1' Course Description:

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1. Course Description The purpose of this
course is to introduce Operations Research
(OR)/Management Science (MS) techniques for
manufacturing, services, and public sector.
OR/MS includes a variety of techniques used in
modeling business applications for both better
understanding the system in question and making
best decisions.
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OR/MS techniques have been applied in many
situations, ranging from inventory management in
manufacturing firms to capital budgeting in large
and small organizations. Public and Private
Sector Applications
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The main objective of this course is to provide
engineers with a variety of decisional tools
available for modeling and solving problems in
industrial systems, businesses and/or nonprofit
context. In this class, each individual will
explore how to make various industrial models and
how to solve them effectively.
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Model Development

• Models are representations of real objects or
  situations
• Mathematical models - represent real world
  problems through a system of mathematical
  formulas and expressions based on key
  assumptions, estimates, or statistical analyses

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Advantages of Models

• Generally, experimenting with models (compared to
  experimenting with the real situation)
• requires less time
• is less expensive
• involves less risk

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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.

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Mathematical Models

• 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.

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Body of Knowledge

• The body of knowledge involving quantitative
  approaches to decision making is referred to as
• Management Science
• Operations research
• Decision science
• It had its early roots in World War II and is
  flourishing in business and industry with the aid
  of computers

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Transforming Model Inputs into Output
Uncontrollable Inputs (Environmental Factors)
Output (Projected Results)
Controllable Inputs (Decision Variables)
Mathematical Model
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Example Project Scheduling

• Consider the construction of a 250-unit
  apartment
• complex. The project consists of hundreds of
  activities involving excavating,
• framing, wiring, plastering, painting,
• land-scaping, 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.).

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Example Project Scheduling

• 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?
• Answer 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.

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Example Project Scheduling

• Question What would be the decision variables of
  the mathematical model? The objective function?
  The constraints?
• Answer
• Decision variables which activities to expedite
  and by how much, and when to start each activity
• Objective function minimize project completion
  time
• Constraints do not violate any activity
  precedence relationships and do not expedite in
  excess of the funds available.

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Example Project Scheduling

• Question
• Is the model deterministic or stochastic?
• Answer
• 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.

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Example Project Scheduling

• Question
• Suggest assumptions that could be made to
  simplify the model.
• Answer
• 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.

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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.

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Model Solution

• 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.

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Computer Software

• A variety of software packages are available for
  solving mathematical models.
• a) Hillier Liebermans Softwares in CD
• b) QSB and Spreadsheet packages such as Microsoft
  Excel
• c) GAMS, LINDO, CPLEX, MINOS etc.

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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

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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)

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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.

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Linear Programming (LP) A mathematical method
that consists of an objective function and many
constraints. LP involves the planning of
activities to obtain an optimal result, using a
mathematical model, in which all the functions
are expressed by a linear relation.
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A standard Linear Programming Problem
Maximize subject to
Applications Man Power Design, Portfolio Analysis
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Simplex method A remarkably efficient solution
procedure for solving various LP problems.
Extensions and variations of the simplex method
are used to perform postoptimality analysis
(including sensitivity analysis).
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(a) Algebraic Form
(0)
(1)
(2)
(3)
(b) Tabular Form
Coefficient of
Basic Variable
Eq.
Right Side
Z
(0)
1 -3 -5 0 0 0 0 0 1 0 1
0 0 0 0 2 0 0 1 0 12 0
3 2 0 0 1 18
(1)
(2)
(3)
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Duality Theory An important discovery in the
early development of LP is Duality Theory. Each
LP problem, referred to as a primal problem is
associated with another LP problem called a dual
problem. One of the key uses of duality theory
lies in the interpretation and implementation of
sensitivity analysis.
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Introduction to MS/OR MS Management Science OR
Operations Research Key components (a)
Modeling/Formulation (b)
Algorithm (c) Application
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Management Science (OR/MS) (1) A discipline that
attempts to aid managerial decision making by
applying a scientific approach to managerial
problems that involve quantitative factors. (2)
OR/MS is based upon mathematics, computer science
and other social sciences like economics and
business.
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General Steps of OR/MS Step 1 Define problem and
gather data Step 2 Formulate a mathematical
model to represent the problem Step
3 Develop a computer based procedure
for deriving a solution(s) to the
problem
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Step 4 Test the model and refine it as
needed Step 5 Apply the model to analyze the
problem and make recommendation
for management Step 6 Help implementation
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Linear Programming (LP)
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Linear Programming (LP) Problem

• The maximization or minimization of some quantity
  is the objective in all linear programming
  problems.
• All LP problems have constraints that limit the
  degree to which the objective can be pursued.
• A feasible solution satisfies all the problem's
  constraints.
• An optimal solution is a feasible solution that
  results in the largest possible objective
  function value when maximizing (or smallest when
  minimizing).
• A graphical solution method can be used to solve
  a linear program with two variables.

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Linear Programming (LP) Problem

• If both the objective function and the
  constraints are linear, the problem is referred
  to as a linear programming problem.
• Linear functions are functions in which each
  variable appears in a separate term raised to the
  first power and is multiplied by a constant
  (which could be 0).
• Linear constraints are linear functions that are
  restricted to be "less than or equal to", "equal
  to", or "greater than or equal to" a constant.

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Problem Formulation

• Problem formulation or modeling is the process of
  translating a verbal statement of a problem into
  a mathematical statement.

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1 LP Formulation (a) Decision Variables
All the decision variables are non-negative. (b)
Objective Function Min or Max (c) Constraints
s.t. subject to
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Guidelines for Model Formulation

• Understand the problem thoroughly.
• Describe the objective.
• Describe each constraint.
• Define the decision variables.
• Write the objective in terms of the decision
  variables.
• Write the constraints in terms of the decision
  variables.

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2 Example
A company has three plants, Plant 1, Plant 2,
Plant 3. Because of declining earnings, top
management has decided to revamp the companys
product line. Product 1 It requires some of
production capacity in Plants
1 and 3. Product 2 It needs Plants 2 and 3.
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The marketing division has concluded that the
company could sell as much as could be produced
by these plants. However, because both products
would be competing for the same production
capacity in Plant 3, it is not clear which mix of
the two products would be most profitable.
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The data needed to be gathered 1. Number of
hours of production time available per week in
each plant for these new products. (The available
capacity for the new products is quite
limited.) 2. Production time used in each plant
for each batch to yield each new product. 3.
There is a profit per batch from a new product.
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Production Time per Batch, Hours
Production Time Available per Week, Hours
Product
1 2
Plant
1 2 3
4 12 18
1 0 0 2 3 2
Profit per batch
3,000 5,000
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of batches of product 1 produced per week
of batches of product 2 produced per week
the total profit per week Maximize subject
to
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Graphic Solution
10
8
6
4
Feasible region
2
0 2 4 6 8
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10
8
6
4
Feasible region
2
0 2 4 6 8
43
10
8
6
4
Feasible region
2
0 2 4 6 8
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10
8
6
4
Feasible region
2
0 2 4 6 8
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Maximize
Slope-intercept form
8
6
4
2
0 2 4 6 8 10
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Summary of the Graphical Solution Procedurefor
Maximization Problems

• Prepare a graph of the feasible solutions for
  each of the constraints.
• Determine the feasible region that satisfies all
  the constraints simultaneously..
• Draw an objective function line.
• Move parallel objective function lines toward
  larger objective function values without entirely
  leaving the feasible region.
• Any feasible solution on the objective function
  line with the largest value is an optimal
  solution.

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4 Standard Form of LP Model
Maximize
s.t.
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5 Other Forms The other LP forms are the
following 1. Minimizing the objective
function 2. Greater-than-or-equal-to
constraints
Minimize
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3. Some functional constraints in equation
form 4. Deleting the nonnegativity constraints
for some decision variables
unrestricted in sign
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6 Key Terminology (a) A feasible solution is a
solution for which all constraints are
satisfied (b) An infeasible solution is a
solution for which at least one constraint
is violated (c) A feasible region is a
collection of all feasible solutions
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(d) An optimal solution is a feasible solution
that has the most favorable value of the
objective function (e) Multiple optimal
solutions have an infinite number of
solutions with the same optimal objective
value
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Multiple optimal solutions
Example
Maximize
Subject to
and
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8
Multiple optimal solutions
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Every point on this red line segment is optimal,
each with Z18.
4
2
Feasible region
0 2 4 6 8 10
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(f) An unbounded solution occurs when the
constraints do not prevent improving the
value of the objective function.
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Case Study - Personal Scheduling
UNION AIRWAYS needs to hire additional customer
service agents. Management recognizes the need
for cost control while also consistently
providing a satisfactory level of service to
customers. Based on the new schedule of flights,
an analysis has been made of the minimum number
of customer service agents that need to be on
duty at different times of the day to provide a
satisfactory level of service.
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Time Period Covered
Minimum of Agents needed
Shift
Time Period
1 2 3 4 5
48 79 65 87 64 73 82 43 52 15
600 am to 800 am 800 am to1000 am 1000 am to
noon Noon to 200 pm 200 pm to 400 pm 400 pm
to 600 pm 600 pm to 800 pm 800 pm to 1000
pm 1000 pm to midnight Midnight to 600 am




Daily cost per agent
170 160 175 180 195
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The problem is to determine how many agents
should be assigned to the respective shifts each
day to minimize the total personnel cost for
agents, while meeting (or surpassing) the service
requirements. Activities correspond to shifts,
where the level of each activity is the number of
agents assigned to that shift. This problem
involves finding the best mix of shift sizes.
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of agents for shift 1 (6AM - 2PM) of
agents for shift 2 (8AM - 4PM) of agents for
shift 3 (Noon - 8PM) of agents for shift 4
(4PM - Midnight) of agents for shift 5 (10PM
- 6AM)
The objective is to minimize the total cost of
the agents assigned to the five shifts.
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Min s.t.
all
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Total Personal Cost 30,610
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Slack and Surplus Variables

• A linear program in which all the variables are
  non-negative and all the constraints are
  equalities is said to be in standard form.
• Standard form is attained by adding slack
  variables to "less than or equal to" constraints,
  and by subtracting surplus variables from
  "greater than or equal to" constraints.
• Slack and surplus variables represent the
  difference between the left and right sides of
  the constraints.
• Slack and surplus variables have objective
  function coefficients equal to 0.

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Example 1 Standard Form

• Max 5x1 7x2 0s1 0s2 0s3
• s.t. x1
  s1 6
• 2x1 3x2
  s2 19
• x1 x2
  s3 8
• x1, x2 ,
  s1 , s2 , s3 gt 0

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Interpretation of Computer Output

• In this chapter we will discuss the following
  output
• objective function value
• values of the decision variables
• reduced costs
• slack/surplus
• In the next chapter we will discuss how an
  optimal solution is affected by a change in
• a coefficient of the objective function
• the right-hand side value of a constraint

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Example 1 Spreadsheet Solution

• Partial Spreadsheet Showing Solution

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Example 1 Spreadsheet Solution

• Reduced Costs