Decision Science
BMGT 825
Fall 2006, UNK
Professor: Ron Konecny Ph.D.
University of Nebraska at Kearney
1
Table of Contents
Class Syllabus
Homework Assignments
Daily Discussion Topics
GP/LP Basics
Sample Problems
2
Contact Information
Office Hours:
249W West Center
Phone 865-8366
email: konecnyr@unk.edu
Office hours: M-TH 2:45-3:45
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.
3
Required Course Materials
1.
2.
3.
4.
5.
An Introduction to Management Science, 11th ed., 2005, Anderson,
Sweeney, Williams
The Goal, 3rd revised edition, 2004, Goldratt and Cox, North River Press
Decision Science 2005 Computer software (Provided by instructor –free-)
Access to Windows NT/ME/XP platform computer with internet access and
Excel 2002 or newer.
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.
4
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.
5
Course Evaluation
Course Performance Evaluation
300 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://Platteriver.unk.edu/BMGT825
6
Course Evaluation - continued
Course Outline: Test Schedule
• Exam 1: Due September 20 - Chapters 1 - 6, 8, Introduction to
Management Science, linear programming, sensitivity analysis,
integer programming, and applications.
• Exam 2: Due November 8 - Chapters 7, 15 & handouts, transportation
& assignment methods, multi-objective programming, integer multiobjective programming, set notation.
• Final: December 13 - Chapters 4, 10, 13 data envelopment analysis,
program evaluation and review technique, critical path method,
simulation.
7
Homework Assignment, due 8/30
Graphical & Algebraic solutions
 Chapter 2, Problem 21
 Chapter 2, Problem 34
 Chapter 2, Problem 50
-
Include the these three hand solved problems in the notebook.
Reading Assignment
•Chapter 1, pg. 1-17
•Chapter 2, all sections
•The Goal, chapters 1-20
8
Homework Assignment, due 9/6
Linear Programming
 Chapter 2, Problem 21 – use software
Reading Assignment
 Chapter 2, Problem 34 – use software
 Chapter 2, Problem 50 – use software
 Chapter 4, Problem 1
 Chapter 4, Problem 2 (modified a and c)
•Chapter 3
•Chapter 4, sections 1-3
•The Goal – Chapters 21-40
 Chapter 4, Problem 3
 Chapter 4, Problem 8
All homework problems Include the following printouts
•model sheet
•output sheet
•variable sheet
9
Homework Assignment, due 9/13
Linear Programming
 Chapter 4, Problem 14
 Chapter 4, Problem 16
 Chapter 4, Problem 17
 Margaret Black Farm - on Platteriver website
Multi-period Problems
 Antelope Endowment Fund (AEF)
10
Homework Assignment, due 9/20
Exam #1 is due. Electronic submission due by 5:59pm on the
http://platteriver.unk.edu/bmgt825 web server.
Academic Dishonesty.
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.
11
Homework Assignment, due 9/27
Transportation & Assignment Problems
– without set notation



Chapter 7, Problem 2 (transportation)
Chapter 7, Problem 20 (assignment)
Chapter 7, Problem 26 (transshipment)
Reading Assignment
•Chapter 7, sections 1-4
12
Homework Assignment, due 10/4
- with set notation





Chapter 7, Problem 2 (transportation)
Chapter 7, Problem 20 (assignment)
Chapter 7, Problem 26 (transshipment)
Chapter 7, Problem 29 (transshipment)
Chapter 4, Problem 8 (cyclical set)
Reading Assignment
•Chapter 7, sections 1-4
13
Homework Assignment, due 10/11
Set Notation Problems



Loper Dairy
Bank Location Problem Section 8.3
–
The bank location problem is on page 396. You do not need to solve as an integer problem. You must use set
notation. Hint: Variable X is only one dimensional.
Production Scheduling, Section 4.3
Production Scheduling Hints (pages 173 – 179)
1.You will need to create a variable name set for March, April, May, and June. March has the previous inventory.
Remember, the For Every and Sum statements may contain a condition such as
•For Every t, t>1 :
•The above statement type of statement will permit you to look back to the previous month such as
•For Every t, t>1 : INVENTORY.Comp802B.(t-1) + …
2. You will need to set the ending inventory for March for both the Component332A and the Component802B individually. For example:
•Inventory.Comp802B.March = 500
•Inventory.Comp322A.March = 200
3. You can construct the Machine and Labor capacity constraints on page 176 with one “For Every” statement.
Treat the storage capacity separately.
4. You will need variables for production, ending inventory, and two variables for inventory change.
•One variable (the book uses I1 ) for month to month increases
•One variable (the book used D1) for month to month decreases (see page 177).
5. The objective function (see page 175) has four summation statements.
6. The answer on page 178 shows two-dimensional variables and one-dimensional variables.
14
Homework Assignment, due 10/18
All problems must use set notation
 Production and Inventory Application, Section 7.4 (page 374)
 Blending Problem, Section 4.4 (page 183)
 Multi-Commodity Transportation
Mixed Integer Linear Programs
 Chapter 8, problem 13 , a and b
 American Settlement Problem
Reading Assignment
•Chapter 8
15
Homework Assignment, due 10/20
Set Notation Problems
 Queens on a chess board (bonus) or
 Sudoku (bonus)
16
Homework Assignment, due 10/25
Multi-criteria Decision Problems Reading Assignment
(or Goal Programming)
•Chapter 15, sec 1, 2
 Chapter 15, Problem 2





Chapter 15, Problem 3
Chapter 15, Problem 4
Chapter 15, Problem 5
Chapter 15, Problem 7
Kearney City Council
•Chapter 4, sec 5
•“Data Envelopment Analysis: Partial
Survey and Applications for Management
Accounting,” Callen, Jeffrey L, Journal of
Management Accounting Research, Vol 3,
Fall 1991
17
Homework Assignment, due 11/8
Exam #2 is due. Class will meet at 5:59 on 11/9/2005
18
Homework Assignment, due 11/29
Project Scheduling – Microsoft Project
- Critical Path Method
 Chapter 10, Problem 6
 Chapter 10, Problem 7
 Chapter 10, Problem 14 (PERT)
 Chapter 10, Problem 20
 Chapter 10, Problem 22
Reading Assignment
• Chapter 10
19
Homework Assignment, due 12/6
PERT, by simulation
Reading Assignment
 Chapter 10, Problem 14
– Use Excel, with Normal Distribution
– Use Excel, with Beta Distribution
•Chapter 10, review
•Handout material
Notebooks are Due at the end of class.
In class final is next week.
20
Discussion Topics
(8/23, Week 1)
August 23
– Introduction to Management Science
– Graphic solution
• Loper Dairy
• Millworks Plywood
– Algebraic Solution
– RHS and objective coefficient ranging
21
Discussion Topics
(8/30 Week 2)
August 30
– 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
22
Discussion Topics
(9/6 Week 3)
September 6
– Discuss The Goal and homework problems
– Identity equations
• Efficiency
• Profit, revenue, and cost functions
– Multi-period investment problems
– Natural resource problems
23
Discussion Topics
(9/13 Week 4)
September 13
– Discuss The Goal and homework problems
Distribute Exam #1
• covers material in first 4 weeks of class
24
Discussion Topics
(9/20 Week 5)
September 20
– Transportation and assignment examples
Discuss Exam #1
25
Discussion Topics
(9/27 Week 6)
September 27
– Discuss transportation homework
– Summation notation exercise
– Introduction to set notation
•
•
•
•
name sets
variable sets
coefficient sets
ordered and cyclical sets
– summation notation
– ‘for every’ notation
– Software utilization of set notation
26
Discussion Topics
(10/4 Week 7)
October 4
– Discuss homework
– Coefficient tables
– Applications
• Bank location example
• Old Farms
27
Discussion Topics
(10/11 Week 8)
October 11
–
–
–
–
Discuss homework
Generalized LP forms using set notation
Pattern usage with set notation
Advanced set referencing
• Queens on a chess board
• Knights on a chess board
• subsets
– Mixed Integer Linear Programming
• Discrete variables
• Fixed costs
• Multiple choice and Mutually exclusive constraints
28
Discussion Topics
(10/18 Week 9)
October 18
– Homework problems
– Introduction Goal Programming (multiple-objective)
• Kearney City Council
– Integer Goal Programming
29
Discussion Topics
(10/25 Week 10)
October 25
– Homework problems
– Integer & multiple-objective programming using sets
– Introduction to Data Envelopment Analysis
• Callen reading
• facets, production frontier, relative efficiency, scale efficiency,
most productive scale size
30
Discussion Topics
(11/1 Week 11)
November 2
– Discuss homework
– Introduction to Networking models
• Project Network Graph (digraph),
• Critical Path Method – CPM
– Distribute Exam 2
31
Discussion Topics
(11/8 Week 12)
November 8
– Networking models Continued
•
•
•
•
Critical Path Method (CPM)
CPM with crashing
Activity schedule
LP/GP solutions
32
Discussion Topics
(11/15 Week 13)
November 15
– Other Networking models
• Program Evaluation and Review Technique(PERT)
• simulation
33
Discussion Topics
(11/29 Week 14)
November 29
– Evaluation of PERT & CPM
34
Discussion Topics
(12/6 Week 15)
December 6
– Homework notebooks due
– Advances in Management Science
• Genetic Algorithms
• Expert Systems
35
Discussion Topics
(12/14 Week 16)
December 14
– Final Exam, in class
36
An Introduction to Linear Programming
Linear Programming Problem
Problem Formulation
A Maximization Problem
Graphical Solution Procedure
37
Requirements of a Linear Programming
Problem
 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.
 The quantity of the objective is limited by a system of
constrains.
– Land, labor, capital, prices, contracts, limited resources
 There must exist multiple alternatives. However, some of
these alternatives may give rather poor results.
 The objective function and constraints in an LP must be
expressed in terms of linear equations or inequalities
38
Problem Formulation
Problem formulation or modeling is the
process of translating a verbal statement of
a problem into a mathematical statement.
39
Guidelines for Model Formulation
 Understand the problem thoroughly.
 Write a verbal description of the objective.
 Write a verbal description of each constraint.
 Define the decision variables.
 Write the objective in terms of the decision
variables.
 Write the constraints in terms of the decision
variables.
40
Linear Programming using
the Decision Science software
Creating,
Saving, and
Printing a file
is performed
from the menu
bar.
Equation label
double click on
the box to create
or change a label.
Equation Editing
You may edit an equation by
placing the cursor on a cell and
pressing the F2 key. If you do
not want to keep the change
just press the ESC key before
leaving the cell.
Duplication
By dragging the square dot on the
right bottom a cell you my
duplicate a line numerous times.
41
Goal Programming using
the Decision Science software
Adding Priorities
You may add an extra priority
by placing the cursor in the
priority list and pressing Edit Insert Row or ALT-E-I.
You may also remove a priority
by pressing Edit Remove Row
Documentation
By starting a line with a double
quotation a line is considered
strictly as documentation.
Documentation lines appear in
green.
To create a new Goal Program/Multi-criteria Decision Model select File-NewGoal program from the menu bar.
42
Sample Problems












Loper Dairy (linear program)
Millworks Plywood (linear program)
Antelope Development Fund (set notation)
Old Farms (set notation)
Queens on a chess board (set notation)
Knights on a chess board (set notation)
Kearney City Council (goal program)
Shortest Route - map (transshipment, set)
Multi-commodity Shipping (3D set)
American Settlement Problem (set)
Suncoast Office Supplies (goal program)
Hospital Evaluation (Data Envelopment Analysis)
43
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.
44
Loper Dairy
Resource Requirements
Yogurt
Cheese
Machine
Labor
Revenue
Profit
Cost
Availability
machine hours per
machine hours per
$16/machine hour 120 machine hours
2
Case of Yogurt
Case of Cheese
labor hours per
labor hours per
$10/labor hour
80 labor hours
1
2
Case of Yogurt
Case of Cheese
2
$50/Case of Yogurt
$64/Case of Cheese
$8/Case of Yogurt
$12/Case of Cheese
Yogurt Revenue
$50
Cheese Revenue
$64
Machine Cost
-$32
Machine Cost
-$32
Labor Cost
Yogurt Profit
-$10
$8
Labor Cost
Cheese Profit
-$20
$12
45
Loper Dairy – LP model
•Identify the decision variables
•What can change in the model?
•Labor hours? Machine hours? Cases of yogurt
produced? Cases of cheese produced?
• Y - Cases of yogurt produced
• C - Cases of cheese produced.
•Write the objective function
•What do we want to do?
Max Profit = 8 Y + 12 C
Max Profit($)= 8
$
case of yogurt
Ycases of yogurt + 12 cases of$ cheese C cases of cheese
46
Loper Dairy – LP model
•Write the constraints
2Y+2C
1Y+2C
2
1
120
80
Machine hours
case of yogurt
Ycases of yogurt + 2
Labor hours
case of yogurt
Ycases of yogurt + 2
(machine hours)
(labor hours)
Machine hours
cases of cheese
Labor hours
cases of cheese
C cases of cheese
120
C cases of cheese 80
machine
hours
labor
hours
47
Yogurt
Loper Dairy – Graphical Solution
80
2 Y + 2 C ≤ 120
(machine hours)
60
40
20
0
0
20
40
60
80
Cheese
48
Yogurt
Loper Dairy – Graphical Solution
80
1 Y + 2 C ≤ 80
60
(labor hours)
40
20
0
0
20
40
60
80
Cheese
49
Yogurt
Loper Dairy – Graphical Solution
80
2 Y + 2 C ≤ 120
1 Y + 2 C ≤ 80
(machine hours)
(labor hours)
60
40
20
0
0
20
40
60
80
Cheese
50
Yogurt
Loper Dairy – Graphical Solution
80
2 Y + 2 C ≤ 120
1 Y + 2 C ≤ 80
(machine hours)
(labor hours)
60
Infeasible
40
20
0
0
Feasible
Region
20
40
60
80
Cheese
51
Yogurt
Loper Dairy – Corner Point Principle
80
Max Profit= 8 Y + 12 C
Subject to:
60
(0, 60)
(20, 40)
40
20
Feasible
Region
(0, 0)
0
20
0
2 Y + 2 C ≤ 120 (machine hours)
1 Y + 2 C ≤ 80 (labor hours)
( C, Y ) Profit
$0
$480
$480
$560
40 (40, 0) 60
80
Cheese
52
Yogurt
Loper Dairy – Sensitivity Analysis
80
Max Profit = 8 Y+ 12 C
Subject to
2.0 Y+ 2.0 C 120 (machine hours)
60
1.0
Y+ 2.0
C
(labor hours)
Profit= 656
40
20
0
104
0
Y=16.0
C=44.0
Feasible
Region
20
40
60
80
Cheese
53
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 54
error.
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.
55
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
- Machine Cost ($32)
- Labor Cost
($10)
Profit
$8
per case
2 hours at $16/hour
1 hour at $10/hour
per case
56
Loper Dairy - Decreasing Cost
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.
Resource Requirements
Yogurt
Cheese
Machine
Labor
Revenue
machine hours per
machine hours per
2
Case of Yogurt
Case of Cheese
labor hours per
labor hours per
1
2
Case of Yogurt
Case of Cheese
2
$50/Case of Yogurt
$64/Case of Cheese
57
Competing Technologies
2400
2200
2000
1800
1600
1400
$20/hr
1200
$15/hr+200
1000
$10/hr+600
800
600
400
200
0
0
20
40
60
80
100
120
58
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.
59
Hartman Company, Problem 4.2
Modify the information in the problem to reflect the changes below.
Product(hours/unit) 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
Do not use the Profit contribution/unit row as presented in
the text problem.
60
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 Tbills 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.
61
AEF - Investment Possibilities
Year 1
Year 2
Year 3
Year 4
Stock1
Stock2
Stock3
Stock4
TBill1
TBill2
TBill3
CD1
CD2
Cash1
Cash2
Cash3
Cash4
62
Hart Manufacturing, Prob 8.11
Modify the information in the problem to reflect the changes below.
Product (hours/unit)
Department Prod 1 Prod 2 Prod 3 Available
A
1.50
3.00
2.00
450
B
2.00
1.00
2.50
350
C
0.25
0.25
0.25
50
Revenue
38.00 40.00 46.00
Cost/
Hour
2.00
4.00
8.00
63
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.
•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?
Corn
Alfalfa
Soybeans
Sale Price
Sale Price
(harvest)
(spring)
$2.00/bushel $2.50/bushel
$42.00/ton
$51.00/ton
$4.50/bushel $5.25/bushel
64
Old Farms - continued
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.
Expected Dry Land Yields
Field\Crop
Corn
Alfalfa Soybeans
Southeast 110 bu/acre 1.0 tons/acre 35 bu/acre
North
110 bu/acre 0.9 tons/acre 38 bu/acre
Northwest
90 bu/acre 1.5 tons/acre 39 bu/acre
West
105 bu/acre 1.1 tons/acre 30 bu/acre
Southwest 95 bu/acre 1.2 tons/acre 27 bu/acre
Expected Irrigated Yields
Corn
Alfalfa Soybeans
180 bu/acre 1.6 tons/acre 47 bu/acre
200 bu/acre 1.5 tons/acre 51 bu/acre
210 bu/acre 1.5 tons/acre 53 bu/acre
190 bu/acre 1.4 tons/acre 41 bu/acre
160 bu/acre 1.5 tons/acre 40 bu/acre
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 requirements by Crop
Field\Crop Corn
Alfalfa
Soybeans
Southeast 1.5 acre-ft
2.3 acre-ft
0.8 acre-ft
North
1.4 acre-ft
0.0 acre-ft
0.7 acre-ft
Northwest 1.2 acre-ft
2.1 acre-ft
0.8 acre-ft
West
1.6 acre-ft
2.6 acre-ft
0.9 acre-ft
Southwest 1.6 acre-ft
0.0 acre-ft
0.8 acre-ft
Water Limit
Acre-ft/field
1500 acre-ft
1700 acre-ft
1300 acre-ft
800 acre-ft
200 acre-ft
Field Size
1920 acres
2240 acres
820 acres
1280 acres
640 acres
65
Old Farms Linear Program (set)
66
Old Farms - Coefficient Tables
67
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.
68
69
diagA.r2:
diagA.r3:
diagA.r4:
diagA.r5:
diagA.r6:
diagA.r7:
diagA.r8:
Q.r2.c1 +
Q.r3.c1 +
Q.r4.c1 +
Q.r5.c1 +
Q.r6.c1 +
Q.r7.c1 +
Q.r8.c1 +
Q.r1.c2 <= 1
Q.r2.c2 + Q.r1.c3 <= 1
Q.r3.c2 + Q.r2.c3 + Q.r1.c4 <= 1
Q.r4.c2 + Q.r3.c3 + Q.r2.c4 + Q.r1.c5 <= 1
Q.r5.c2 + Q.r4.c3 + Q.r3.c4 + Q.r2.c5 + Q.r1.c6 <= 1
Q.r6.c2 + Q.r5.c3 + Q.r4.c4 + Q.r3.c5 + Q.r2.c6 + Q.r1.c7 <= 1
Q.r7.c2 + Q.r6.c3 + Q.r5.c4 + Q.r4.c5 + Q.r3.c6 + Q.r2.c7 + Q.r1.c8 <= 1
diagB.r1:
diagB.r2:
diagB.r3:
diagB.r4:
diagB.r5:
diagB.r6:
diagB.r7:
Q.r1.c1 +
Q.r2.c1 +
Q.r3.c1 +
Q.r4.c1 +
Q.r5.c1 +
Q.r6.c1 +
Q.r7.c1 +
Q.r2.c2 + Q.r3.c3 + Q.r4.c4 + Q.r5.c5 + Q.r6.c6 + Q.r7.c7 + Q.r8.c8 <= 1
Q.r3.c2 + Q.r4.c3 + Q.r5.c4 + Q.r6.c5 + Q.r7.c6 + Q.r8.c7 <= 1
Q.r4.c2 + Q.r5.c3 + Q.r6.c4 + Q.r7.c5 + Q.r8.c6 <= 1
Q.r5.c2 + Q.r6.c3 + Q.r7.c4 + Q.r8.c5 <= 1
Q.r6.c2 + Q.r7.c3 + Q.r8.c4 <= 1
Q.r7.c2 + Q.r8.c3 <= 1
Q.r8.c2 <= 1
diagC.c2:
diagC.c3:
diagC.c4:
diagC.c5:
diagC.c6:
diagC.c7:
Q.r8.c2 +
Q.r8.c3 +
Q.r8.c4 +
Q.r8.c5 +
Q.r8.c6 +
Q.r8.c7 +
Q.r7.c3 + Q.r6.c4 + Q.r5.c5 + Q.r4.c6 + Q.r3.c7 + Q.r2.c8 <= 1
Q.r7.c4 + Q.r6.c5 + Q.r5.c6 + Q.r4.c7 + Q.r3.c8 <= 1
Q.r7.c5 + Q.r6.c6 + Q.r5.c7 + Q.r4.c8 <= 1
Q.r7.c6 + Q.r6.c7 + Q.r5.c8 <= 1
Q.r7.c7 + Q.r6.c8 <= 1
Q.r7.c8 <= 1
diagD.c2:
diagD.c3:
diagD.c4:
diagD.c5:
diagD.c6:
diagD.c7:
Q.r1.c2 +
Q.r1.c3 +
Q.r1.c4 +
Q.r1.c5 +
Q.r1.c6 +
Q.r1.c7 +
Q.r2.c3 + Q.r3.c4 + Q.r4.c5 + Q.r5.c6 + Q.r6.c7 + Q.r7.c8 <= 1
Q.r2.c4 + Q.r3.c5 + Q.r4.c6 + Q.r5.c7 + Q.r6.c8 <= 1
Q.r2.c5 + Q.r3.c6 + Q.r4.c7 + Q.r5.c8 <= 1
Q.r2.c6 + Q.r3.c7 + Q.r4.c8 <= 1
Q.r2.c7 + Q.r3.c8 <= 1
Q.r2.c8 <= 1
70
Queens - output table
71
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.
72
73
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.
74
Multi-commodity Transportation
The 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.
75
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.
76
Multi-commodity Transportation- Model
77
American Settlement Problem







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.
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.
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.
Community Requirements
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.
In order for the community to survive it must have 30 organization points, 40 contentment points,
and 20 food points.
Maximize the number of people in the community.
78
American Settlement Problem
Persons in the community produce food, contentment, and organization for the community.
Negative values represent the creation of discontentment and disorganization.
Contentment
People
administrators
carpenters
clergy
farmers
fishermen
forgers
millers
soldiers
teachers
Structures
church
bakery
boat
carpentry shop
fence
granary
house
market
pier
school
wagons
Animals
chickens
horses
mules
sheep
Organization
1
2
3
0
-2
1
0
-1
2
4
1
2
-1
0
0
0
1
2
10
1
2
Food
-2
-3
-2
4
2
-3
1
-2
-1
2
1
2
1
2
1
3
5
1
1
1
2
2
1
1
2
1
2
1
79
American Settlement Problem
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.
Big
Brown
People
administrators
carpenters
clergy
farmers
fishermen
forgers
millers
soldiers
teachers
Structures
church
bakery
boat
carpentry shop
fence
granary
house
market
pier
school
wagons
Animals
chickens
horses
mules
sheep
Small
Brown
White
Red
Tan
1
1
1
1
1
1
4
1
2
2
1
3
2
3
3
2
1
2
1
2
2
1
1
2
1
1
1
1
1
1
1
1
1
Blue
Hat
Green
Hat
Black
Hat
rods
1
1
2
1
1
3
2
1
1
1
1
1
1
1
2
1
2
1
1
1
1
1
3
Body
2
1
1
2
1
1
1
2
2
1
2
1
2
1
1
3
1
3
2
1
80
Kearney City Council
A 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:
 Priority 1: Build at least 500 family units.
 Priority 2: Add at least 60 million dollars to the property tax base.
 Priority 3: The amount services financed by the city must remain under $250,000.
 Priority 4: Provide space for at least a 5 acre park
 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
81
82
Kearney City Council - Variable List
integer
integer
integer
83
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
84
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.
85
Shortest Route - tables
Determine the shortest route from Cheyenne, WY to Dallas, TX.
86
Shortest Route - tables
Determine the shortest route from Cheyenne, WY to Dallas, TX.
The optimal route is stored in the ROUTES table as well as the OUTPUT table.
87
Suncoast Office Supplies
 See section 15.2 in Anderson, Sweeney, and Williams
88
Suncoast Office Supplies - Variable List
89
Hospital Evaluation - DEA
 See section 4.5 in Anderson, Sweeney, Williams
The DEA variable identification table
90
Hospital Evaluation - continued
The DEA data input table. This is found on the ‘Model’ spreadsheet
91
Hospital Evaluation - output
92
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.
93
Summation Notation - 2 dimensional
See table 7.3 in text
Completion Times
Variable Names
Client1
X1,1
Carle
X2,1
McClymonds
X3,1
Terry
Client2
Client3
X1,2
X2,2
X3,2
X1,3
X2,3
X3,3
Client1
Terry
Carle
McClymonds
10
9
6
Client2
15
18
14
Client3
9
5
3
94
‘For Every’ Notation
This example continues from the ‘Summation Notation’ . Example from ASW text, figure 7.4
Completion Times
Variable Names
Client1
X1,1
Carle
X2,1
McClymonds
X3,1
Terry
Client2
Client3
X1,2
X2,2
X3,2
X1,3
X2,3
X3,3
Client1
Terry
Carle
McClymonds
10
9
6
Client2
15
18
14
Client3
9
5
3
95
F-117
Stealth Fighter
B-2 Bomber
F-22
Raptor
96
Survey
Name of your favorite cartoon character: _______________
1.
How many DVDs have you rented this month?
2.
How many parking tickets from UNK have you received this year?
3.
Chocolate chip recipe:

_______________________

_______________________

_______________________

_______________________
97