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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