Models.

    Warning: Computer solutions for the network and integer problems are valid, however the produced sensitivity results may not be valid. This is due to the facts that, among other things, these problems are Integer-LPs. Moreover, in the case of network models anyone constraint in any of these models is always redundant. Therefore, simply ignore the sensitivity analysis of the printouts. Exercise for Network Models

    Verification of Integer LP Example in Chapter 13.

    1. Simplex method for larger problems

    Primal and Dual Final Tableau

    Integer LP: Read Chapter 13 (Sections 13.1-13.2) Use Lindo Integer package to find and verify the optimal solutions given for each Numerical Examples in Chapter 13.
    Read What Can Go Wrong in Integer Programming?

    Verification of Integer LP Example in Chapter 13. Walking Through Integer Models.

    Visit the following Web site:
    A Tutorial on Integer.

    Some Good Questions for You: Check them all

    Optimization Online is a repository of e-prints about optimization and related topics

    What About Nonlinear Optimization? Read Ch. 10

    Essentials of Linear Programming for Managers

  7. Decision Analysis: Read Chapter 15 (All sections), Decision Analysis Summary, and the course lecture notes. How stable is your decision? Implement the numerical examples by using Decision Analysis JavaScript

    Homework do the following problems: Decision Analysis Homework.

    Decision Analysis Solution Set with Decision Tree for problem no. 3

    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. Walking Through Decision Analysis Models

    Some More Good Questions for You: Check them all

    Game Theory for Business: Unification of Linear Program and Decision Analysis
    Measuring Risk: Mean, Variance, Coefficient of Variation
    Multinomial Distributions Calculator: Mean, Variance, Coefficient of Variation Calculator

  8. Monte Carlo methods: Read Chapter 16 (section 1), Read Modern Simulation Overview   

    Monte Carlo methods rely on random sampling -- the computer-based equivalent of a coin toss, dice roll, or roulette wheel.  The numbers from random sampling are "plugged into" a mathematical model and used to calculate outcomes of interest.  This process is repeated many -- typically thousands of -- times.  With the aid of software, we can obtain statistics and view charts and graphs of the results.

    Monte Carlo simulation is especially helpful when there are several different sources of uncertainty that interact to produce an outcome.  For example, if we're dealing with uncertain market demand, competitors' pricing, and variable production and raw materials costs at the same time, it can be very difficult to estimate the impacts of these factors -- in combination -- on Net Profit.  Monte Carlo simulation can quickly analyze thousands 'what-if' scenarios, often yielding surprising insights into what can go right, what can go wrong, and what we can do about it.

    Walking Through Monte Carlo    

    Monte Carlo Applications as a Learning Object for Statistics:

    1. Proof of the Central Limit Theorem   
    2. Sampling Distribution of Sample Mean   
    3. The Let's Make a Deal Applet  It discusses the pros and cons of switching doors after the first selection.
    Monte Carlo Estimation for Pi: A Modification, Pi = 4 (Number of Darts in a Quarter of a Circle) / (Number of Darts in First Square)   

    Monte Carlo Application

  9. Review: Homework: Prepare your summary-sheets Your Exam is in-class, open book and open-notes. You may bring a few prepared summary-sheets.

    Preparation for the Exam: 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.

    To prepare yourself for the actual test, you are advised to review all the topics we have covered, to review past homework assignment, and then prepare your own few pages of a summary sheet. The process of producing a summary sheet, helps you to crystallize your mind to be reflective and responsive to any question posed to you about the topics you've learned in this course, it also helps you to reinforce the wholeness of the topics in your mind.

    To view an LP Summary-Sheets prepared by one of your classmates, click here. If you think you have prepared a better Summary-Sheets, kindly send it to me via an attached email. Thank you.

    To view a Summary-Sheets on Decision Analysis, click here.

    Exercise Your Knowledge on this Past Final Exam:

    Past LP-Part Exam
    Past Comprehensive Final Exam (Word.Doc)

  10. Final Examination is a comprehensive one. Read the Examination Facts. 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 (a 2- 1/2 hours session).


             Instructions for Homework Assignment

    • Homework: Homework Assignments will be collected and graded. Your homework assignment consists of two parts:

      Part I: Reading the lecture notes, Readings and problem solving from your textbook.
    • Part II: Compute Implementation (whenever applicable) using LINDO, Excel, JavaScripts, etc., since without a computer package one cannot perform any realistic decision-making process and analysis.

      You do not have to type your homework assignments; you may save some times in spending more on learning than typing. In this case please, please hand-in your eligible homework on time.

    • Students should attempt as many of the problems in each chapter as possible. At least one problem representative of each topic covered in the classroom should be attempted. Suggested problems would include those for which solutions are available in the back of your textbook. Problem formulation and solving are an important aspect of learning Decision Science. It is therefore important that you regularly do your homework assignment selected from the text.
    • Homework Due Date: Meeting deadlines and even sending me material before deadlines are very important, since you will be engaging in group learning activities where time is crucial. So you should make every effort to complete your work on time. Therefore, late homework submissions carry no credit at all.

    • The readings from the textbook will be supplemented by this course Web site materials.

    • Students should attempt as many of the problems in each chapter (covered) as possible. At least one problem representative of each topic covered in the classroom should be attempted. Suggested problems would include those for which solutions are available in the back of your textbook. Problem formulation and solving are an important aspect of learning statistics. It is therefore important that you regularly do your homework assignment selected from the text.
    • The use of a scientific calculator is required for the course and should be brought to each class meeting.

    • Keep a copy of your complete homework and any other material before submitting for grading. Keep the copy until you receive a grade notification form me. These steps will ensure the safety of any material that is lost or unduly delayed. If some material are delayed or lost, i.e., not received on time, you will be ask to resubmit another copy. In the unlikely event that you are unable to resubmit another copy, you will be required to redo it.


    Learning Style For This Course

    This course requires a particular learning style known as learning-to-learn. Effective and efficient learning includes completing weekly homework assignment and learning from feedback. Knowledge conquered by thinking for yourself becomes a possession -- a property entirely our own.

    Unfortunately, most classroom courses are not learning systems. Instructors attempt to help their students acquire skills and knowledge with lectures, tests and memorization. Instructors "tell," which doesn't translate into usable skills. We learn by doing, failing and practicing. Computer assisted learning serves this purpose.

    The change in learning in this course over the years is less emphasis on strategic solution algorithms and more on modeling processes, applications and software. This trend continues as more students with diverse backgrounds seek Business degrees without too much theory and mathematics. Our approach is middle-of-the-road: no excess of math or software. We learn how to formulate problems prior to software usage. You should learn how to model a decision problem, first by hand and then by using software. The software should be used for two purposes:

    1. Computer-assisted learning concepts and techniques, and

    2. For large problems that are too difficult to solve by hand.

    What are the most critical challenges in learning for this course?

    1. To refresh your high school math including linear algebra, basic statistics, and probability.

    2. To learn the new technology, mainly the use of software within a reasonable amount of time. The learning curve of the software we will be using is very sharp.

    3. To link the course materials with other courses in your Business program.

    4. What is Management Science? A rational, structured approach to problem solving. It is the study of developing procedures that are used in decision-making and planning. An objective measure of performance must be identified to measure success. The objective must represent the decision-makers goal and serve as a starting point for developing a model for the problem.
    5. The context of modeling: What is a management science model? There are two types of models: Deterministic and Probabilistic. Deterministic modeling is linear programming for optimization while decision analysis is a probabilistic modeling tool used for problems under uncertainty.

    6. Model design, selection and setup: This includes justifying model selection (validation), setting assumptions, parameters, advantages and limitations of various models, considering the effects of data quality and accessibility, regulations, implicit versus explicit assumptions, computer models (verification), and sensitivity analysis.

    7. Input data selection and analysis: How to find data and the balance between quality, accessibility, credibility, and relevance. How to evaluate data quality and understand its impact.
    8. Analysis of results: How to tell if results are reasonable, sensitivity of output to changes in input, recognition of the useful life of a model.
    9. Cases and applications: Word Problem Formulation, Pure Integer/ Mixed-integer Linear Programs, Transportation Problem, Assignment Problem, Shortest Path Problem, Max Flow Problem, Critical Path Method in Project Management, Decision Analysis Cases.

    10. Communication: Clearly and accurately communicate the process and result by understanding the nature of the audience, effects of standards and regulations, use of appropriate format and media and maintenance of internal documentation.

    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. I am sure you will be enthusiastic about the topics covered in this course throughout the semester and beyond. Enthusiasm is one of the most powerful engines of success; enthusiasm changes problems into challenges. When you study for this course, put your whole mind into it. Stamp your work with your own personality when it is submitting. Be active, energetic, honest, and remember: learning-to-learn was never achieved without enthusiasm.


    Solving Linear Programs by Excel
    Some of these examples can be modified for other types problems

    1. Excel Add-Ins
    2. Excel LP Solver
    3. Excel Tutorial (Recommended)
    4. Short Note on Excel
    5. Carpenter I
    6. Carpenter II
    7. Carpenter III
    8. Your Linear Programming HomeWork: The Wilson Problem
    9. Wilson Problem I
    10. Wilson Problem II
    11. Wilson Sensitivity
    12. Wilson Formulation
    13. Wilson Problem with New Product
    14. LP: Non-Standard-Form
    15. Shadow Prices
    16. Shadow Prices 3-Constraint
    17. Degenerate LP
    18. Other Excel Examples
    19. Excel for Allocation Golf Problem Page 242
    20. Excel for Nutrition Problem page 247


    Back to
    Decision Science: Making Good Strategic Decisions
    main Web site.


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