Introduction to Linear Programming

1. OPERATION RESEARCH
By
MITKU A.
(CHAPTER TWO:LINEAR PROGRAMMING)
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2. Introduction
 Linear programming (LP): model enable
users to find optimal solutions to certain
problems in which the solutions must
satisfy a given set of requirements or
constraints.
 A large number of decision problems faced
by a business manager involve allocation of
resources to various activities, with the
objective of increasing profits or decreasing
costs, or both.
 Thus, the manager has to take a decision
as to how best the resources be allocated
among the various activities.
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3.  Mathematical programming involves
optimization of a certain function, called
the objective function, subject to certain
constraints.
 Linear Programming (LP) is a
mathematical modeling technique useful
for economic allocation of “scarce” or
“limited” resources, such as labour,
material, machine, time, warehouse,
space, capital, etc. to several competing
activities such as products, services,
jobs, new equipment, projects etc, on the
basis of a given criterion of optimality.11/9/2018

4. Structure of Linear
Programming Model
The general structure of LP model consists
of three basic elements or
components.These are:
◦ The decision variables, and parameters
◦ The objective function and
◦ The constraints
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5.  Decision variables (Activities): The
activities are usually denoted by
decision variables .
 The values of these activities represent
the extent to which each of these is
performed.
 The values of certain variables may or
may not be under the decision –maker’s
control.
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6. The objective function:
 the objective (goal) function of each LP
problem is expressed in terms of
decision variables to optimize the
criterion of optimality (the measure of
performance)
 such as profit, cost, revenue, distance,
etc.
 In its general form it is expressed as:
Optimize (Maximize or Minimize) Z=
c1x1+c2x2+------cnxn
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7. The constraints:
 There are always certain limitations
(or constraints) on the use of
resources, e.g. labor, Machine, Raw
material, space, money, etc.
 These limit the degree to which an
objective can be achieved.
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8. Assumptions Underlying Linear
Programming
The basic assumptions of linear programming
include the following:
Proportionality: the measure of effectiveness
(profit or loss) in the objective function and
amount of each resource used must be
proportional to the value of each decision
variable considered individually.
 Additivity: It states that the sum of
resources used by different activities must
be equal to the total quantity of resources
used by each activity for all resources
individually and collectively.
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9.  Divisibility: This assumption implies
that solutions need not be in whole
numbers (integers). Instead, they are
divisible and may take any fractional
value.
 Certainty: the coefficients in the
objective function and constrains are
completely known (deterministic) and do
not change during the period being
studied.
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10.  Finiteness: An optimum solution
cannot be computed in the situations
where there are infinite number of
alternative activities and resource
restrictions.
 Optimality: In linear programming, the
maximum profit solution or the
minimum cost solution always occur at
the corner point of the set of feasible
solutions.
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11. Formulations of Linear
Programming Problem
 formulation is the process of converting
the verbal description and numerical
data into mathematical expressions
which represents the relevant
relationship among decision factors,
objectives and restrictions on the use of
resources
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12. General form of LPM
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13. 11/9/2018 Mekuria B.

14. The basic steps in formulating a linear
programming model are as follows-
Step I. Identification of the Decision
Variables
Step II. Identification of the Constraints
Step III. Identification of the Objective
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15.  Example 1:-A firm is engaged in
producing two products, A and B. Each
unit of product A requires 2 kg of raw
material and 4 labour hours for
processing, whereas each unit of
product B requires 3 Kg of raw material
and 3 hours of labour, of the same type.
Every week, the firm has an availability
of 60 kg of raw material and 96 labour
hours. One unit of product A sold yields
Birr 40 and one unit of product B sold
gives Birr 35 as profit.
 Formulate this problem as a linear11/9/2018

16. Example 2:-
High Quality furniture Ltd. manufactures two
products, tables and chairs. Both the
products have to be processed through two
machines:-Ml andM2.The total machine-
hours available are 200 hours of M1 and 400
hours of M2 respectively.
 Time in hours required for producing a chair
and a table on both the machines is as
follows:
Time in hours
Machine TableChair
M1 7 4
M2 5 5
 Profit from Sale of table is Br.50 and sale of
a chair is Br. 30
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17. Example 3
A Juice Company has available two kinds of
food Juices: Orange Juice and Grape Juice.
The company produces two types of
punches: Punch A and Punch B. One bottle
of punch A requires 20 liters of Orange Juice
and 5 liters of Grape Juice. One bottle of
punch B requires 10 liters of Orange Juice
and 15 liters of Grape Juice.
 From each of bottle of Punch A profit of $4
is made and from each bottle of Punch B a
profit of $3 is made .Suppose that the
company has 230 liters of Orange Juice
and 120 liters of Grape Juice available
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18. Example 4:-
The Agricultural Research institute suggested to a
farmer to spread out at least 4800kg of a special
phosphate fertilizer and not less than 7200 kg of a
special nitrogen fertilizer to raise productivity of crops in
his fields. There are two sources for obtaining these-
mixtures A and B. Both of these are available in bags
weighing 100kg each and they cost Birr 40 and Birr 24
respectively. Mixture A contains phosphate and nitrogen
equivalent of 20 kg and 80 kg respectively, while
mixture B contains these ingredients equivalent of 50 kg
each.
Write this as a linear programming
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19. Example: 5
Suppose that a machine shop has two
different types of machines; machine 1 and
machine 2, which can be used to make a
single product . These machine vary in the
amount of product produced per hr., in the
amount of labor used and in the cost of
operation.
Assume that at least a certain amount of
product must be produced and that we
would like to utilize at least the regular labor
force. How much should we utilize each
machine in order to utilize total costs and still
meets the requirement?11/9/2018

20. ________________________________________________________________
Resourceused
Machine1(X1) Machine(X2) Minimum requiredhours
_____________________________________________________________________
Productproduced/hr 20 15 100
Labor/hr 2 3 15________
OperationCost $25 $30_____________________________
11/9/2018 Mitku A.
Formulate Linear Programming Model

21.  Example 6
A company owns two flour mills (A and B) which
have different production capacities for HIGH,
MEDIUM and LOW grade flour. This company
has entered contract supply flour to a firm every
week with 12, 8, and 24 quintals of HIGH,
MEDIUM and LOW grade respectively. It costs
the Co. $1000 and $800 per day to run mill A and
mill B respectively. On a day, mill A produces 6,
2, and 4 quintals of HIGH, MEDIUM and LOW
grade flour respectively. Mill B produces 2, 2 and
12 quintals of HIGH, MEDIUM and LOW grade
flour respectively. How many days per week
should each mill be operated in order to meet the
contract order most economically standardize?
Formulate Linear Programming Model
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22. Methods of Solving Linear Programming
Problems
 They can be solved by graphic method or by applying
algebraic method, called the simplex Method.
 The graphic approach is restricted in application that it can
be used only when two variables are involved.
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23. Graphical Method
 The graphic solution procedure is one
of the methods of solving two variable
linear programming problems. It
consists of the following steps:-
 Step I: Defining the problem.
 Step II: Plot the constraints graphically.
 Step III: Locate the solution space
Use corner point method
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24.  Example 1:-A firm is engaged in
producing two products, A and B. Each
unit of product A requires 2 kg of raw
material and 4 labour hours for
processing, whereas each unit of
product B requires 3 Kg of raw material
and 3 hours of labour, of the same type.
Every week, the firm has an availability
of 60 kg of raw material and 96 labour
hours. One unit of product A sold yields
Birr 40 and one unit of product B sold
gives Birr 35 as profit.
 Formulate this problem as a linear11/9/2018

25. Example 2.4: A firm that assembles computer and computer equipment is about to start
production of two new microcomputers. Each type of microcomputer will require assembly time,
inspection time and storage space. The amount of each of these resources that can be devoted to
the production of the microcomputers is limited. The manager of the firm would like to
determine the quantity of each microcomputer to produce in order to maximize the profit
generated by the seal of these microcomputers.
Additional information
In order to develop a suitable model of the problem, the manager has met with design and
manufacturing personnel. As the result of those meetings, the manager has obtained the
following information:
Type 1 Type 2
Profit per unit $60 $50
Assembly time per unit 4 hours 10 hours
Inspection time per unit 2 hours 1 hours
Storage space per unit 3 cubic feet 3 cubic feet
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26. The manager has also acquired information on the availability of company resources. These
amounts are:
Resource Amount available
Assembly time 100 hours
Inspection time 22 hours
Storage space 39 cubic feet
The manager also met with the firm’s marketing manager and learned that demand for the
microcomputers was such that whatever combination of these two types of microcomputers is
produced, all of the outputs can be sold.
Required:
a. Formulate the LP model (problem)
b. Find the product mix so that the company can make a maximum profit.
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27. Example2.5
Minimize
Subjectto
Z=40x1 +24x2
20x1 +50x2 ≥4800
80x1 +50x2 ≥7200
x1,x2≥0
TotalCost
PhosphateRequirement
NitrogenRequirement
Herethe decision variables x1 andx2 represent, respectively, the number of bags of mixture A
andofmixtureB,tobebought.
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28.  Example 6
A company owns two flour mills (A and B) which
have different production capacities for HIGH,
MEDIUM and LOW grade flour. This company
has entered contract supply flour to a firm every
week with 12, 8, and 24 quintals of HIGH,
MEDIUM and LOW grade respectively. It costs
the Co. $1000 and $800 per day to run mill A and
mill B respectively. On a day, mill A produces 6,
2, and 4 quintals of HIGH, MEDIUM and LOW
grade flour respectively. Mill B produces 2, 2 and
12 quintals of HIGH, MEDIUM and LOW grade
flour respectively. How many days per week
should each mill be operated in order to meet the
contract order most economically standardize?
Formulate Linear Programming Model
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29. Some Special Cases of Linear
Programming
1. Multiple Optimal Solutions: - Situations arise
when there are more than one optimal solution.
2. Unbounded solution: - This is a situation that
occurs whenever a maximization LPP has
unbounded solution space. No finite solution can
be determined.
3. Infeasible Solution: - This indicates that
there may be a solution which can satisfy each
constraint but there is no feasible solution.
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30. Multiple Optimal Solutions
Example: Solve the following linear programming problem.
0,
3
3605020
2
9
110X110Z
21
2
21
1
21
21






XX
x
xx
X
XX
toSubject
XMaximize
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31. Unbounded solution
Example:-solvethefollowinglinearprogrammingproblemusinggraphicmethod.
MaximizeZ=3x1+2x2
Subjectto: x1-x2≥1
x1+x2≥3
x1,x2≥0
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32. Infeasible Solution
Example: - solve the following linear programming problem
using graphic method.
0,
81
5
46Zmax
21
21
21




xx
x
xx
toSubject
xximize
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33. Will be Continued -------
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