Linear programming is an optimization technique used to maximize or minimize a linear objective function subject to linear constraints. It involves defining variables, constraints, and an objective function in terms of those variables. The optimal solution is found by systematically considering all extreme points of the feasible region defined by the constraints. Linear programming has wide applications in fields like production, transportation, finance, and resource allocation.

1. ALAMELU V
DFK1211
DEPT. OF FISHERIES MICROBIOLOGY

2.  LP - problem of maximizing or minimizing a
linear function subject to linear constraints.
The constraints may be equalities or
inequalities - maximizing profit or
minimizing costs in business.
 Developed by George B. Denting in 1947
 LP - technique for making decisions under
certainty i.e.; when all the courses of options
available to an organisation are known & the
objective of the firm along with its constraints
are quantified.

3.  Rothschid and Balsiger – 1971 – to allocate the
catch of sock eye salmon in bristol bay
 Sieger (1979) – to maximise catches of new
england otter trawl fishery subject to total
allowable catch, proc., and harvesting capacity
 Application of LP to economic- envt systems –
diverse ranging from forest manage. Envt qty
models, petroleum refining, electric power
generation to complex regional and national
models for optimal utilization of water resources .
 Due to fast paced devop. In math. Programming
techiques – LP application both in fisheries and
coastal envts are few

4.  Linear Programming is the analysis of problems in
which a Linear function of a number of variables is
to be optimized (maximized or minimized) when
whose variables are subject to a number of
constraints in the mathematical near inequalities.
 From the above definitions, it is clear that:
 (i) LP - is an optimization technique, where the
underlying objective is either to maximize the
profits or to minim is the Cost
 (ii) It deals with the problem of allocation of
finite limited resources amongst different
competiting activities in the most optimal
manner.

5.  (iil) It generates solutions based on the feature
and characteristics of the actual problem or
situation. Hence the scope of linear programming
is very wide as it finds application in such diverse
fields as marketing, production, finance &
personnel etc.
 (iv) Linear Programming has been highly
successful in solving the following types of
problems :
 (a) Product-mix problems
 (b) Investment planning problems
 (c) Blending strategy formulations and
 (d) Marketing & Distribution management.

6.  (v) Even though LP has wide & diverse’ applications,
yet all LP problems have the following properties in
common:
 (a)The objective is always the same (i.e.; profit
maximization or cost minimization).
 (b) Presence of constraints which limit the extent to
which the objective can be pursued/achieved.
 (c) Availability of alternatives i.e.; different courses of
action to choose from, and
 (d) The objectives and constraints can be expressed
in the form of linear relation.
 (VI) Regardless of the size or complexity, all LP
problems take the same form

7.  Objectives of business decisions frequently
involve maximizing profit or minimizing
costs
 Linear programming uses linear algebraic
relationships to represent a firm’s decisions,
given a business objective, and resource
constraints

8.  Decision variables- mathematical symbols representing
levels of activity of an operation
• Objective function :
– a linear relationship reflecting the objective of business
decisions
– most frequent objective of business firms is to maximize
profit
– most frequent objective of individual operational units (such
as a production or packaging department) is to minimize
cost
 Constraints:
– a linear relationship representing a restriction on decision
making
 Parameters - numerical coefficients and constants used in
the objective function and constraints

9.  Step 1 : Clearly define the decision variables
 Step 2 : Construct the objective function
 Step 3 : Formulate the constraints
Linear programming requires that all the
mathematical functions in the model to be linear
functions.
◦ Conversion of stated problem into a linear mathematical
model which involves all the essential elements of the
problem.
◦ Exploration of different solutions of the problem.
◦ Finding out the most suitable or optimum solution.

10. Let: X1, X2, X3, ………, Xn = decision variables
Z = Objective function or linear function
Requirement: Maximization of the linear function Z.
Z = c1X1 + c2X2 + c3X3 + ………+ cnXn …..Eq (1)
subject to the following constraints:
…..Eq (2)
where aij, bi, and cj are given constants.

11. Two products: Chairs and Tables for the
Auditorium
Decision: How many of each to make this month?
Objective: Maximize profit

12. Tables
(per table)
Chairs
(per chair)
Hours
Available
Profit
Contribution
$7 $5
Carpentry 3 hrs 4 hrs 2400
Painting 2 hrs 1 hr 1000
Other Limitations:
• Make not more than 450 chairs
• Make at least 100 tables

13. Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize $7 T + $5 C

14.  Have 2400 hours of carpentry time available
3 T + 4 C < 2400 (hours)
 Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)

15. More Constraints:
 Make not more than 450 chairs
C < 450 (num. chairs)
 Make at least 100 tables
T > 100 (num. tables)
Nonnegativity:
Cannot make a negative number of chairs or
tables
T > 0
C > 0

16. Maximize Z = 7T + 5C
(profit)
Subject to the constraints:
3T + 4C < 2400(carpentry hrs)
2T + 1C < 1000(painting hrs)
C < 450(max # chairs)
T > 100 (min # tables)
T, C > 0
(nonnegativity)

17.  Graphing an LP model helps provide insight
into LP models and their solutions.
 While this can only be done in two
dimensions, the same properties apply to all
LP models and solutions.

18.  Feasible Region: The set of points that
satisfies all constraints
 Corner Point Property: An optimal solution
must lie at one or more corner points
 Optimal Solution: The corner point with the
best objective function value is optimal

19.  1. Decision or Activity Variables & Their Inter-Relationship.
 2. Finite Objective Functions – clearly defined, unambigous objective
 3. Limited Factors/Constraints – availability of machines, hours, labors
 4. Presence of Different Alternatives – should be present
 5. Non-Negative Restrictions – negative – no value – must assume
nonnegativity
 6. Linearity Criterion – decision variable – must be direct proportional
 7. Additivity –profit exactly equal to sum of all individal
 8. Mutually Exclusive Criterion – occurrence of one variable rules out
the simultaneous occur. Of such variable
 9. Divisibility. - factional values – need not be whole no.
 10. Certainty- relevant parameters – fully and completely known
 11. Finiteness – assume finite no. of activities or constraints – must –
w/o this – not possible for optimal solution

20.  Simplicity and easy way of understanding.
 Linear programming makes use of available
resources
 To solve many diverse combination problems
 Helps in Re-evaluation process- linear
programming helps in changing condition of
the process or system.
 LP - adaptive and more flexibility
to analyze the problems.
 The better quality of decision is provided

21.  LP - works only with the variables that are
linear.
 The idea is static, it does not consider
change and evolution of variables.
 Non linear function cannot be solved over
here.
 Impossibility of solving some problem
which has more than two variables in
graphical method.

22.  Plan Formulation – 5 year plan
 Railways – allocation site for rail route
 Agriculture Sector – crop rotation pattern, food crop, fertilizer
minimization
 Aviation Industry – allocation of air crafts for various routes
 Commercial Institutions – oil refineries – correct blending and
mixing of oil mix for improvement of final product
 Process Industries. - location of ware house and product mix –
paint industry
 Steel Industry – optimal combination for final products – bars,
plates, sheets
 Corporate Houses – distribution of goods for consumers
throughout the country

23.  Military Applications - selecting an air weapon system against the
enemy
 Agriculture. - farm economics and farm management. – allocating
scarce resources
 Environmental Protection - handling wastes and hazardous materials
 Facilities Location - location nonpublic health care facilities
 Product-Mix. - the existence of various products that the company
can produce and sell.
 Production. - will maximize output and minimize the costs.
 Mixing or Blending. - determine the minimum cost blend or mix
 Transportation & Trans-Shipment - the best possible channels of
distribution available to an organisation for its finished product sat
minimum total cost of transportation or shipping from company's

24.  Portfolio Selection - Selection of desired and specific
investments out of a large number of investment'
options
 Profit Planning & Contract - to maximize the profit
margin
 Traveling Salesmen Problem - problem of a salesman
to find the shortest route originating from a particular
city
 Staffing - allocating the optimum employees
 Job Analysis - evaluation of jobs in an organisation –
matching right job
 Wages and Salary Administration- Determination of
equitable salaries and various incentives and perks

25.  Linear Relationship
 Constant Value of objective & Constraint Equations.
 No Scope for Fractional Value Solutions.
 Degree Complexity
 Multiplicity of Goals
 Flexibility