Module 01.ppt simulation anh optimization

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
CHE2034IU:
Simulation and Optimization
Spring 2022 (Sem-II)
International University-HCMC
Department of Chemical
Engineering

Administrative Details
• Location: A2-310
• Section:
• Classes : Thurday 13:15 -15:30 pm

Contact Information
• Instructor:
Khanh B. Vu <vbkhanh@hcmiu.edu.vn>
Office: LA1-707
(tel: 0977348831)
• Course Assistant:

Textbook
• Text:
– Introduction to Operations Research, 7th
ed., by Hillier &
Lieberman, McGraw-Hill, 11th
Edition (2021).
• Reference:
– Schaum’s Outline Series: Operations Research by
R.Bronson, McGraw-Hill, 2nd
Edition (1997).
– Hamdy A. Taha, Operation Research: An Introduction,
Pearson,10th
Edition (2017).
– Wayne L. Winston and Munirpallam Venkataramanan,
Introduction to Mathematical Programming: Operations
Research: Vol. 1, Thomson, 4th
Edition (2002).

Course Outline
PART I: THEORY (25/01/2021-02/05/2021), 10 WEEKS
•Lecture 1: An introduction to Model Building (today)
•Lecture 2: Formulating linear programming (LP) problems
•Lecture 3: Solution of an LP: Graphical Solution
•Lecture 4: Solution of an LP: Simplex Method, Standard
Form, Degeneracy, Alternate Solutions,
Unbounded LP, Infeasible LP.
•Lecture 5: Solution of an LP: Finding an initial feasible
solution, Big-M Method, Two-Phase Method,
solution of an LP using a software package –
LINDO.

Course Outline
• Lecture 7: Revised Simplex Method, Simplex Formulas,
Shadow Price, Reduce Cost
• Midterm Exam
• Lecture 8: Sensitivity Analysis
• Lecture 9: Duality Theorem, Finding the dual of an LP,
Economic Interpretation of the Dual Problem
and Dual Variables, Lagrange Multipliers,
Complementary Slackness, Dual Simplex
Method, How to Read the LINDO Output.
• Lecture 10: Integer Programming

Course Outline
PART II: LAB (19/04/2021-30/05/2021), 6 WEEKS
•Lab 1: TBD
•Lab 2: TBD
•Lab 3: TBD
•Lab 4: TBD
•Lab 5: TBD
•Lab 6: Lab Exam
•…
•Theory Final Exam

Grade Allocation
• Midterm: 30%
• Final Exam: 40%
• Homework, contribution, attendance: 30%

Pre-Requisite: MATH
• Linear Algebra
• Differential Equations

Optimization is everywhere
It is embedded in language, and part of the
way we think.
•Firms want to maximize value to shareholders
•People want to make the best choices
•We want the highest quality at the lowest price
•When playing games, we want the best strategy
•When we have too much to do, we want to optimize the
use of our time
•Etc.

Optimization examples
A manufacturer of computer system components assembles two types of
graphics terminals, model A and model B. The amounts of materials and labor
required for each assembly, and the total amounts available, are shown in the
table below. The profits that can be realized from the sale of each terminal are
$22 and $28 for the model A and B, respectively, and we assume there is a
market for as many terminals as can be manufactured
Resourced required per unit Resources available
A B
Materials 8 10 3400
Labor 2 3 960
The manufacturer would like to determine how many of each model to assemble
in order to maximize profits.

A company wishes to minimize its combined costs of production and inventory
over a 4-week time period. An item produced in a given week is available for
consumption during that week, or it may be kept in inventory for use in later
weeks. Initial inventory at the beginning of week 1 is 250 units. The minimum
allowed inventory carried from one week to the next is 50 units. Unit production
cost is $15, and the cost of storing a unit from one week to the next is $3. The
following table shows production capacities and the demands that must be met
during each week.
Production Period Production Capacity Demand
1 800 900
2 700 600
3 600 800
4 800 600
A minimum production of 500 items per week must be maintained. Inventory
costs are not applied to items remaining at the end of the 4th production period,
nor is the minimum inventory restriction applied after this final period.

A mixture of freeze-dried vegetables is to be composed of beans, corn, broccoli,
cabbage, and potatoes. The mixture is to contain (by weight) at most 40% beans and
at most 32% potatoes. The mixture should contain at least 5 grams iron, 36 grams
phosphorus, and 28 grams calcium. The nutrients in each vegetable and the costs
are shown in the table.
Vegetable Milligrams Nutrient per Pound of Vegetable Cost per Pound
(cents)
Iron Phosphorus Calcium
Beans 0.5 10 200 20
Corn 0.5 20 280 18
Broccoli 1.2 40 800 32
Cabbage 0.3 30 420 28
Potatoes 0.4 50 360 16
Determine the amount of each vegetable to include so that the cost of the mixture is
minimized.

A manufacturer wishes to produce an alloy (blend) that is 30 percent lead, 30
percent zinc, and 40 percent tin. Suppose there are on the market alloys j = 1, . . . , 9
with the percent composition (of lead, zinc, and tin) and prices as shown in the
display below. How much of each type of alloy should be purchased in order to
minimize costs per pound of blend?

What is Operations Research?
• Before: application of mathematics and the scientific
method to military operations
• Today: Operations Research and Management Science
mean “the use of mathematical models in providing
guidelines to managers for making effective decisions
within the state of the current information, or in seeking
further information if current knowledge is insufficient
to reach a proper decision.”

Evolution of OR
• World War II : British military leaders asked
scientists and engineers to analyze several military
problems
-Deployment of radar
-Management of convoy, bombing, antisubmarine, and
mining operations.
-The result was called Military Operations Research, later
Operations Research
• MIT was one of the birthplaces of OR
• Professor Morse at MIT was a pioneer in the US
• Founded MIT OR Center and helped found ORSA

Career Opportunities
• Accounting
• Actuarial Work
• Computer Services
• Corporate Planning
• Economic Analysis
• Financial Modeling
• Industrial Engineering
• Investment Analysis
• Logistics
• Manufacturing Services
• Management Consulting
• Management Training
• Market Research
• Operations Research
• Policy Planning
• Production Engineering
• Quantitative Methods
• Strategic Planning
• Systems Analysis
• Transportation

Lecture 1
An introduction to Model Building

1.1 - An Introduction to Modeling
 Operations Research and Management Science
mean “the use of mathematical models in providing
guidelines to managers for making effective decisions
within the state of the current information, or in seeking
further information if current knowledge is insufficient
to reach a proper decision.”
 A system is an organization of interdependent
components that work together to accomplish the goal
of the system.

The term operations research was coined during WW II
when leaders asked scientists and engineers to analyze
several military problems.
The scientific approach to decision making requires the
use of one or more mathematical models. A
mathematical model is a mathematical representation of
the actual situation that may be used to make better
decisions or clarify the situation.

A Modeling Example
Eli Daisy produces the drug Wozac in huge batches by
heating a chemical mixture in a pressurized container.
Each time a batch is produced, a different amount of
Wozac is produced. The amount produced is the
process yield (measured in pounds).
Daisy is interested in understanding the factors that
influence the yield of Wozac production process.
The solution on subsequent slides describes a model
building process for this situation.

Daisy is interested in determining the factors that
influence the process yield. This would be referred to as
a descriptive model since it describes the behavior of
the actual yield as a function of various factors.
Daisy might determine the following factors influence
yield:
• Container volume in liters (V)
• Container pressure in milliliters (P)
• Container temperature in degrees centigrade (T)
• Chemical composition of the processed mixture

Letting A, B, and C be the percentage of the mixture made
up of chemical A, B, and C, then Daisy might find , for
example, that:
Yield = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T
2
–
0.001P
2
+ 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C –
9.6B*C

To determine this relationship, the yield of the process would
have to measured for many different combinations of the
previously listed factors. Knowledge of this equation would
enable Daisy to describe the yield of the production process
once volume, pressure, temperature, and chemical
composition were known.

Prescriptive or Optimization Models
Prescriptive models “prescribes” behavior for an organization
that will enables it to best meet its goals. Components of this
model include:
• objective function(s)
• decision variables
• constraints
An optimization model seeks to find values of the decision
variables that optimize (maximize or minimize) an objective
function among the set of all values for the decision variables
that satisfy the given constraints.

The Objective Function
The Daisy example seeks maximize the yield for the
production process. In most models, there will be a
function we wish to maximize or minimize. This function is
called the model’s objective function. To maximize the
process yield we need to find the values of V, P, T, A, B,
and C that make the yield equation (below) as large as
possible.
Yield = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2
– 0.001P2
+ 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C – 9.6B*C

In many situations, an organization may have more than
more objective.
For example, in assigning students to the two high schools
in Bloomington, Indiana, the Monroe County School Board
stated that the assignment of students involve the following
objectives:
• equalize the number of students at the two high schools
• minimize the average distance students travel to school
• have a diverse student body at both high schools

The Decision Variables
Variables whose values are under our control and influence
system performance are called decision variables. In the
Daisy example, V, P, T, A, B, and C are decision variables.
Constraints
In most situations, only certain values of the decision
variables are possible. For example, certain volume,
pressure, and temperature conditions might be unsafe.
Also, A, B, and C must be nonnegative numbers that sum to
one. These restrictions on the decision variable values are
called constraints.

Suppose the Daisy example has the following
constraints:
1. Volume must be between 1 and 5 liters
2. Pressure must be between 200 and 400 mmHg
3. Temperature must be between 100 and 200 degrees
centigrade
4. Mixture must be made up entirely of A, B, and C
5. For the drug to perform properly, only half the mixture
at most can be product A.

Mathematically, these constraints can be expressed:
V ≤ 5
V ≥ 1
P ≤ 400
P ≥ 200
T ≤ 200
T ≥ 100
A ≥ 0
B ≥ 0
C ≥ 0
A + B + C = 1.0
A ≤ 0.5

The Complete Daisy Optimization Model
Letting z represent the value of the objection function (the yield), the
entire optimization model may be written as:
maximize z = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2
– 0.001P2
+ 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C – 9.6B*C
V ≤ 5
V ≥ 1
P ≤ 400
P ≥ 200
T ≤ 200
T ≥ 100
A + B + C = 1.0
A ≤ 0.5
A ≥ 0
B ≥ 0
C ≥ 0
Subject to (s.t.)

Any specification of the decision variables that satisfies
all the model’s constraints is said to be in the feasible
region. For example, V = 2, P = 300, T = 150, A = 0.4,
B = 0.3 and C = 0.3 is in the feasible region. An
optimal solution to an optimization model any point in
the feasible region that optimizes (in this case
maximizes) the objective function.
It can be determined that the optimal solution to its
model is V = 5, P = 200, T = 100, A = 0.294, B = 0, C =
0.706, and z = 209.384 as shown on the next slides.

Static and Dynamic Models (based on time reference)
A static model is one in which the decision variables do
not involve sequences of decisions over multiple periods.
A dynamic model is a model in which the decision
variables do involve sequences of decisions over multiple
periods. In a static model, we solve a one shot problem
whose solutions are prescribe optimal values of the
decision variables at all points in time.
The Daisy problem is an example of a static model.

Static and Dynamic Models
For a dynamic model, consider a company (SailCo) that
must determine how to minimize the cost of meeting (on-
time) the demand for sail boats it produces during the next
year. SailCo must determine the number of sail boats to
produce during each of the next four quarters. SailCo’s
decisions must be made over multiple periods and thus
posses a dynamic model.

Linear and Nonlinear models
Suppose that when ever decision variables appear in the
objective function and in the constraints of an optimization model
the decision variables are always multiplied by constants and
then added together. Such a model is a linear model.
f(x1, x2, …, xn) = c1x1 + c2x2 + … + cnxn
The Daisy example is a nonlinear model. While the decision
variables in the constraints are linear, the objective function is
nonlinear since the objective function terms:
0.001T*P, - 0.01T2
, – 0.001P2
, 19A*B, 11.4A*C, and – 9.6B*C
are nonlinear. In general, nonlinear models are much harder to
solve.

Integer and Noninteger Models
If one or more of the decision variables must be integer,
then we say that an optimization model is an integer model.
If all the decision variables are free to assume fractional
values, then an optimization model is a noninteger model.
The Daisy example:
maximize z = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2
– 0.001P
2
+ 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C –
9.6B*C

Integer and Noninteger Models
Suppose we consider producing chairs and tables using only 21 m2
of wood. Each chair (table) requires 6 (7) m2
of wood. Each chair is
sold at $12 (×10) and each table is sold at $13 (×10). Let T and C
denote the number of tables and chairs produced
The IP formulation below maximizes the revenue:

Deterministic and Stochastic Models
Suppose that for any value of the decision variables the
value of the objective function and whether or not the
constraints are satisfied is known with certainty. We
then have a deterministic model. If this is not the case,
then we have a stochastic model.
If we view the Daisy example as a deterministic model,
then we are making the assumption that for given values
of V, P, T, A, B, and C the process yield will always be
the same. Since this is unlikely, the objective function can
be viewed as the average yield of the process for given
decision variable values.

1.2 – The Seven-Step Model-Building Process
Formulate
Problem
Observe
System
Verify
model and
use it for
Prediction
Formulate a
Mathematical
Model of
Problem Select a
Suitable
Alternative
Present
Results to
Organization
Implement
and Evaluate
Recommendations

Formulate
Problem
Observe
System
Verify
model and
use it for
Prediction
Formulate
Mathematical
Model, Gather
Data Select a
Suitable
Alternative
Present
Results to
Organization
Implement
and Evaluate
Recommendations

Operations research used to solve an organization’s
problem follows a seven-step model building procedure:
1. Formulate the Problem
• Define the problem.
• Specify objectives.
• Determine parts of the organization to be studied.
2. Observe the System
• Determines parameters affecting the problem.
• Collect data to estimate values of the parameters.

3. Formulate a Mathematical Model of the Problem
4. Verify the Model and Use the Model for Prediction
• Does the model yield results for values of decision
variables not used to develop the model?
• What eventualities might cause the model to
become invalid?
5. Select a Suitable Alternative
Given a model and a set of alternative solutions,
determine which solution best meets the organizations
objectives.

6. Present the Results and Conclusion(s) of the
Study to the Organization
• Present the results to the decision maker(s)
• If necessary, prepare several alternative
solutions and permit the organization to chose the
one that best meets their needs.
• Any non-approval of the study’s
recommendations may have stemmed from an
incorrect problem definition or failure to involve the
decision maker(s) from the start of the project. In
such a case, return to step 1, 2, or 3.

7. Implement and Evaluate Recommendations
Upon acceptance of the study by the organization, the
analyst:
• Assists in implementing the recommendations.
• Monitors and dynamically updates the system as
the environment and parameters change to ensure
that recommendations enable the organization to
meet its goals.
An example of the application of these steps are shown on
subsequent slides (Section 1.3).

1.3 – The CITGO Petroleum Problem
Klingman et. Al. (1987) applied a variety management-
science techniques to CITGO Petroleum. Their work
saved the company an estimated $70 million per year.
CITGO is an oil-refining and marketing company that was
purchased by Southland Corporation (7-11 Stores).
The following slides focus on two aspects of the CITGO’s
team’s work:
1. A mathematical model to optimize the operation of CITGO’s
refineries.
2. A mathematical model – supply, distribution and marketing
(SDM) system – used to develop an 11- week supply,
distribution and marketing plan for the entire business.

Optimizing Refinery Operations
Step 1 (Formulate the Problem) Klingman et. al. wanted to
minimize the cost of CITGO’s refineries.
Step 2 (Observe the System) The Lake Charles, Louisiana,
refinery was closely observed in an attempt to estimate key
relationships such as:
• How the cost of producing each of CITGO’s products
(motor fuel, no. 2 fuel oil, turbine fuel, naphtha, and several
blended motor fuels) depends upon the inputs used to
produce each product.
• The amount of energy needed to produce each product
(requiring the installation of a new metering system).

• The yield associated with each input – output
combination. For example, if 1 gallon of crude oil would
yield 0.52 gallons of motor fuel, then the yield would be
52%.
• To reduce maintenance costs, data were collected on
parts inventories and equipment breakdowns. Obtaining
accurate data required the installation of a new data based -
management system and integrated maintenance
information system. Additionally, a process control system
was also installed to accurately monitor the inputs and
resources used to manufacture each product.
Step 2 (continued)

Step 3 (Formulate a Mathematical Model of the Problem)
Using linear programming (LP), a model was developed to
optimize refinery operations.
The model:
• Determines the cost-minimizing method for mixing or
blending together inputs to obtain desired results.
• Contains constraints that ensure that inputs are blended
to produce desired results.
• Includes constraints that ensure inputs are blended so
that each output is of the desired quality.
• Other model constraints ensure plant capacities are not
exceeded and allow for inventory for each end product.

Step 4 (Verify the Model and Use the Model for
Prediction) To validate the model, inputs and outputs
from the refinery were collected for one month. Given
actual inputs used at the refinery during that month,
actual outputs were compared to those predicted by the
model. After extensive changes, the model’s predicted
outputs were close to actual outputs.

Step 5 (Select Suitable Alternative Solutions) Running
the LP yielded a daily strategy for running the refinery.
For instance, the model might say, produce 400,000
gallons of turbine fuel using 300,000 gallons of crude 1
and 200,000 of crude 2.

Step 6 (Present the Results and Conclusions)
and
Step 7 (Implement and Evaluate Recommendations)
Once the data base and process control were in
place, the model was used to guide day-to-day
operations. CITGO estimated that the overall
benefits of the refinery system exceeded $50
million annually.

The Supply Distribution Marketing (SDM) System
Step 1 (Formulate the Problem) CITGO wanted a
mathematical model that could be used to make supply,
distribution, and marketing decisions such as:
• Where should the crude oil be purchased?
• Where should products be sold?
• What price should be charged for products?
• How much of each product should be held in inventory?
The goal was to maximize profitability associated with
these decisions.

Step 2 (Observe the System) A database that kept track of
sales, inventory, trades, and exchanges of all refined goods
was installed. Also regression analysis was used to develop
forecasts of for wholesale prices and wholesale demand for
each CITGO product.
Step 3 (Formulate a Mathematical Model of the Problem)
and Step 5 (Select Suitable Alterative Solutions)
A minimum-cost network flow model (MCNFM) is used a
determine an 11-week supply, marketing, and distribution
strategy. The model makes all decisions discussed in Step 1.
A typical model run involved 3,000 equations and 15,000
decision variables required only 30 seconds on an IBM 4381.

Step 4 (Verify the Model and Use the Model for Prediction)
The forecasting models are continuously evaluated to
ensure that they continue to give accurate forecasts.
Step 6 (Present the Results and Conclusions)
and
Step 7 (Implement and Evaluate Recommendations)
Implementing the SDM required several organizational
changes. A new vice-president was appointed to coordinate
the operation of the SDM and refinery LP model. The product
supply and product scheduling departments were combined to
improve communications and information flow.

General Comment on Model Building
Model Building is a prime example of so called
Interdisciplinary work that spans various engineering,
scientific (primarily mathematics), and business
fields.
It is a very productive and an exciting area to work in.
This text takes us there.

General Comment on Model Building
Mathematical Model
• An idealized representation of a real world
problem
• Objective function: z = c1x1+c2x2+…+cnxn
• Decision variables: (x1,x2,…,xn)
• Constraints: a11x1+a13x3 ≤ b1
• Goal: Choose values of the decision variables that
maximize/minimize the objective function subject
to the constraints.

Module 01.ppt simulation anh optimization

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