Linear Programming Formulation Graphical Method by yrs83496

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									                                                                   RB/OR/LPP/CN/1
Linear Programming Formulation & Graphical Method

Aim: To optimally utilize the scarce resources- e.g. money, man power, machinery
etc.

LP is the most popular and widely accepted deterministic technique of
Mathematical programming.

The credit for its development goes to George B Dentzig in 1947.


Basic Terminology, Requirements, Assumptions of LP

The word linear stress on directly proportional relationship among the variables
while the world programming refers to the systematic procedure of solution.

Basic Requirements:
   1.   Decision variables and their relationship
   2.   Objective Function
   3.   Constraints
   4.   Alternative courses of actions
   5.   Non negativity restriction
   6.   Linearity

Basic Assumptions
   1. Proportionality : The amount of each resource used and associated
      contribution to profit (or cost) in the OF must be proportional.
   2. Divisibility: Continuous values of decision variables and resources must be
      permissible in obtaining optimal solution.
   3. Additivity: The total profitability and the total amount of each resource
      utilized must be equal to sum of the respective individual amounts.
   4. Certainty: All coefficients and parameters in LP model are known with
      certainty.

Advantages of LP
   1. LP helps in attaining the optimum use of productive factors.
   2. Improve the quality of decisions
   3. Provides possible and practical solutions and keeps room open for
      modification.
   4. Highlights the problem areas, bottlenecks and constraints operating upon the
      problem in hand.
                                                                    RB/OR/LPP/CN/2
Limitations of LP

  1. Treat all relationships leaner which usually does not happen in realty.
  2. Integral solution is not guaranteed.
  3. Does not take into consideration the effect of time and uncertainty.
  4. Sometimes, It is unable to solve large scale problems and demands for
     decomposition.
  5. Deals with only a single objective.

Application Areas of LPP

       Agriculture: Allocation of scarce resources-acreage, labour, water,
       capital…

       Military : Various optimizations pertaining to allocation, transportation,
       strategies or so.

       Production Management
          Product Mix
          Production Planning
          Assembly Line Balancing
          Blending Problems
          Trim Loss Prevention

       Financial Management
           Portfolio Selection
           Profit Planning

       Marketing Management
          Media Selection
          Traveling Salesman Problem
          Physical Distribution

       Personnel Management
           Staffing Problem
           Determination of Equitable Salaries
           Job Evaluation and Selection

Formulation of a LP Model
  1. Identify the decision variables and express them in algebraic symbols
  2. Identify all the constraints or limitations and express as equations
  3. Identify the Objective Function and express it as a linear function.
                                                                   RB/OR/LPP/CN/3


General Mathematical Formulation of LPP

Optimize (Maximize or Minimize)
     Z = c1 x1 + c2 x2+…+cn xn
Subject to:
     a11 x1 + a12 x2 +…+ a1n xn (<=, =, >= ) b1
     a21 x1 + a22 x2 +…+ a2n xn (<=, =, >= ) b2

      .
      .
      am1 x1 + am2 x2 +…+ amn xn (<=, =, >= ) bm

      and x1, x2, …xn >=0


The above formulation may also be expressed with the following notations:

Optimize (Max. or Min.)         z = Σ cj x j   for j = 1..n,
                                (Objective Function)
Subject to :
      Σ a ij xj (<=, =, >=) bi ; for j = 1 ..n, i = 1,2, …m
               (Constraints)
and xj   >=0 ; j= 1, 2, …, m
(Non negativity restrictions)

Some Definitions

   1. Solution: pertains to the values of decision variables that satisfies
      constraints
   2. Feasible solution: Any solution that also satisfies the non negativity
      restrictions
   3. Basic Solution: For a set of m simultaneous equations in n unknowns
      (n>m), a solution obtained by setting n- m of the variables equal to zero and
      solving the m equation in m unknowns is called basic solution.
   4. Basic Feasible solution: A feasible solution that is also basic.
   5. Optimum Feasible solution: Any basic feasible solution which optimizes
      the objective function
   6. Degenerate Solution: If one or more basic variable becomes equal to zero.
                                                                    RB/OR/LPP/CN/4
Steps for Graphical Solution

     1.   Define the problem mathematically
     2.   Graph by constraints by treating each inequality as equality.
     3.   Locate the feasible region and the corner points.
     4.   Find out the value of objective function at these points.
     5.   Find out the optimal solution and the optimal value of O.F.

Some special cases:

     1. Infeasible Solution
     2. Multiple optimum solution
     3. Unbounded solution


Simplex Method of Solution of LPP

  Steps:
          1.   Formulation of the mathematical model
          2.   Set up an initial solution
          3.   Test for Optimality (variable entry criteria)
          4.   Test for feasibility (variable leaving criteria)
          5.   Identify the key element
          6.   Determine the new solution
          7.   Revise the solution until optimization.

								
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