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1 Decision Science BMGT 825 Fall 2009, UNK Professor: Ron Konecny Ph.D. University of Nebraska at Kearney
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Decision Science
BMGT 825
Fall 2009, UNK
Professor: Ron Konecny Ph.D.
University of Nebraska at Kearney
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Table of Contents
4Class Syllabus
4Homework Assignments
4Daily Discussion Topics
4GP/LP Basics
4Sample Problems
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Contact Information
Office Hours:
249W West CenterPhone 865-8366
email: [email protected]
Office hours: M-TH 10:00-11:00
Appointments can be arranged for times outside of normal office hours. Students are encouraged to seek extra assistance if needed. Please do not delay visiting during office hours if you have questions on the material. Please call ahead for an appointment outside of normal office hours.
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Required Course Materials
1. An Introduction to Management Science, 12th ed., 2008, Anderson, Sweeney, Williams
2. The Goal, 3rd revised edition, 2004, Goldratt and Cox, North River Press
3. The World is Flat, Release 3, 2007 Thomas Friedman, Picador Press
4. Decision Science 2007 Computer software (Provided by instructor –free-)
5. Access to Windows Vista/XP platform computer with internet access and Excel 2007.
6. The Department of Management is a member of the Microsoft Academic Alliance. This permits all MBA students taking BMGT 825 to receive free software. The list is extensive. You may download and use full versions of any/all of these software packages beyond the end of this class. You will receive an email from the MSDNAA (Microsoft Developer Network Academic Alliance) giving you a user name and a password.
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Course Description
Course Description:
Recent developments relating to business application of linear
programming, simplex method, transportation method, post-optimality
analysis, game theory, utility theory, PERT-CPM, queuing theory,
dynamic programming, Markov chains, Decision tree analysis, time
series, analysis and forecasting.
Course Objectives:
To introduce students to some quantitative methods and techniques of
management science. To cultivate their skill in the application of those
methods and techniques. To encourage students to apply the learned
tools in business applications. To present state-of-the-art modeling
techniques.
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Course Evaluation
Course Performance Evaluation300 points : 3 examinations: 100 points each
100 points : homework submission
50 points : Notebook & short papers
450 points total
Grade assignments will correspond to standard UNK policy,Notebooks will be graded on completeness, organization, presentation, and neatness.
A detailed listing of homework assignments and daily discussion topics is contained on the Decision Science Software CD.
Other supporting information for the class may be found at http://Management.unk.edu/BMGT825
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Course Evaluation - continued
Course Outline: Test Schedule
• Exam 1: Due October 1 - Chapters 1 - 6, 8, Introduction to
Management Science, linear programming, sensitivity analysis,
integer programming, and applications.The Goal reading.
• Exam 2: Due November 12 - Chapters 9, 10, 11, 12 Scheduling,
inventory models, waiting line models, and simulation.
• Final: December 17 - Chapters 4, 14, 15 data envelopment analysis,
multi-criteria decision making, and forecasting.
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http://www.unk.edu/offices/aaeo/index.php?id=1542
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ACADEMIC POLICY FOR STUDENTS WITH DISABILITIES
The college is committed to providing support for students with disabilities. Any student with physical, learning, emotional, or psychological disabilities is encouraged to stop by room 163 in the Memorial Student Affairs Building, The Academic Success Office and make an appointment with the Disabilities Coordinator. If you have an accommodation plan please see me as soon as possible, so we can make any arrangements necessary for your learning. No accommodations can be provided until a Reasonable Accommodation Plan is in place. Please remember, plans are not retroactive and cannot be used for assignments prior to the date of my signature.
A detailed list of the UNK Academic Policy for Students with Disabilities is available the following website.
ACADEMIC INTEGRITYAll students at the University of Nebraska at Kearney are expected to conduct their academic affairs in an honest and responsible manner. Any student found guilty of dishonesty in academic work shall be subject to disciplinary actions. Acts ofacademic dishonesty include, but are not limited to:
•plagiarism, i.e., the intentional appropriation of the work, be it ideas or phrasing of words, of another without crediting the source.
•cheating, i.e., unauthorized collaboration or use of external information during examinations;
•assisting fellow students in committing an act of cheating;
•falsely obtaining, distributing, using or receiving test materials or academic research materials;
•submitting examinations, themes, reports, drawings, laboratory notes, research papers or other work as one's own when such work has been prepared by another person or copied from another person (by placing his/her own name on a paper, the student is certifying that it is his/her own work);
•improperly altering and/or inducing another to improperly alter any academic record.
Additionally, graduate students are more likely to assume roles as active scholars. With these roles come added responsibilitiesfor academic honesty. For such individuals academic honesty requires an active pursuit of truth, not just an avoidance of falsehood. This pursuit includes but is not limited to:
providing a full and a complete representation of any scholarly findings, be it experimental data or information retrieved from archives;
•taking care that the resources of the University (e.g., library materials, computer, or laboratory equipment) are used for theirintended academic purposes and that they are used in a manner that minimizes the likelihood of damage or unnecessary wear;
•assuring that one's co-workers are given due credit for their contributions to any scholarly endeavor;
•respecting a diversity of opinion and defending one's colleagues as well as one's own academic freedom;
•respecting the rights of other students who may come under the tutelage of the graduate student and being fair and impartial in grading and other forms of evaluation; and
•seeking permission from an instructor when submitting work that has been used in other courses.
In cases of alleged academic dishonesty, the instructor shall attempt to discuss the matter with the student and explain the sanction(s) which he/she plans to impose. In the event that the student challenges the allegation of academic dishonesty, or is not satisfied with the sanction(s) imposed by the instructor, the student may file an appeal according to the approved appealpolicies of the University of Nebraska Graduate College.
20 Jul 2006
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Graphical , Algebraic, and computer solutions
4 Chapter 2, Problem 14
4 Chapter 2, Problem 36
4 Chapter 2, Problem 50
- Include the these three hand solved problems in the notebook as well as the computer
solution.
Reading Assignment
•Chapter 1, pg. 1-17
•Chapter 2, all sections
Homework Assignment, due 9/2
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Homework Assignment, due 9/9
Linear Programming
- Use the DS software on all problems
4 Chapter 2, Problem 14
4 Chapter 2, Problem 36
4 Chapter 2, Problem 50
4 Chapter 4, Problem 1
4 Chapter 4, Problem 2 (modified a and c)
4 Chapter 4, Problem 3
4 Chapter 4, Problem 8
Reading Assignment
•Chapter 3
•Chapter 4, sections 1-3
•The Goal – Chapters 1-20
You must submit all homework problems on the Management.unk.edu
webserver before the beginning of class on the due date. The model must
include at least the:
• model sheet
•output sheet
•variable sheet
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Homework Assignment, due 9/17
Linear Programming
4 Chapter 4, Problem 14
4 Chapter 4, Problem 16
4 Chapter 4, Problem 17
4 Margaret Black Farm - on Platteriver website
Multi-period Problems
4 Antelope Endowment Fund (AEF)
4 The Goal – Chapters 21-40
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Homework Assignment, due 9/24
Simulation 4 Loper Dairy simulation
– Solve a series linear programs by changing the price (revenue) of a case of cheese to the following values: 58, 60, 62 ,64, 66, 68, 70, 72
– Specify the variables yogurt and cheese for output with the simulation
Transportation & Assignment Problems
– without set notation4 Chapter 6, Problem 2 (transportation)
4 Chapter 6, Problem 14 (assignment)
4 Chapter 6, Problem 19 (transshipment)
Reading Assignment
•Chapter 6, all sections
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Homework Assignment, due 10/1
Exam #1 is due. Electronic submission due by 5:59pm on the http://management.unk.edu/bmgt825 web server.
Academic Honesty. The maintenance of academic honesty and integrity is a vital concern of the University community. Any student found guilty of academic
dishonesty shall be subject to both academic and disciplinary sanctions. Academic dishonesty includes, but is not limited to, the following:
1. Cheating. Copying or attempting to copy from an academic test or examination of another student; using or attempting to use unauthorized
materials, information, notes, study aids or other devices for any academic test, examination or exercise; engaging or attempting to engage the
assistance of another individual in misrepresenting the academic performance of a student; or communicating information in an unauthorized
manner to another person for an academic test, examination or exercise.
2. Fabrication and Falsification. Falsifying or fabricating any information or citation in any academic exercise, work, speech, test or examination.
Falsification is the alteration of information, while fabrication is the invention or counterfeiting of information.
3. Plagiarism. Presenting the work of another as one's own (i.e., without proper acknowledgment of the source) and submitting examinations,
theses, reports, speeches, drawings, laboratory notes or other academic work in whole or in part as one's own when such work has been prepared by
another person or copied from another person.
4. Abuse of Academic Materials. Destroying, defacing, stealing, or making inaccessible library or other academic resource material.
5. Complicity in Academic Dishonesty. Helping or attempting to help another student to commit an act of academic dishonesty.
6. Falsifying Grade Reports. Changing or destroying grades, scores, or markings on an examination or in an instructor's records.
7. Misrepresentation to Avoid Academic Work. Misrepresentation by fabricating an otherwise justifiable excuse such as illness, injury, accident,
etc., in order to avoid or delay timely submission of academic work or to avoid or delay the taking of a test or examination.
8. Other. Academic units and members of the faculty may prescribe and give student prior notice of additional standards of conduct for academic
honesty in a particular course, and violation of any such standard of conduct shall constitute misconduct under this Code of Conduct and the
University Disciplinary Procedures.
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Discussion Topics (8/25, Week 1)
August 26
– Introduction to Management Science
– Graphic solution
• Loper Dairy
– Algebraic Solution
– RHS and objective coefficient ranging
– Introduction to software
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Discussion Topics (9/2 Week 2)
September 2– Discuss homework
• shadow prices
• RHS and objective coefficient ranging
– Features of DS software
– Introduction to Decision Science Software• installing
• Editing, Saving, Printing Simplex method
– Discuss problem types• Production mix
• Ingredient mix
• Production Scheduling
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Discussion Topics (9/9 Week 3)
September 9
– Discuss The Goal and homework problems
– Identity equations
• Efficiency
• Profit, revenue, and cost functions
– Multi-period investment problems
– Natural resource problems
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Discussion Topics (9/16 Week 4)
September 16
– Discuss The Goal and homework problems
– Rudimentary simulation
– Transportation and assignment examples
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Discussion Topics (9/23 Week 5)
September 23
Discuss transportation homework
Discuss Loper Dairy simulation
Distribute Exam #1
• covers material in first 5 weeks of class
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An Introduction to Linear Programming
4Linear Programming Problem
4Problem Formulation
4A Maximization Problem
4Graphical Solution Procedure
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Requirements of a Linear Programming
Problem
4 All linear programming (LP) problems seek to maximize
or minimize some quantity, such as profit or cost. This is
called Optimization of the Objective Function.
4 The quantity of the objective is limited by a system of
constrains.
– Land, labor, capital, prices, contracts, limited resources
4 There must exist multiple alternatives. However, some of
these alternatives may give rather poor results.
4 The objective function and constraints in an LP must be
expressed in terms of linear equations or inequalities
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Problem Formulation
4Problem formulation or modeling is the
process of translating a verbal statement of
a problem into a mathematical statement.
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Guidelines for Model Formulation
4Understand the problem thoroughly.
4Write a verbal description of the objective.
4Write a verbal description of each constraint.
4Define the decision variables.
4Write the objective in terms of the decision
variables.
4Write the constraints in terms of the decision
variables.
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Sample Problems
4 Loper Dairy (linear program)
4 Millworks Plywood (linear program)
4 Antelope Development Fund (set notation)
4 Old Farms (set notation)
4 Queens on a chess board (set notation)
4 Knights on a chess board (set notation)
4 Kearney City Council (goal program)
4 Shortest Route - map (transshipment, set)
4 Multi-commodity Shipping (3D set)
4 American Settlement Problem (set)
4 Suncoast Office Supplies (goal program)
4 Hospital Evaluation (Data Envelopment Analysis)
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Loper Dairy
At Loper Dairy specialty yogurt and cheese are produced and
sold nationally.
< The production of one case of yogurt requires 2 machine
hours and 1 labor hour. Profit for selling one case of yogurt
is $8.00.
< The production of one case of cheese requires 2 machine
hours and 2 labor hours. Profit for selling one case of
cheese is $12.00
< There are only 120 machine hours and 80 labor hours
available.
< Profits equal revenue minus costs.
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Loper Dairy
Cost Availability machine hours per machine hours per
Case of Yogurt Case of Cheese
labor hours per labor hours per
Case of Yogurt Case of Cheese
Revenue
Profit
Resource Requirements
$16/machine hour
$10/labor hour
120 machine hours
80 labor hours
Yogurt Cheese
$50/Case of Yogurt $64/Case of Cheese
$8/Case of Yogurt $12/Case of Cheese
Machine
Labor
2 2
1 2
Yogurt Revenue $50
Machine Cost -$32
Labor Cost -$10
$8Yogurt Profit
Cheese Revenue $64
Machine Cost -$32
Labor Cost -$20
$12Cheese Profit
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Loper Dairy – LP model
•Identify the decision variables
•What can change in the model?
•Labor hours? Machine hours? Cases of yogurt produced? Cases of cheese produced?
• Y - Cases of yogurt produced
• C - Cases of cheese produced.
•Write the objective function
•What do we want to do?
Max Profit = 8 Y + 12 C
cheeseofcasesCcheeseofcases
$+ 12of yogurt casesY
yogurtofcase
$Max Profit($)= 8
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Loper Dairy – LP model
•Write the constraints
≤ 120 (machine hours)2 Y + 2 C
cheeseofcasesCcheeseofcases
Machine hours+ 2of yogurt casesY
yogurtofcase
Machine hours2 ≤ 120 machine
hours
≤ 80 (labor hours)1 Y + 2 C
cheeseofcasesCcheeseofcases
Labor hours+ 2of yogurt casesY
yogurtofcase
Labor hours1 ≤ 80 labor
hours
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Loper Dairy – Graphical Solution
20 40 60 80
20
40
60
80
00
Cheese
Yogurt ≤ 120 (machine hours)2 Y + 2 C
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Loper Dairy – Graphical Solution
20 40 60 80
20
40
60
80
00
Cheese
Yogurt
≤ 80 (labor hours)1 Y + 2 C
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Loper Dairy – Graphical Solution
20 40 60 80
20
40
60
80
00
Cheese
Yogurt
≤ 80 (labor hours)1 Y + 2 C
≤ 120 (machine hours)2 Y + 2 C
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Loper Dairy – Graphical Solution
20 40 60 80
20
40
60
80
00
Cheese
Yogurt
≤ 80 (labor hours)1 Y + 2 C
≤ 120 (machine hours)2 Y + 2 C
Feasible
Region
Infeasible
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Loper Dairy – Corner Point Principle
20 40 60 80
20
40
60
80
00
Cheese
Yogurt
1 Y + 2 C ≤ 80 (labor hours)
2 Y + 2 C ≤ 120 (machine hours)
Feasible
Region
Max Profit= 8 Y + 12 CSubject to:
( C, Y ) Profit
(0, 0)
(40, 0)
(0, 60)
(20, 40) $0$480$480$560
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120
80
560
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Loper Dairy – Sensitivity Analysis
Y=40.0
C=20.0
2.0 2.0
1.0 2.0
Y+ C ≤
Y+ C ≤
Y+ C
Feasible
Region
Profit=
Max Profit =Subject to
Cheese
Yogurt
20 40 60 800
20
40
60
80
0
(machine hours)
(labor hours)
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Loper Dairy - Linear Program
The objective function multiplies the
profit for each item sold by the number
of items sold.
The variable “Yogurt” represents the
number of cases of yogurt produced
and sold. The variable “Cheese”
represents the number of cases of
cheese produced and sold.
The “LaborHours” constraint illustrates
that when one case of yogurt is
produced then one labor hours are used.
Similarly, when one case of cheese is
produced then two labor hours are used.
Examining the “Variables” table is
useful in making sure the model
describes the problem correctly. The
“count” column shows how many times
a variable is used in the model. If the
count equals one for a variable then
most likely there is a spelling or logic
error.
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Loper Dairy - simplex tableau
Initial Simplex Tableau - Selecting the “Tableau” sheet activates the simplex presentation. The cursor
is placed on the pivot element (the intersection of the pivot column and pivot row)
The “Final Tableau” can be viewed by selecting “final” under the “Step” menu option or by clicking on the “Tableau”
sheet after running solving the problem using “Run Local Solution.” The cursor will appear over the word “Basis”
when the solution is optimal.
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Loper Dairy - identity variable and equations
Determining profit for an individual product can be quite
tedious. The table at right shows how the profit for one
case of yogurt is determined. If you were to determine the
profit for selling an automobile with thousands of parts
the calculation would be nearly impossible, especially if
the purchase prices changed often.
Revenue $50 per case
- Machine Cost ($32) 2 hours at $16/hour
- Labor Cost ($10) 1 hour at $10/hour
Profit $8 per case
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Loper Dairy - Decreasing Cost
machine hours per machine hours per
Case of Yogurt Case of Cheese
labor hours per labor hours per
Case of Yogurt Case of Cheese
Revenue
Machine
Labor
2 2
1 2
Resource Requirements
$50/Case of Yogurt $64/Case of Cheese
Yogurt Cheese
At Loper Dairy specialty yogurt and cheese are produced and sold nationally. As a member of the
production management team you are responsible for determining the proper allocation of the
resources of machinery (capital) and labor to produce a mix of yogurt and cheese that results in the
maximum profit.
Labor is $10/hour for the first 80 hours and $15/hour for the next 20 hours (overtime). There are
three possible contracts for the machine time. Contract #1 has a machine costs of $20/hour. Contract
#2 has a machine cost of $15/hour plus a $200 start fee. Contract #3 has a machine cost of $10/hour
plus a $600 start fee. There are only 120 hours of machine time available for any contract.
Production resource requirements are listed in the table below.
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0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0 20 40 60 80 100 120
$20/hr
$15/hr+200
$10/hr+600
Competing Technologies
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Millworks Plywood
At Millworks Plywood has a contract to produce 800 sheets of Grade A plywood and 600
sheets of Grade B plywood. Two separate production lines are available to produce
plywood.
< The first production line, Alpha Line, can produce 10 sheets of grade A plywood and 10
sheets of grade B plywood in one hour at a machine cost of $5.00 per hour.
< The second production line, Beta Line, can produce 20 sheets of grade A plywood and
10 sheets of grade B plywood in one hour as a machine cost of $7.00 per hour.
Minimize the cost to Millworks Plywood in meeting the contract.
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Hartman Company, Problem 4.2
Modify the information in the problem to reflect the changes below.
Labor-Hours
Department Prod 1 Prod 2 Available Cost/Hour
A 1 0.35 100 12
B 0.3 0.2 36 15
C 0.2 0.5 50 8
Revenue 48.10 26.20
Product(hours/unit)
Do not use the Profit contribution/unit row as presented in
the text problem.
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Antelope Endowment Fund
The Antelope Endowment Fund has five million dollars to invest. The AEF gives $400,000
in scholarships to university students at the beginning of each year. The AEF can invest
the available funds in common stock, treasury bills (T-bills), and local bank certificate
of deposits (CD‟s).
For every dollar invested in common stock a profit of $.10 is expected after one year and T-
bills are expected to earn $.14 on the dollar at the end of the second year. The T-bills
should be held for two years to avoid excess transactions costs. CD‟s must be held for 3
three years for a return of $.25 for every dollar invested.
As Chief Executive Officer you plan to maximize the total value of the fund at the end of
your term. Since your term expires at the end of four years, you plan to sell all assets at
the end of the forth year for the next CEO to invest at their discretion. As a risk averse
organization, the AEF board desires to hold at least 30% of all monies invest in T-Bills
and at least 25% in CD‟s. Further, investments in stocks in the third year must be at least
10% above investments in stocks in the second year. Investments in stocks in the fourth
year should be less than 80% of the monies invested in stock in the third year. Finally,
AEF must hold $200,000 in cash on hand for emergency purposes. Maximize the AEF
assets.
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AEF - Investment Possibilities
Year 1 Year 2 Year 3 Year 4
Stock1
Stock2
Stock3
Stock4
TBill1
TBill2
TBill3
CD1
CD2
Cash1
Cash2
Cash3
Cash4
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Hart Manufacturing, Prob 8.11
Modify the information in the problem to reflect the changes below.
Cost/
Department Prod 1 Prod 2 Prod 3 Available Hour
A 1.50 3.00 2.00 450 2.00
B 2.00 1.00 2.50 350 4.00
C 0.25 0.25 0.25 50 8.00
Revenue 38.00 40.00 46.00
Product (hours/unit)
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Old Farms
Mark is a crop consultant for Old Farms near Loop River. The farm raises three crops; corn, alfalfa, and
soybeans. For simplicity assume that it is possible to raise dry-land crops or irrigated crops in any field. The
Old Farm may sell as much of corn, alfalfa, or soybeans that they can raise (there is no sales limit). However,
when the crop is sold may greatly effect the farm's profit. Futures prices for the spring are higher than the
expected price for harvest time. It is possible to store a limited amount corn, alfalfa, and soybeans for the
spring. The consultant has recommended that you store no more than 50% of your alfalfa yield for spring sale.
Water limitations differ from field to field. There is, however, an overall limitation of 4200 acre-ft of water that
Old Farms can use for the entire growing season.
Expected
Sale Price
Std deviation
Sale Price
Expected
Sale Price
Std deviation
Sale Price
Corn $3.40/bushel $0.50 $3.85/bushel $0.80
Alfalfa $42.00/ton $8.00 $51.00/ton $12.00
Soybeans $9.50/bushel $1.25 $10.25/bushel $2.20
HARVEST SPRING
•What is the best production plan for each section?
•What is the value of one more acre-foot of water?
•What is the marginal value of the storage capacity of each crop?
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Storage Capacity
Since Old Farm stores alfalfa in round bails there is no limitation regarding storage. Storage capacity for corn
and soybeans combined is 150,000 bushels.
Dry-land yields and irrigated yields vary from one parcel of land to another. There is no quality difference
between dry-land and irrigated crops. The following table represents the potential yields by crop in each field.
Field\Crop Corn Alfalfa Soybeans Corn Alfalfa Soybeans
Southeast 110 bu/acre 1.0 tons/acre 35 bu/acre 180 bu/acre 1.6 tons/acre 47 bu/acre
North 110 bu/acre 0.9 tons/acre 38 bu/acre 200 bu/acre 1.5 tons/acre 51 bu/acre
Northwest 90 bu/acre 1.5 tons/acre 39 bu/acre 210 bu/acre 1.5 tons/acre 53 bu/acre
West 105 bu/acre 1.1 tons/acre 30 bu/acre 190 bu/acre 1.4 tons/acre 41 bu/acre
Southwest 95 bu/acre 1.2 tons/acre 27 bu/acre 160 bu/acre 1.5 tons/acre 40 bu/acre
Expected Dry Land Yields Expected Irrigated Yields
Water requirement for each crop also differs by parcel. The following table gives the water requirement for each crop
planted in each field. The NRD (Natural Resource District) limits the amount of water allocated to each field
Water Limit Field Size
Field\Crop Corn Alfalfa Soybeans Acre-ft/field
Southeast 1.5 acre-ft 2.3 acre-ft 0.8 acre-ft 1500 acre-ft 1920 acres
North 1.4 acre-ft 0.0 acre-ft 0.7 acre-ft 1700 acre-ft 2240 acres
Northwest 1.2 acre-ft 2.1 acre-ft 0.8 acre-ft 1300 acre-ft 820 acres
West 1.6 acre-ft 2.6 acre-ft 0.9 acre-ft 800 acre-ft 1280 acres
Southwest 1.6 acre-ft 0.0 acre-ft 0.8 acre-ft 200 acre-ft 640 acres
Water requirements by Crop
Old Farms - continued
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Queens on a Chess Board
The objective is to place as
many queens on a chess
board as possible with one
stipulation. At most one
queen my lay on any row,
any column, and any
diagonal.
The purpose of this exercise is to demonstrate advanced
features in variable indexing.
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71
diagA.r2: Q.r2.c1 + Q.r1.c2 <= 1
diagA.r3: Q.r3.c1 + Q.r2.c2 + Q.r1.c3 <= 1
diagA.r4: Q.r4.c1 + Q.r3.c2 + Q.r2.c3 + Q.r1.c4 <= 1
diagA.r5: Q.r5.c1 + Q.r4.c2 + Q.r3.c3 + Q.r2.c4 + Q.r1.c5 <= 1
diagA.r6: Q.r6.c1 + Q.r5.c2 + Q.r4.c3 + Q.r3.c4 + Q.r2.c5 + Q.r1.c6 <= 1
diagA.r7: Q.r7.c1 + Q.r6.c2 + Q.r5.c3 + Q.r4.c4 + Q.r3.c5 + Q.r2.c6 + Q.r1.c7 <= 1
diagA.r8: Q.r8.c1 + Q.r7.c2 + Q.r6.c3 + Q.r5.c4 + Q.r4.c5 + Q.r3.c6 + Q.r2.c7 + Q.r1.c8 <= 1
diagB.r1: Q.r1.c1 + Q.r2.c2 + Q.r3.c3 + Q.r4.c4 + Q.r5.c5 + Q.r6.c6 + Q.r7.c7 + Q.r8.c8 <= 1
diagB.r2: Q.r2.c1 + Q.r3.c2 + Q.r4.c3 + Q.r5.c4 + Q.r6.c5 + Q.r7.c6 + Q.r8.c7 <= 1
diagB.r3: Q.r3.c1 + Q.r4.c2 + Q.r5.c3 + Q.r6.c4 + Q.r7.c5 + Q.r8.c6 <= 1
diagB.r4: Q.r4.c1 + Q.r5.c2 + Q.r6.c3 + Q.r7.c4 + Q.r8.c5 <= 1
diagB.r5: Q.r5.c1 + Q.r6.c2 + Q.r7.c3 + Q.r8.c4 <= 1
diagB.r6: Q.r6.c1 + Q.r7.c2 + Q.r8.c3 <= 1
diagB.r7: Q.r7.c1 + Q.r8.c2 <= 1
diagC.c2: Q.r8.c2 + Q.r7.c3 + Q.r6.c4 + Q.r5.c5 + Q.r4.c6 + Q.r3.c7 + Q.r2.c8 <= 1
diagC.c3: Q.r8.c3 + Q.r7.c4 + Q.r6.c5 + Q.r5.c6 + Q.r4.c7 + Q.r3.c8 <= 1
diagC.c4: Q.r8.c4 + Q.r7.c5 + Q.r6.c6 + Q.r5.c7 + Q.r4.c8 <= 1
diagC.c5: Q.r8.c5 + Q.r7.c6 + Q.r6.c7 + Q.r5.c8 <= 1
diagC.c6: Q.r8.c6 + Q.r7.c7 + Q.r6.c8 <= 1
diagC.c7: Q.r8.c7 + Q.r7.c8 <= 1
diagD.c2: Q.r1.c2 + Q.r2.c3 + Q.r3.c4 + Q.r4.c5 + Q.r5.c6 + Q.r6.c7 + Q.r7.c8 <= 1
diagD.c3: Q.r1.c3 + Q.r2.c4 + Q.r3.c5 + Q.r4.c6 + Q.r5.c7 + Q.r6.c8 <= 1
diagD.c4: Q.r1.c4 + Q.r2.c5 + Q.r3.c6 + Q.r4.c7 + Q.r5.c8 <= 1
diagD.c5: Q.r1.c5 + Q.r2.c6 + Q.r3.c7 + Q.r4.c8 <= 1
diagD.c6: Q.r1.c6 + Q.r2.c7 + Q.r3.c8 <= 1
diagD.c7: Q.r1.c7 + Q.r2.c8 <= 1
72
Queens - output table
73
Knights on a Chess Board
The objective is to place as
many knights on a chess
board as possible without
any knight jeopardizing
any other knight.
The purpose of this exercise is to demonstrate advanced
features in variable indexing and output table formatting.
74
75
Spreadsheet Formatting
Formatting Cells
Formatting spreadsheet cell font size, font color,
background color, boarders, alignment, and numeric
presentation is performed by double clicking on the right
mouse button. The “Workbook Designer” appears
permitting Excel type format changes. When done editing
press alt-F4 or click on the (x) in the top right corner to
return to the Decision Science screen.
Equations in Cells
DS permits Excel type equations in any of the
coefficient tables. Equations may reference cells in
other tables. In the example at left, an “if” statement
places a “K” in a cell if there is a “1” in the
corresponding cell in the “Knight” table.
76
Multi-commodity TransportationThe purpose of this problem is to demonstrate the use of three-dimensional indexing.
Multi-commodity transportation problems can become extremely large with hundreds
of thousands of constraints and variables. However, with the use of set notation such
problems are manageable.
Suppose that a steel firm ship produces Bands, Plates, and Coils at three different
foundries; Gary, Cleveland, and Pittsburgh. These item are shipped to Framingham,
Detroit, Lansing, Windsor, St. Louis, Fremont, and Lafayette. Supply from each steel
firm and demand for each city are listed below.
77
Multi-commodity Transportation- Costs
Transportation costs (variable costs) of shipping products from the supplier to the
destination are given in the following tables. There is also a limitation that no more that 625
units (all items combined) are permitted to ship along any one route.
78
Multi-commodity Transportation- Model
79
American Settlement Problem4 Your niece asked you to help construct an early American settlement for a science project from her
oversized box of Legos. Her box contains 20 big brown pieces, 30 small brown pieces, 22 white pieces, 15 red pieces, 40 tan pieces, 20 body pieces, 4 blue hats, 4 green hats, 4 black hats, and 20 rods.
4 In the community there are nine types of people: farmers, carpenters, forgers (includes all smiths and metal workers), clergy, teachers, fishermen, grain millers (includes bakers), soldiers (includes hunters), and administrators. Each member of the community offers unique physical, mental, and spiritual insights.
4 In the town you may build eleven types of structures: churches, schools, houses, fences, boat piers, boats, wagons, granaries, market, bakery shops (serves also as a grain mill), and carpentry shops (serves also as a lumber mill). You may also construct four types of animals: horses, mules, sheep, and chickens.
4 Community Requirements
4 If a bakery is built then there must be at least one miller. If a carpentry shop is built then there must be at least one carpenter. If a church is built then there must be at least one clergy. At most, only one church can be built. The number of fishermen must be greater than or equal to the number of boats. The number of boats must be greater than or equal to the number of piers. The number of schools must be less than or equal to the number of teachers. The community only needs one school. The number of farmers must be greater than or equal to the number of wagons. A fence protects the animals from predators and from running away, therefore, the number of fences built must be less than or equal to the number of animals. Each person unit represents one family, which must have housing. One house can hold one or two families.
4 In order for the community to survive it must have 30 organization points, 40 contentment points, and 20 food points.
4 Maximize the number of people in the community.
80
American Settlement Problem
Contentment Organization Food
People
administrators 1 4 -2
carpenters 2 1 -3
clergy 3 2 -2
farmers 0 -1 4
fishermen -2 0 2
forgers 1 0 -3
millers 0 0 1
soldiers -1 1 -2
teachers 2 2 -1
Structures
church 10 2
bakery 1 2
boat 1
carpentry shop 2
fence 1
granary 2
house 1 1
market 3 2
pier 1 1
school 5 1
wagons 2 1
Animals
chickens 2
horses 1
mules 2
sheep 1
Persons in the community produce food, contentment, and organization for the community.
Negative values represent the creation of discontentment and disorganization.
81
American Settlement Problem
Big Small Blue Green Black
Brown Brown White Red Tan Body Hat Hat Hat rods
People
administrators 1 1
carpenters 1 1 1 2
clergy 1 1 1
farmers 1 1 3
fishermen 1 1 1 2
forgers 1 1 1
millers 1 1 1
soldiers 1 1 1
teachers 1 1 1
Structures
church 4 3 2 1 1
bakery 2 2 2
boat 1 1 1 2
carpentry shop 2 1 1
fence 1
granary 2 1 2
house 1 1 1 1
market 3 3 1 1 1 1
pier 2 2
school 3 2 2 2
wagons 1 1 2
Animals
chickens 2 1
horses 1 1 3
mules 2 1 2
sheep 3 1
Construction Requirements - Construction of each community member, animal, and building requires a different
combination of Legos pieces. The table below lists the strengths of each member and the Lego piece requirement.
82
Kearney City CouncilA proposal to develop 80 acres of land was presented to the Kearney City Council.
Negotiations with the developers of Shipwreck Point has lead to the following goals:
4 Priority 1: Build at least 500 family units.
4 Priority 2: Add at least 60 million dollars to the property tax base.
4 Priority 3: The amount services financed by the city must remain under $250,000.
4 Priority 4: Provide space for at least a 5 acre park
4 Priority 5: At least 40% of the family units must live in single family dwellings
Housing Statistics applying to this project are given in the following table
Single Deluxe
Family Condo Apartment
Land Usage per dwelling .25 acre .8 acre .75 acre
Family Units per dwelling 1 4 6
Tax Base per dwelling 200,000$ 640,000$ 600,000$
Required Services per Dwelling 1,500$ 1,200$ 2,500$
83
84
Kearney City Council - Variable List
integer
integer
integer
85
Shortest Route using capacitated transshipment
Major Cities•Kearney, NE
•Albuquerque, NM
•Cheyenne, WY
•Dallas, TX
•Denver, CO
•Des Moines, IA
•Kansas City, MO
•Minneapolis, MN
•Oklahoma City, OK
•Omaha, NE
Find the shortest route
between any two cities
86
Shortest Route Map - model
The miles.i.j<>‟null‟ statement permits exclusion of routes. Traditionally, route prohibitions are created by
listing the miles between the two prohibited routes as a very large value. If mileage in the MILES table
from i to j is not listed then the route is considered prohibited. Any given mileage, even 0, is considered
valid and the route is included. The miles.i.j in constraints StartCity and StopCity is not used directly in
the constraint equation but is used to limit the routes possible.
87
Shortest Route - tablesDetermine the shortest route from Cheyenne, WY to Dallas, TX.
88
Shortest Route - tablesDetermine the shortest route from Cheyenne, WY to Dallas, TX.
The optimal route is stored in the ROUTES table as well as the OUTPUT table.
89
Suncoast Office Supplies
4 See section 15.2 in Anderson, Sweeney, and Williams
90
Suncoast Office Supplies - Variable List
91
Hospital Evaluation - DEA
4 See section 4.5 in Anderson, Sweeney, Williams
The DEA variable identification table
92
Hospital Evaluation - continued
The DEA data input table. This is found on the „Model‟ spreadsheet
93
Hospital Evaluation - output
94
Summation Notation - one dimensional
The use of summation notation greatly simplifies the addition of a range of values.
The following summation shows a one dimension summation of the variable X.
95
Summation Notation - 2 dimensional
Client1 Client2 Client3
Terry X1,1 X1,2 X1,3
Carle X2,1 X2,2 X2,3
McClymondsX3,1 X3,2 X3,3
Variable NamesClient1 Client2 Client3
Terry 10 15 9
Carle 9 18 5
McClymonds 6 14 3
Completion Times
See table 7.3 in text
96
„For Every‟ Notation
Client1 Client2 Client3
Terry X1,1 X1,2 X1,3
Carle X2,1 X2,2 X2,3
McClymondsX3,1 X3,2 X3,3
Variable NamesClient1 Client2 Client3
Terry 10 15 9
Carle 9 18 5
McClymonds 6 14 3
Completion Times
This example continues from the ‘Summation Notation’ . Example from ASW text, figure 7.4
97
F-117
Stealth Fighter
B-2 Bomber
F-22
Raptor
