ALAMELU VDFK1211

DEPT. OF FISHERIES MICROBIOLOGY

LP - problem of maximizing or minimizing alinear function subject to linear constraints.The constraints may be equalities orinequalities - maximizing profit orminimizing costs in business.

Developed by George B. Denting in 1947

LP - technique for making decisions undercertainty i.e.; when all the courses of optionsavailable to an organisation are known & theobjective of the firm along with its constraintsare quantified.

Rothschid and Balsiger – 1971 – to allocate thecatch of sock eye salmon in bristol bay

Sieger (1979) – to maximise catches of newengland otter trawl fishery subject to totalallowable catch, proc., and harvesting capacity

Application of LP to economic- envt systems –diverse ranging from forest manage. Envt qtymodels, petroleum refining, electric powergeneration to complex regional and nationalmodels for optimal utilization of water resources .

Due to fast paced devop. In math. Programmingtechiques – LP application both in fisheries andcoastal envts are few

Linear Programming is the analysis of problems inwhich a Linear function of a number of variables isto be optimized (maximized or minimized) whenwhose variables are subject to a number ofconstraints in the mathematical near inequalities.

From the above definitions, it is clear that:

(i) LP - is an optimization technique, where theunderlying objective is either to maximize theprofits or to minim is the Cost

(ii) It deals with the problem of allocation offinite limited resources amongst differentcompetiting activities in the most optimalmanner.

(iil) It generates solutions based on the featureand characteristics of the actual problem orsituation. Hence the scope of linear programmingis very wide as it finds application in such diversefields as marketing, production, finance &personnel etc.

(iv) Linear Programming has been highlysuccessful in solving the following types ofproblems :

(a) Product-mix problems (b) Investment planning problems (c) Blending strategy formulations and (d) Marketing & Distribution management.

(v) Even though LP has wide & diverse’ applications,yet all LP problems have the following properties incommon:

(a)The objective is always the same (i.e.; profitmaximization or cost minimization).

(b) Presence of constraints which limit the extent towhich the objective can be pursued/achieved.

(c) Availability of alternatives i.e.; different courses ofaction to choose from, and

(d) The objectives and constraints can be expressedin the form of linear relation.

(VI) Regardless of the size or complexity, all LPproblems take the same form

Objectives of business decisions frequentlyinvolve maximizing profit or minimizingcosts

Linear programming uses linear algebraicrelationships to represent a firm’s decisions,given a business objective, and resourceconstraints

Decision variables- mathematical symbols representing

levels of activity of an operation

• Objective function :

– a linear relationship reflecting the objective of businessdecisions

–most frequent objective of business firms is to maximizeprofit

–most frequent objective of individual operational units (suchas a production or packaging department) is to minimizecost

Constraints:

– a linear relationship representing a restriction on decision

making

Parameters - numerical coefficients and constants used in

the objective function and constraints

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.

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.

Two products: Chairs and Tables for the Auditorium

Decision: How many of each to make this month?

Objective: Maximize profit

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

Decision Variables:

T = Num. of tables to make

C = Num. of chairs to make

Objective Function: Maximize Profit

Maximize $7 T + $5 C

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)

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

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)

Graphing an LP model helps provide insightinto LP models and their solutions.

While this can only be done in twodimensions, the same properties apply to allLP models and solutions.

Feasible Region: The set of points thatsatisfies all constraints

Corner Point Property: An optimal solutionmust lie at one or more corner points

Optimal Solution: The corner point with thebest objective function value is optimal

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

Simplicity and easy way of understanding.

Linear programming makes use of availableresources

To solve many diverse combination problems

Helps in Re-evaluation process- linearprogramming helps in changing condition ofthe process or system.

LP - adaptive and more flexibilityto analyze the problems.

The better quality of decision is provided

LP - works only with the variables that arelinear.

The idea is static, it does not considerchange and evolution of variables.

Non linear function cannot be solved overhere.

Impossibility of solving some problemwhich has more than two variables ingraphical method.

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

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 ofdistribution available to an organisation for its finished product satminimum total cost of transportation or shipping from company'sgodown

Portfolio Selection - Selection of desired and specificinvestments 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

Linear Relationship

Constant Value of objective & Constraint Equations.

No Scope for Fractional Value Solutions.

Degree Complexity

Multiplicity of Goals

Flexibility