Introduction to OR
OR6205 –Sep13, 2023
Application of Analytics
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Descriptive analytics (analyzing data to create informative descriptions of what has happened in the
past or is happening in the present).
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Predictive analytics (using models to create predictions of what is likely to happen in the future).
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Prescriptive analytics (using decision models, including optimization models, to create and/or advise
managerial decision making).
Applying Prescriptive Analytics
Basic Steps for Applying Prescriptive Analytics
1.
Formulating a mathematical model to begin applying prescriptive analytics
2.
Learning how to derive solutions from the model
3.
Testing the model
4.
Preparing to apply the model
5.
Implementation
Mathematical Models
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Mathematical model of a business problem is the system of equations and related mathematical expressions that
describe the essence of the problem.
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There are n related quantifiable decisions to be made, they are represented as decision variables
( 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )
whose respective values are to be determined.
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Objective Function: measure of performance (e.g., profit) is expressed as a mathematical function of decision
variables
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(Z = 3 𝑥1 + 2 𝑥2 + . . . + 5 𝑥𝑛 )
Constraints: any restrictions on the values that can be assigned to these decision variables are also expressed
mathematically, typically by means of inequalities or equations
(4 𝑥1 + 5 𝑥2 ≤ 10)
The constants (namely, the coefficients and right-hand sides) in the constraints and the objective function are called
the parameters of the model.
Graphical LP Solution
WYNDOR GLASS CO - Prototype Example
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The WYNDOR GLASS CO. produces high-quality glass products, including windows and glass doors. It has
three plants. Aluminum frames and hardware are made in Plant 1, wood frames are made in Plant 2, and
Plant 3 produces the glass and assembles the products.
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Because of declining earnings, top management has decided to revamp the company’s product line.
Unprofitable products are being discontinued, releasing production capacity to launch two new products
having large sales potential:
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Product 1: An 8-foot glass door with aluminum framing
Product 2: A 4 × 6 foot double-hung wood-framed window
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Product 1 requires some of the production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs
only Plants 2 and 3. The marketing division has concluded that the company could sell as much of either
product 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.
Therefore, an OR team has been formed to study this question.
Discussions with Upper Management to Identify
Management’s Objectives for the Study
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Determine what the production rates should be for the two products in order to maximize their total profit,
subject to the restrictions imposed by the limited production capacities available in the three plants.
(Each product will be produced in batches of 20, so the production rate is defined as the number of
batches produced per week.)
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Because the work on the current batch of a particular product commonly is only partially completed at
the end of a given week, the production rate can be either an integer or noninteger number. Any
combination of production rates that satisfies the restrictions imposed by the limited production
capacities is permitted, including producing none of one product and as much as possible of the other.
The OR team also Identified the Data that Needed
to be Gathered:
1.
2.
3.
Number of hours of production time available per week in each plant for these new products. (Most of
the time in these plants already is committed to current products, so the available capacity for the new
products is quite limited.)
Number of hours of production time used in each plant for each batch produced of each new product.
Profit per batch produced of each new product. (Profit per batch produced was chosen as an appropriate
measure after the team concluded that the incremental profit from each additional batch produced would
be roughly constant regardless of the total number of batches produced. Because no substantial costs
will be incurred to initiate the production and marketing of these new products, the total profit from each
one is approximately this profit per batch produced times the number of batches produced .)
WYNDOR GLASS CO Example
This problem is a classic example
of a resource-allocation problem,
the most common type of linear
programming problem.
Graphical Solution
Graphical Solution
Graphical Method
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Three lines just constructed are
parallel.
The Reddy Mikks Company – Example
Optimum solution of the Reddy Mikks model
Standard Form of the Model
Common Terminology for Linear Programming
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The key terms are resources and activities, where m denotes the number of different kinds of
resources that can be used and n denotes the number of activities being considered.
Some typical resources are money and particular kinds of machines, equipment, vehicles,
and personnel.
Examples of activities include investing in particular projects, advertising in particular
media, shipping goods from a particular source to a particular destination, and so forth.
Data for Allocation of Resources to Activities
A Standard Form of the Model
Objective Function
Constraints
Nonnegativity Constraints
Terminology for Solutions of the Model
Terminology for Solutions of the Model
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A feasible solution is a solution for which all the constraints are satisfied.
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An infeasible solution is a solution for which at least one constraint is violated.
no feasible solutions
Optimal Solution
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Given that there are feasible solutions, the goal of linear programming is to find a best feasible
solution, as measured by the value of the objective function in the model.
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An optimal solution is a feasible solution that has the most favorable value of the objective
function.
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The most favorable value is the largest value if the objective function is to be maximized, whereas it is
the smallest value if the objective function is to be minimized.
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Most problems will have just one optimal solution. However, it is possible to have more than one.
Any problem having multiple optimal solutions will have an infinite number of them, each with the
same optimal value of the objective function.
Example of Multiple Optimal Solutions
objective function were changed
to Z = 3x1 + 2x2
No Optimal Solutions
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Another possibility is that a problem has no optimal solutions .
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No optimal solutions occurs only if:
1.
2.
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it has no feasible solutions
the constraints do not prevent improving the value of the objective function (Z) indefinitely in the
favorable direction (positive or negative).
The latter case is referred to as having an unbounded Z or an unbounded objective.
Example of No Optimal Solutions
This problem would have no optimal solutions if the
only functional constraint were x1 ≤ 4, because x2
then could be increased indefinitely in the feasible
region without ever reaching the maximum value of
Z = 3x1 + 5x2.
A Corner-point Feasible (CPF) Solution
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A corner-point feasible (CPF) solution is a solution that lies at a corner of the feasible region.
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CPF solutions are commonly referred to as extreme points (or vertices) by OR professionals, but we
prefer the more suggestive corner-point terminology in an introductory course.
CPF plays the key role when the
simplex method searches for an
optimal solution.
Relationship between Optimal Solutions and CPF
Solutions
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Consider any linear programming problem with feasible solutions and a bounded feasible region.
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The problem must possess CPF solutions and at least one optimal solution. Furthermore, the
best CPF solution must be an optimal solution.
Thus, if a problem has exactly one optimal solution, it must be a CPF solution.
If the problem has multiple optimal solutions, at least two must be CPF solutions.
Linear Inequality
Assumptions of Linear Programming
From, a mathematical viewpoint, the assumptions simply are that the model must have a linear objective
function subject to linear constraints.
However, from a modeling viewpoint, these mathematical properties of a linear programming model imply that
certain assumptions must hold about the activities and data of the problem being modeled.
1.
Proportionality : Contribution of a variable is proportional to its value
2.
Additivity: Contribution of variables are independent
3.
Divisibility: Decision variables can take fractional values
4.
Certainty: Each parameter is known with certainty
Proportionality
● Proportionality is an assumption about both the objective function and the functional constraints
● Contribution of a variable is proportional to its value
Additivity
● Proportionality is an assumption about both the objective function and the functional constraints
● Contribution of variables is independent
Additivity assumption: Every function in a linear programming model (whether the objective
function or the function on the left-hand side of a functional constraint) is the sum of the individual
contributions of the respective activities.
Divisibility
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Concerns the values allowed for the decision variables
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Decision variables can take fractional values
Divisibility assumption: Decision variables in a linear programming model are allowed to have any
values, including noninteger values, that satisfy the functional and nonnegativity constraints. Thus, these
variables are not restricted to just integer values. Since each decision variable represents the level of
some activity, it is being assumed that the activities can be run at fractional levels.
Certainty
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Concerns the parameters of the model, namely, the coefficients in the objective function cj, the
coefficients in the functional constraints aij, and the right-hand sides of the functional constraints bi.
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Each parameter is known with certainty
Certainty assumption: The value assigned to each parameter of a linear programming model is
assumed to be a known constant.
Additional Example – Investment
Additional Example – Investment
● Multitudes of investment opportunities are available to today’s investor.
● Examples of investment problems are capital budgeting for projects, bond investment strategy, stock
portfolio selection, and establishment of bank loan policy.
● Using LP to maximize the return
Example – Bank Loan Model
● Bank One is in the process of devising a loan policy that involves a maximum of $12 million. The
following table provides the pertinent data for available loans.
Bad debts are unrecoverable and produce no
interest revenue.
Competition with other financial institutions
dictates the allocation of at least 40% of the
funds to farm and commercial loans. To assist
the housing industry in the area, home loans
equals at least 50% of personal, car, and
home loans combined. The bank limits the
overall ratio of bad debts on all loans to at most
4%.
Solution- Mathematical Model
• The objective of the Bank One is to
maximize net return
• The difference between interest revenue
and lost bad debts.
• Interest revenue is accrued on loans in
good standing. For example, when 10%
of personal loans are lost to bad debt,
the bank receives interest on 90% of the
loan—that is, 14% interest on .9 x1 of
the original loan x1
Solution- Mathematical Model
1.
Total fund not exceed $12 (million)
2.
Farm and commercial loans equal at least 40% of all loans:
Solution- Mathematical Model
3.
Home loans should equal at least 50% of combined personal, car, and home loans:
4.
Bad debts do not exceed 4% of all loans:
5.
Nonnegativity:
Additional Example – Bus Scheduling Model
Example - Bus Scheduling Model
● The bus transportation system in Progress City
operates under the three traditional 8-hour
shifts, starting daily at 8:00 a.m., 4:00 p.m., and
midnight (12:01 a.m.).
● The city is studying the feasibility of revamping
its bus schedule with the goal of reducing
carbon footprint.
● The study seeks the minimum number of buses
that can handle public transportation needs.
● Gathered data shows that the minimum number
of buses needed to meet the transportation
demand can be approximated over successive
4-hour intervals.
Solution- Mathematical Model
Solution- Mathematical Model
Excel Solution
Excel Solution
LP Transformation Techniques
Ads Example
● The goal is to minimize the cost of reaching 1.5 million people using ads of different types.
Introduce a non-linear constraint
● We are now going to introduce a non-linear constraint. Suppose that we require that the total of ads from
the electronic media is within 5 of the number of ads of paper-based media. This can be modeled as
follows: |x1 + x2 – x3 – x4 | ≤ 5
Transformation Techniques
● The constraint “|x1 + x2 – x3 – x4 | ≤ 5” is not a linear constraint. However, the constraint can be
transformed into linear constraints using a simple trick/ “technique”.
● The constraint “|x1 + x2 – x3 – x4 | ≤ 5” is equivalent to the following two constraints:
1.
2.
x1 + x2 – x3 – x4 ≤ 5
-x1 - x2 + x3 + x4 ≤ 5
The feasible regions are exactly the same
Can we always use the trick to transform problems
involving absolute values to a linear program?
● Unfortunately, we can’t. Consider the case in which we want the number of radio and TV ads to differ by
at least 2. This corresponds to the constraint “|x1 – x2 | ≥ 2.” This is equivalent to “x1 – x2 ≥ 2 OR –x1 +
x2 ≥ 2”. But it cannot be made linear.
The feasible region is in yellow. It’s in
two separate pieces. But a linear
programming feasible region is always
connected. In fact, it’s always
convex. That is, if two points are
feasible, then so is the line segment
joining the two points.
A Minimax Objective Functions
,
,
Transforming a Minimax Objective
● The minimax objective can be transformed by including an additional decision variable z, which
represents the maximum costs:
● In order to establish this relationship, the following extra constraints must be imposed:
The Equivalent Linear Program
Simplex Method
Solving Linear Programming Problems: The
Simplex Method
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Simplex method: a general procedure for solving linear programming problems.
Developed by the brilliant George Dantzig in 1947
It has proved to be a remarkably efficient method that is used routinely to solve
huge problems on today’s computers.
The simplex method is an algebraic procedure. However, its underlying
concepts are geometric.
The Essence of the SIMPLEX Method
Constraint boundary: is a line that forms the boundary of
what is permitted by the corresponding constraint.
The points of intersection are the corner-point solutions
of the problem.
The five that lie on the corners of the feasible region—(0,
0), (0, 6), (2, 6), (4, 3), and (4, 0)—are the corner-point
feasible solutions (CPF solutions).
The other three—(0, 9), (4, 6), and (6, 0)—are called
corner-point infeasible solutions.
The Essence of the SIMPLEX Method
Each corner-point solution lies at the intersection of two
constraint boundaries.
For a linear programming problem with n decision
variables, each of corner-point solutions lies at the
intersection of n constraint boundaries.
Certain pairs of the CPF solutions share a constraint
boundary, and other pairs do not.
For any linear programming problem with n decision
variables, two CPF solutions are adjacent to each other
if they share n − 1 constraint boundaries. The two
adjacent CPF solutions are connected by a line segment
that lies on these same shared constraint boundaries.
Such a line segment is referred to as an edge of the
feasible region.
Since n = 2 in the example, two of its CPF solutions are adjacent if they share one
constraint boundary; for example, (0, 0) and (0, 6) are adjacent because they share the
x1 = 0 constraint boundary.
One Reason for our Interest in Adjacent CPF
● Optimality test: Consider any linear programming problem that possesses at least one optimal
solution. If a CPF solution has no adjacent CPF solutions that are better (as measured by Z), then
it must be an optimal solution.
Thus, for the example, (2, 6) must be optimal simply
because its Z = 36 is larger than Z = 30 for (0, 6) and Z =
27 for (4, 3). This optimality test is the one used by the
simplex method for determining when an optimal solution
has been reached.
Simplex Method Example- Geometric Viewpoint
● Initialization: Choose (0, 0) as the initial CPF solution to examine.
(This is a convenient choice because no calculations are required
to identify this CPF solution.)
● Optimality Test: Conclude that (0, 0) is not an optimal solution.
(Adjacent CPF solutions are better.)
● Iteration 1: Move to a better adjacent CPF solution, (0, 6), by
performing the following three steps.
1.
Considering the two edges of the feasible region that
emanate from (0, 0), choose to move along the edge that
leads up the x2 axis.
2.
Stop at the first new constraint boundary: 2 x2 = 12
3.
Solve for the intersection of the new set of constraint
boundaries: (0, 6).
Simplex Method Example- Geometric Viewpoint
Optimality Test: Conclude that (0, 6) is not an optimal solution. (An adjacent CPF solution is
better.)
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1.
Iteration 2: Move to a better adjacent CPF solution, (2, 6), by performing the following
three steps:
Considering the two edges of the feasible region that emanate from (0, 6), choose to
move along the edge that leads to the right.
2. Stop at the first new constraint boundary encountered when moving in that direction: 3
x1 + 2 x2 = 18. (Moving farther in the direction selected in step 1 leaves the feasible
region.)
3. Solve for the intersection of the new set of constraint boundaries: (2, 6). (The equations
for these constraint boundaries, 3 x1 + 2 x2 = 18 and 2 x2 = 12, immediately yield this
solution.)
Optimality Test: Conclude that (2, 6) is an optimal solution, so stop. (None of the adjacent
CPF solutions are better.)