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ECS716/QMT710: Operations Research
Pn Pa’ezah Hamzah
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Part 1:
What is Linear Programming (LP)?
General form of LP model
Assumptions
of
LP
model
Formulation of LP model
Graphical Solution ‐ only for LP problems in 2 variables
Part 2:
Simplex Solution
Computer Solution
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At the end of this class, students should be able
to: describe the basic approach of LP
explain the
types
of
problems
that
can
be
solved using LP formulate LP problems solve LP problems in 2 varibles by the
graphical method
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Explain the meaning of the terms:▪ Linear programming
▪ Objective function
▪
Constraints▪ Feasible solutions
▪ Optimal solution
▪ Equality and inequality ▪ Graphical method
▪ Isoprofit/isocost function
▪ Corner point (extreme point)
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Linear programming is a mathematical programming
and
an
optimization
technique
that has been widely used by business and
industry for
many
years.
It involves allocating scarce resources on the
basis of a given criterion of optimality. Most often, this criterion is either maximum profit
or minimum
cost.
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Examples of problems solved by LP: allocation
of
production
facilities,
production
mix
determination, blending, manpower allocation and
assignment, personnel allocation, transportation
planning, scheduling, advertising budget allocation.
LP uses mathematical model to describe the
problem of concern. The
word
linear implies
that
all
mathematical
relationships in the model are linear. The word programming does not refer to
computer programming;
it
refers
to
planning.
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Every LP problem consists of:
decisions that must be made
an objective to be achieved
a set
of
restrictions (or constraints) to
consider
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The general LP model in variables X i for allocating mresources with n constraints can be formulated as follows:
MAXIMIZE (or MINIMIZE) Z = c1 X 1 + c2 X 2 + … + cn X n
subject to: a11 X 1 + a12 X 2 + … + a1n X n ≤ b1
a21 X 1 + a22 X 2 + … + a2n X n ≤ b2ak1 X 1 + ak2 X 2 + … + akn X n ≤ bk …am1 X 1 + am2 X 2 + … + amn X n ≤ bm
X i ≥ 0 where i =1,…,n
constraints
non‐negativity constraints
the objective function
Note:
All mathematical relationships are linear. “ >” and “ < “ relationships are not allowed (since LP model is deterministic.)
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If
all
the
functions in
the
model
are
linear
⇒ Linear Programming (LP) problem.
If all
the
variables in
the
LP
model
are
integer
⇒ Integer Linear Programming (ILP) problem.
If some
of
the
variables are
integer
⇒Mixed Integer Linear Programming (MILP) problem.
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1. Certainty ‐ all parameters of the model are known with certainty.
2. Additivity ‐the
total
consumption
of
each
resource,
as
well
as
the
overall
objective value are the aggregates of the resource consumptions and the contributions to the problem objective, resulted from
carrying out each activity independently.
3. Proportionality ‐ the contribution of each variable in the objective function or its
usage of the resources are directly proportional to the value of the variable.
4. Divisibility ‐ the decision variables can take on any fractional values.
5. Nonnegativity ‐the
decision
variables
can
take
on
any
value
greater
than
or
equal
to zero.
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Step 1: Identify and define the decision variables
(unknowns) of
the
problem.
Step 2: Write the objective function to be optimized
(maximize/minimize) as a linear function of
the decision
variables.
Step 3: Translate the requirements, restrictions, or
wishes,
that
are
in
narrative
form
to
linear
equations or inequalities in terms of the
decision variables.
Step 4:
Add
the
non
‐negativity
constraints.
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ABC Inc. produces two types of chandelier: Design 1 and Design 2. The critical resources are available labor hours, crystals and
switches for the next production cycle. The following table outlines
usage factors
and
unit
profit:
There are 200 switches, 1566 hours of labor, and 2880 pieces of crystals available.
Formulate an
LP
model
to
determine
the
quantity
of
each
type
of
chandelier to produce in order to maximize profit.
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Design 1 Design 2Switch 1 1
Labor 9 hours 6 hours
Crystal 12 pieces 16 pieces
Unit Profit RM350 RM300
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Maxi Design Sdn. Bhd. has been awarded a contract to design a label for a new
computer produced by KC Computer Company. Maxi Design estimates that at least 160 hours will be required to complete the project. Three of the firm’s
graphic designers
are
available
for
assignment
to
this
project.
Aida,
a senior
team
leader; Bakri, a senior designer; and Carol, a junior designer.
Since Aida has worked on several projects for KC, the management has specified
that Aida must be assigned with at least 45% of the total number of hours
assigned to
the
two
senior
designers.
To
provide
label
‐designing
experience
for
Carol, Carol must be assigned at least 20% of the total project time. However, the
number of hours assigned to Carol must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Aida has a maximum of 50 hours available to work on this project.
Hourly salary rates are RM40 for Aida, RM30 for Bakri, and RM20 for Carol.
Formulate an LP model to determine the number of hours each graphic designer should be assigned to the project in order to minimize the total salary.
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There are TWO methods to solve by graphing:
A. Isoprofit /Isocost Line Method
▪isoprofit line
method
for
maximization
problems.
▪ isocost line method for minimization problems.
B. Corner (Extreme)Point Method▪ for both maximization and minimization problems.
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Step 1: Formulate the problem.
Step 2: For each
constraint,
draw
a graph
of
the
feasible
solutions.
The solution set of the LP problem is that region (or set of ordered pairs), which satisfies ALL the constraints
simultaneously. This region is called the area of feasible solution (or feasible
region.)
Step 3: Draw an
objective
function
line.
Step 4: Determine the optimum point.
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Step 4: Determine the optimum point. For maximization problems:
Move parallel objective function lines toward
larger objective function values without entirely
leaving the
feasible
region.
For
minimization
problems:Move parallel objective function lines toward
smaller objective function values without entirely
leaving
the
feasible
region.16
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Objective Function
x
y
Feasible
Region
Optimum point
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Step 1: Graph the constraints.
Step 2:
Locate all
the
corner
points
of
the
graph.
The coordinates of the corners will be determined algebraically.
It is important to note that the optimal point is obtained at the
boundary
of
the
feasible
region
and
furthermore
at
the
corner
points.
For linear programs, it can be shown that the optimal point will always be obtained at corner points.
Step 3: Determine the optimal value.Test all the corner points to see which one yields the optimum
value for the objective function.
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Solve the ABC Co. problem by graphing.a. If the company can sell all the chandeliers it can
produce, how many of each design should it make in
order to maximize the profit?
b. Determine the quantity of each resource used at the
optimal production
level.
Identify
all
resources
that
are
fully utilized.
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A dietician is considering a simple breakfast menu (excluding beverages)
consisting of
scrambled
egg
and
curry
puffs,
for
participants
of
an
orientation program. An adequate amount of the following nutrients in
the breakfast: vitamin A, vitamin B, and iron must be included. Each
small scoop of scrambled egg contains 3 milligrams of vitamin A, 4
milligrams of
vitamin
B,
and
1 milligram
of
iron.
Each
curry
puff
contains
3 milligrams of vitamin A, 2 milligrams of vitamin B, and 2 milligram of
iron. The minimum requirements for vitamin A, vitamin B and iron are 30
milligrams, 24 milligrams and 12 milligrams respectively. Each scoop of
scrambled egg
costs
30
sen
and
each
curry
puff
costs
35
sen.
The
dietician would like to determine how much of each type of food to serve
in order to meet the nutrient requirements and also to minimise the total
cost. The following table summarises the relevant information.
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a. Formulate a linear
programming
model
for
this
problem.
b. Determine the optimal solution. What is the lowest cost that meets the nutrient requirements?
c. Determine the quantity of each nutrient in the optimal solution.
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Nutrient Content (mg.) Nutrient Requirement (mg.)
Nutrient Scrambled Egg Curry Puff
Vitamin A 3 3 30
Vitamin B 4 2 24
Iron 1 2 12
Unit Cost 30 sen 35 sen