Deterministic Models in OR
K327:02
Joseph Khamalah, Ph.D.
Agenda: Week of August 26, 2010
 Introduction
– Syllabus
– Student Info Sheet
– Exams
– Cases/Assignments
– Peer Evaluation
 Group Formation
2
Grade Weights
Within Area
Total
a) INDIVIDUAL WORK (55-75%)…………...……60%
Exams (2@ 20 – 35%) ………40%
Assignments (10 – 20%) …….20%
b) GROUP WORK (25-45%)……………….…..40%
Exams (2@ 10 – 20%)…..…...20%
Cases (4@ 01 – 03) …………..12%
Project (01 – 10%) …....……...08%
Grand Total…………...………..………... 100%
3
Deterministic Models in Operations Research
Introduction – Chapter 1
Learning Objectives







Develop a general understanding of the management science /
operations research approach to decision making.
Realize that quantitative applications begin with a problem situation.
Obtain a brief introduction to quantitative techniques and their
frequency of use in practice.
Understand that managerial problem situations have both quantitative
and qualitative considerations that are important in the decision
making process.
Learn about models in terms of what they are and why they are
useful (the emphasis is on mathematical models).
Identify the step-by-step procedure that is used in most quantitative
approaches to decision making.
Learn about basic models of cost, revenue, and profit and be able to
compute the breakeven point.
5
Introduction to Operations Research
 Operations Research/Management Science
– Winston:
“a scientific approach to decision making,
which seeks to determine how best to design and operate
a system, usually under conditions requiring the allocation
of scarce resources.”
– Kimball & Morse:
“a scientific method of providing
executive departments with a quantitative basis for
decisions regarding the operations under their control.”
6
Introduction to OR cont’
 Provides rational basis for decision making
– Solves the type of complex problems that turn up in
–
–
the modern business environment
Builds mathematical and computer models of
organizational systems composed of people,
machines, and procedures
Uses analytical and numerical techniques to make
predictions and decisions based on these models
7
Introduction to OR cont’
 Draws upon
– engineering, management, mathematics
 Closely related to the "decision sciences"
– applied mathematics, computer science,
economics, industrial engineering and systems
engineering
8
Body of Knowledge


The body of knowledge involving quantitative approaches
to decision making is referred to as
– Management Science
– Operations Research
– Decision Science
– Operations Management
– Industrial Engineering
– Industrial Management
– Systems Management
– Systems Science
It has its early roots in World War II and is flourishing in
business and industry with the aid of computers.
9
What is OR? Cont’
 Origins in the 1940's with the management of the Cold War


armed forces. Currently used in a very broad range of
organizations and systems.
The linear programming model for solving OR problems was
developed in the 1940's. The first computer implementation
to solve linear programs, known as the simplex method was
developed in 1947 by George Dantzig.
The Nobel Prize in Economics was awarded in 1975 to L.V.
Kantorovich and T.C. Koopmans for their work in linear
programming. Linear programming was also used in
portfolio management by H.M. Marcowitz, who won the
Nobel Prize in 1990.
10
Features of OR
 Emphasis on:

– large, complex operations
– mathematical models
– computer implementation
Extensive use in:
– manufacturing
– transportation
– entertainment
– construction/development
– communication
– computer/database systems
– economics/investing
– armed forces
– biology/genomics
11
Examples of OR Problems
 resource allocation
 scheduling
 routing
 inventory management
 system design and construction
 production
 pricing
 forecasting
 capital budgeting
 customer service
12
The role of the OR Analyst
1.
2.
3.
4.
5.
6.
7.
Formulate the Problem: Determine the nature and constraints of the
problem, and the goal(s) of the client for the problem. Determine a
model class appropriate for the problem.
Observe the system: Collect the facts and data necessary for
precisely specifying and solving the model.
Formulate a mathematical model needed for the problem solution
and identify the software tools used to solve this mathematical
model.
Verify the Model: Check the results of the model on known
situations to see if it gives reasonable and predictable answers in
these cases.
Determine a solution to the new case according to the model
developed above.
Present the solution in a form which the client can use.
Assist in the implementation of the solution
13
Problem Solving and Decision Making

7 Steps of Problem Solving
(First 5 steps are the process of decision making)
– Define the problem.
– Identify the set of alternative solutions.
– Determine the criteria for evaluating alternatives.
– Evaluate the alternatives.
– Choose an alternative (make a decision).
--------------------------------------------------------------------– Implement the chosen alternative.
– Evaluate the results.
14
Quantitative Analysis and Decision
Making

Potential Reasons for a Quantitative Analysis
Approach to Decision Making
– The problem is complex.
– The problem is very important.
– The problem is new.
– The problem is repetitive.
15
Quantitative Analysis

Quantitative Analysis Process
– Model Development
– Data Preparation
– Model Solution
– Report Generation
16
Model Development


Models are representations of real objects or situations
Three forms of models are:
– Iconic models - physical replicas (scalar representations) of
–
–
real objects;
Analog models - physical in form, but do not physically
resemble the object being modeled; and
Mathematical models - represent real world problems
through a system of mathematical formulae and expressions
based on key assumptions, estimates, or statistical analyses.
17
Advantages of Models

Generally, experimenting with models (compared to
experimenting with the real situation):
– requires less time
– is less expensive
– involves less risk
18
Mathematical Models


Cost/benefit considerations must be made in
selecting an appropriate mathematical model.
Frequently a less complicated (and perhaps less
precise) model is more appropriate than a more
complex and accurate one due to cost and ease of
solution considerations.
For every complex problem,
there is a simple solution,
which is neat and
wrong!!
19
Mathematical Models, cont’





Relate decision variables (controllable inputs) with fixed or
variable parameters (uncontrollable inputs)
Frequently seek to maximize or minimize some objective
function subject to constraints
Are said to be stochastic if any of the uncontrollable inputs is
subject to variation, otherwise are deterministic
Generally, stochastic models are more difficult to analyze.
The values of the decision variables that provide the
mathematically-best output are referred to as the optimal
solution for the model.
20
Transforming Model Inputs into Output
Uncontrollable Inputs
(Environmental Factors)
Controllable
Inputs
(Decision
Variables)
Mathematical
Model
Output
(Projected
Results)
21
Example: Project Scheduling
Consider the construction of the Phase III student housing
complex. The project consists of hundreds of activities
involving excavating, framing,
wiring, plastering, painting, landscaping, and more. Some of the
activities must be done sequentially
and others can be done at the same
time. Also, some of the activities
can be completed faster than normal
by purchasing additional resources (workers, equipment, etc.).
22
Example: Project Scheduling, cont’
 Question:
What is the best schedule for the activities and
for which activities should additional resources
be purchased? How could management science
be used to solve this problem?
*****
Management science can provide a structured, quantitative
approach for determining the minimum project completion time
based on the activities' normal times and then based on the
activities' expedited (reduced) times.
23
Example: Project Scheduling, cont’

Question:
What would be the uncontrollable inputs?
*****
–Normal and expedited activity completion times
–Activity expediting costs
–Funds available for expediting
–Precedence relationships of the activities
24
Example: Project Scheduling, cont’

Question:
What would be the:
1. objective function of the mathematical model?
2. the decision variables?
3. the constraints?
 *****
1. Objective function: minimize project completion time
2. Decision variables: which activities to expedite and by
how much, and when to start each activity
3. Constraints: do not violate any activity precedence
relationships and do not expedite in excess of the funds
available.
25
Example: Project Scheduling, cont’

Question:
Is the model deterministic or stochastic?
*****
Stochastic. Activity completion times, both normal and
expedited, are uncertain and subject to variation. Activity
expediting costs are uncertain. The number of activities
and their precedence relationships might change before
the project is completed due to a project design change.
26
Example: Project Scheduling, cont’

Question:
Suggest assumptions that could be made to
simplify the model.
*****
Make the model deterministic by assuming normal and
expedited activity times are known with certainty and
are constant. The same assumption might be made
about the other stochastic, uncontrollable inputs.
27
Data Preparation




Data preparation is not a trivial step, due to the time
required and the possibility of data collection errors.
A model with 50 decision variables and 25
constraints could have over 1300 data elements!
Often, a fairly large data base is needed.
Information systems specialists might be needed.
28
Model Solution




The analyst attempts to identify the alternative (the
set of decision variable values) that provides the
“best” output for the model.
The “best” output is the optimal solution.
If the alternative does not satisfy all of the model
constraints, it is rejected as being infeasible,
regardless of the objective function value.
If the alternative satisfies all of the model
constraints, it is feasible and a candidate for the
“best” solution.
29
Model Solution, cont’

One solution approach is trial-and-error.
– Might not provide the best solution
– Inefficient (numerous calculations required)

Special solution procedures have been developed
for specific mathematical models.
– Some small models/problems can be solved by hand
–
calculations
Most practical applications require using a computer
30
Computer Software



OR would not have developed as a field without the computer.
Currently there are many software packages each of which
handles a particular class of models. Examples of decision models
for which there is extensive software available:
–
–
–
–
–
–
linear/integer programming models
network & routing models
decision tree models
simulation models
queuing models
investment models
A variety of these s/w packages are available on IPFW’s network
–
–
–
–
–
Spreadsheet packages such as Microsoft Excel
The Management Scientist, developed by the textbook authors
DS - Decision Science
POM
HOM
31
Quantitative Methods in Practice






Linear Programming
Integer Linear
Programming
PERT/CPM
Inventory models
Waiting Line Models
Simulation





Decision Analysis
Goal Programming
Analytic Hierarchy
Process
Forecasting
Markov-Process Models
32
Model Testing and Validation
 Often, goodness/accuracy of a model cannot be assessed



until solutions are generated.
Small test problems having known, or at least expected,
solutions can be used for model testing and validation.
If the model generates expected solutions, use the model
on the full-scale problem.
If inaccuracies or potential shortcomings inherent in the
model are identified, take corrective action such as:
– Collection of more-accurate input data
– Modification of the model
33
Report Generation



A managerial report, based on the results of the
model, should be prepared.
The report should be easily understood by the
decision maker.
The report should include:
– the recommended decision
– other pertinent information about the results (for example,
how sensitive the model solution is to the assumptions
and data used in the model)
34
Implementation and Follow-Up




Successful implementation of model results is of
critical importance.
Secure as much user involvement as possible
throughout the modeling process.
Continue to monitor the contribution of the model.
It might be necessary to refine or expand the model.
35
Example: Austin Auto Auction



An auctioneer has developed a simple mathematical
model for deciding the starting bid s/he will require
when auctioning a used automobile.
Essentially, s/he sets the starting bid at seventy (70)
percent of what s/he predicts the final winning bid
will (or should) be. S/he predicts the winning bid by
starting with the car's original selling price and
making two deductions, one based on the car's age
and the other based on the car's mileage.
The age deduction is $800 per year and the mileage
deduction is $0.025 per mile.
36
Example: Austin Auto Auction, cont’

Question:
Develop the mathematical model that will give the starting
bid (B) for a car in terms of the car's original price (P),
current age (A) and mileage (M).
*****
The expected winning bid can be expressed as:
P - 800(A) - 0.025(M)
The entire model is:
B = 0.7(expected winning bid)
B = 0.7(P - 800(A) - 0.025(M))
B = 0.7(P) - 560(A) - 0.0175(M) or
B = 0.7P - 560A - 0.0175M
37
Example: Austin Auto Auction, cont’

Question:
Suppose a four-year old car with 60,000 miles on
the odometer is up for auction. If its original price
was $12,500, what starting bid should the auctioneer
require?
*****
B = 0.7(12,500) - 560(4) - 0.0175(60,000)
= 8,750.00 - 2,240.00 – 1,050.00
= $5,460.00
38
Example: Austin Auto Auction, cont’

Question:
The model is based on what assumptions?
*****
The model assumes that the only factors influencing
the value of a used car are the original price, age, and
mileage (not condition, rarity, or other factors).
Also, it is assumed that age and mileage devalue a car
in a linear manner and without limit. (Note, the starting
bid for a very old car might be negative!)
39
Example: Iron Works, Inc.
Iron Works, Inc. manufactures two
products made from steel. It has just received
this month's allocation of b pounds of steel.
It takes a1 pounds of steel to make a unit of product 1
and a2 pounds of steel to make a unit of product 2.
Let x1 and x2 denote this month's production level of
product 1 and product 2, respectively. Denote by p1 and
p2 the unit profits for products 1 and 2, respectively.
Iron Works has a contract calling for at least m units of
product 1 this month. The firm's facilities are such that at
most u units of product 2 may be produced monthly.
40
Example: Iron Works, Inc., cont’

Mathematical Model
– The total monthly profit =
(profit per unit of product 1)
x (monthly production of product 1)
+ (profit per unit of product 2)
x (monthly production of product 2)
= p1x1 + p2x2
We want to maximize total monthly profit:
Max p1x1 + p2x2
41
Example: Iron Works, Inc., cont’

Mathematical Model (continued)
– The total amount of steel used during monthly production
equals:
(steel required per unit of product 1)
x (monthly production of product 1)
+ (steel required per unit of product 2)
x (monthly production of product 2)
= a1x1 + a2x2
This quantity must be less than or equal to the
allocated b pounds of steel:
a1x1 + a2x2 < b
42
Example: Iron Works, Inc., cont’

Mathematical Model (continued)
– The monthly production level of product 1 must
–
–
be
greater than or equal to m :
x1 > m
The monthly production level of product 2 must be
less than or equal to u :
x2 < u
However, the production level for product 2 cannot be
negative:
x2 > 0
43
Example: Iron Works, Inc., cont’

Mathematical Model Summary
Constraints
Max p1x1 + p2x2
Objective
Function
s.t.
a1x1 + a2x2
x1
x2
x2
<
>
<
>
b
m
u
0
“Subject to”
44
Example: Iron Works, Inc., cont’


Question:
Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 =
100, p2 = 200. Rewrite the model with these specific values
for the uncontrollable inputs.
*****
Substituting, the model is:
Max 100x1 + 200x2
s.t.
2x1 + 3x2
x1
x2
x2
< 2000
>
60
< 720
>
0
45
Example: Iron Works, Inc., cont’
 Question:
The optimal solution to the current model is
x1 = 60 and x2 = 626 2/3. If the product were
engines, explain why this is not a true optimal
solution for the "real-life" problem.
*****
One cannot produce and sell 2/3 of an engine. Thus the
problem is further restricted by the fact that both x1 and x2 must
be integers. They could remain fractions if it is assumed these
fractions are work in progress to be completed the next month.
46
Example: Iron Works, Inc., cont’
Uncontrollable Inputs
$100 profit per unit Prod. 1
$200 profit per unit Prod. 2
2 lbs. steel per unit Prod. 1
3 lbs. Steel per unit Prod. 2
2000 lbs. steel allocated
60 units minimum Prod. 1
720 units maximum Prod. 2
0 units minimum Prod. 2
60 units Prod. 1
626.67 units Prod. 2
Controllable Inputs
Max 100(60) + 200(626.67)
s.t. 2(60) + 3(626.67) < 2000
60
> 60
626.67 < 720
626.67 > 0
Mathematical Model
Profit = $131,333.33
Steel Used = 2000
Output
47
Example: Ponderosa Development Corp.


Ponderosa Development Corporation (PDC) is a small
real estate developer that builds only one style house.
The selling price of the house is $115,000.
Land for each house costs $55,000 and lumber,
supplies, and other materials run another $28,000 per
house. Total labor costs are approximately $20,000
per house.
48
Example: Ponderosa Development Corp.


Ponderosa leases office space for $2,000 per month.
The cost of supplies, utilities, and leased equipment
runs another $3,000 per month.
The one salesperson of PDC is paid a commission
of $2,000 on the sale of each house. PDC has seven
permanent office employees whose monthly salaries
are given on the next slide.
49
Example: Ponderosa Development Corp.
Employee
Monthly Salary
President
$10,000
VP, Development
6,000
VP, Marketing
4,500
Project Manager
5,500
Controller
4,000
Office Manager
3,000
Receptionist
2,000
50
Example: Ponderosa Development Corp.

Question:
Identify all costs and denote the marginal cost and
marginal revenue for each house.
*****
–The monthly salaries total $35,000 and monthly office lease and
supply costs total another $5,000. This $40,000 is a monthly
fixed cost.
–The total cost of land, material, labor, and sales commission per
house, $105,000, is the marginal cost for a house.
–The selling price of $115,000 is the marginal revenue per house.
51
Example: Ponderosa Development Corp.

Question:
Write the monthly cost function c (x), revenue
function r (x), and profit function p (x).
*****
c (x) = variable cost + fixed cost = 105,000x + 40,000
r (x) = 115,000x
p (x) = r (x) - c (x) = 10,000x - 40,000
52
Example: Ponderosa Development Corp.

Question:
What is the breakeven point for monthly sales
of the houses?
 *****
r (x ) = c (x )
115,000x = 105,000x + 40,000
Solving for x gives:
115,000x - 105,000x = 40,000
10,000x = 40,000
Therefore, x = 40,000/10,000 = 4.
53
Example: Ponderosa Development Corp.

Question:
What is the monthly profit if 12 houses per
month are built and sold?
*****
Simply plug 12 into the profit formula and solve
p (12) = 10,000(12) - 40,000 = $80,000 monthly profit
54
Example: Ponderosa Development Corp.
Thousands of Dollars

Graph of Break-Even Analysis
1200
Total Revenue =
115,000x
1000
800
600
Total Cost =
40,000 + 105,000x
400
200
0
Break-Even Point = 4 Houses
0
1
2
3
4
5
6
7
8
Number of Houses Sold (x)
9
10
55
More Examples of Optimization Problems:
A Product Mix Problem
 Woody's Furniture Company makes chairs, tables, and desks.

Chairs are made entirely out of pine, and use 8 linear feet of
pine per chair. Tables and desks are made of pine and
mahogany, tables using 12 linear feet of pine and 15 linear
feet of mahogany, and desks use 16 linear feet of pine and 20
linear feet of mahogany. Chairs require 3 hours of labor to
produce, tables 6 hours, and desks 9 hours. Chairs provide
$35 profit, tables $60 profit, and desks $75 profit.
Woody has 120 linear feet of pine and 60 linear feet of
mahogany delivered each day, and has a work force of 6
carpenters, each of whom puts in an 8-hour day. How can
Woody make the best use of these resources?
56
Tabular description of Woody’s problem
Amount of resource/profit per
Chair
Table
Desk
Resource
Available
Pine
8
12
16
120
Mahogany
0
15
20
60
Man-hours
3
6
9
48
Profit
35
60
75
57
A Diet Problem
A dietician wishes to plan a meal using ground beef, potatoes,
and spinach which satisfies minimum daily requirements of
protein, carbohydrates, and iron. The nutritional makeup of each
ounce of foodstuff (in the appropriate nutritional units) is given
below.
Nutrient


Food
Protein
Carbs
Iron
Beef
20
10
5
Potatoes
10
30
6
Spinach
6
7
20
The minimum daily requirements of protein, carbohydrates,
and iron in the diet are 500, 400, and 75 respectively, and the
cost of beef, potatoes, and spinach are $0.35, $0.20, and $0.15
per ounce, respectively.
How should the dietician plan her menu?
58
Tabular description of the diet problem
Amount of resource/cost per
ounce of
Demand
Beef
Potatoes
Spinach
Protein
20
10
6
500
Carbs
10
30
7
400
Iron
5
6
20
75
Cost
$0.35
$0.20
$0.15
59
An Assignment Problem

The OR IDEAS Consulting Company has six crack OR analysts
that it wants to assign to six projects it has contracted. The
specialized skills of each analyst has been carefully evaluated with
respect to each job to determine the amount of profit the company
can expect to earn if the analyst is assigned to that particular job.
The results are given in the table below:
Project
♦
1
2
3
4
5
6
Anderson
10500
9000
8000
10000
12000
11500
Carlucci
12000
8000
7000
11000
11500
11000
Nataraja
10000
9000
6000
9000
12000
10500
Chin
12000
11000
9000
9000
10000
11000
Yohana
9500
7000
8000
8000
9000
7000
Yamanaka
12500
12000
10000
10000
11000
14000
How should OR IDEAS put its analysts to best use?
60
A Transportation Problem

After a week of renting cars that travel all across the country, Avis finds
that it has a shortage of cars in Los Angeles, San Diego, Pittsburgh, New
York, Atlanta, and Chicago, while it has a surplus of cars in San
Francisco, Denver, Miami, and Houston. There is a certain cost of
shipping cars from each supply point to each demand point, as given in the
chart in the following table:
Outlet

Surplus
LA
SD
Pgh
NY
Atl
Chi
SF
5
4
19
16
17
13
6
Denver
4
7
9
7
8
4
250
Miami
7
8
17
10
9
12
20
Houston
5
7
10
8
12
13
13
Shortage
4
16
10
9
7
16
How can Avis return the cars to the correct locations?
61