Linear+programming+basics+part+1
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    Linear+programming+basics+part+1 Linear+programming+basics+part+1 Document Transcript

    • Applied Operations Research Linear Programming Definition: LP is a mathematical modelling technique that helps in resource allocation decisions, and composed of linear equations and inequalities. Uses:  Advertising Budget allocation  Resource allocation in Production Processes  Manpower Planning etc. Example for Reference: A company produces Tables and chairs, each table requires 4 hours of carpentry work and 2 hours of painting. And each chair requires 3 hours of carpentry work and 1 hour of painting. Total carpentry hours available are 240 hours and 100 hours of painting time are available. Profit contribution of each table is $ 7 and $ 5 for each chair. Formulate a LPP keeping in consideration profit maximization. Requirements: Problems seek to maximize or minimize an objective, subject to some constraints: Following are the requirements of the linear system in Operations Research: 1. Objective / Purpose –an objective or purpose defines “why” we are going to solve the problem.(it can be maximization of profit, minimization of cost etc)  It should be understandable  Mathematically expressible , and  Linear in nature 2. Constraints –constraints are the conditions that stop us from undertaking the activity after a certain point. (it can be labor availability constraint, raw material constraint and power constraint etc ex: if raw material is available to make only 100 cars then availability of raw material stops you from producing more then 100 cars)  It should be understandable  Mathematically expressible , and  Linear in nature Hemant Sharma, Assistant Professor, RIMT-SMS Page 1
    • 3. There must be alternatives available 4. As expressed in points (1) and (2) all the expressions must be LINEAR in nature -------------------------------------------------------------------------------------------------------------------------- Steve asked- “What holds a Car together and prevents it from breaking?” John said –“The SCREWS” Assumptions of a Linear System: There are few Properties that we assume to hold true in case of a linear system: These properties are as follows: 1) Certainty 2) Proportionality 3) Additive nature 4) Divisibility 5) Non-negativity Question: Why there is a need to assume the above mentioned properties? Answer: To solve a linear system, we need to perform some mathematical operations (addition, subtraction, multiplication and division), these assumptions are made to: Make sure that the System remains linear while these operations are being performed (to be continued) Example for reference: A Company manufactures Tables and Chairs, Each table requires 4 hours of carpentry time and 2 hours of painting time and each chair requires 3 hours of carpentry and 1 hour of painting time. During the present production cycle, 240 hours of carpentry time and 100 hours of painting time are available. Profit contribution of each table is $ 7 and $ 5 for each chair. Formulate a LPP keeping in consideration profit maximization. Hemant Sharma, Assistant Professor, RIMT-SMS Page 2
    • Explanation of Assumptions 1) Certainty : - It means that the coefficients of all the equations and inequalities are known and remain constant throughout. i.e.- if in the beginning we were told that to make a table we need 4 hours of carpentry and 2 hours of painting , then certainty means that no matter how many tables we make, we make them in the beginning of the production cycle or in the end of the cycle each table will take 4 hours of carpentry time and 2 hours of painting time. Certainty says that we need to maintain the process of production as decided in the beginning 2) Proportionality: It means that all the resources and outcomes vary in a constant proportion (as decided earlier), and there is NO ECONOMIES of scale operating here. i.e. if we need 4 hours of carpentry and 2 hours of painting time to make one table then No of tables Carpentry hours required Painting hours required 2 4 hours/table X 2 tables = 2 hours/table X 2 tables = 8 hours 4 hours 3 4 hours/table X 3 tables = 2 hours/table X 3 tables = 12 hours 6 hours 6 4 hours/table X6 tables 2 hours/table X 6 tables = =24 hours 12 hours Proportionality makes sure that all the equations and inequalities together act as a linear system and there is dependence established (which is linear in nature) 3) Additive Nature:- (2+2=4) - it means that if 1 table gives profit of $ 7 then 2 tables will give profit of $ 14 and 3 tables will give a profit of $ 21 and so on.. i.e. there is NO DISCOUNTING allowed. The resources add up in a linear manner to give away outputs OUTPUT = EXACT sum of INPUTS (4 = 2 + 2) Hemant Sharma, Assistant Professor, RIMT-SMS Page 3
    • 4) Divisibility :- it means that the coefficients can assume fractional values, Why? Suppose we have a system with 2 linear equations 2x + 3y = 15 5/2 X (2x + 3y = 15) we are trying to eliminate “x” from these 5x + 2y = 21 -- 5x + 2y = 21 equations, so as to solve for “y” ------------------------------- 11/2 y = 33/2  “y” = 3 This operation can be performed only, if we allow the “coefficients of “x” and “y”” to assume fractional values, i.e. allow for division 5) Non-negativity :- implies that the Decision variables (here the number of tables and chairs) CANNOT take negative values i.e. Variables >= 0 (since the number of tables and chairs a company can produce CANNOT be negative) Please consult, in case of doubts You can mail me at sharma.hemant@ymail.com, or clarify in the class Hemant Sharma, Assistant Professor, RIMT-SMS Page 4