LINEAR PROGRAMMING PROJECT

V.PAVITHRA SUKANYAH.V.K RIZWANA SULTANA SHILPA JAIN V.PAVITHRA

INTRODUCTION Modern technological advance growth of scientific techniques Operations Research (O.R.) recent addition to scientific tools O.R. new outlook to many conventional management problems Seeks the determination of best (optimum) course of action of a decision problem under the limiting factor of limited resources Modern technological advance growth of scientific techniques Operations Research (O.R.) recent addition to scientific tools O.R. new outlook to many conventional management problems Seeks the determination of best (optimum) course of action of a decision problem under the limiting factor of limited resources

Operational Research can be considered as being the application of scientific method by inter-disciplinary teams to problems involving the control of organized systems so as to provide solutions which best serve the purposes of the organization as a whole. WHAT IS OR?

CHARACTERISTIC NATURE OF OR Inter-disciplinary team approach Systems approach Systems approach Helpful in improving the quality of solution Helpful in improving the quality of solution Scientific method Scientific method Goal oriented optimum solution Goal oriented optimum solution Use of models Use of models Require willing executives Require willing executives Reduces complexity Reduces complexity Inter-disciplinary team approach Systems approach Systems approach Helpful in improving the quality of solution Helpful in improving the quality of solution Scientific method Scientific method Goal oriented optimum solution Goal oriented optimum solution Use of models Use of models Require willing executives Require willing executives Reduces complexity Reduces complexity

PHASES TO OR Judgment phase – Determination of the problem – Establishment of the objectives and values – Determination of suitable measures of effectiveness Research phase – Observation and data collection – Formulation of hypothesis and models – Observation and experimentation to test the hypothesis – Prediction of various results, generalization, consideration of alternative method Action phase – Implementation of the tested results of the model Judgment phase – Determination of the problem – Establishment of the objectives and values – Determination of suitable measures of effectiveness Research phase – Observation and data collection – Formulation of hypothesis and models – Observation and experimentation to test the hypothesis – Prediction of various results, generalization, consideration of alternative method Action phase – Implementation of the tested results of the model

METHODOLOGY OF OR Formulating the problem Constructing the model Deriving the solution – Analytical method – Numerical method – Simulation method Testing the validity Controlling the solution Implementing the result Formulating the problem Constructing the model Deriving the solution – Analytical method – Numerical method – Simulation method Testing the validity Controlling the solution Implementing the result

PROBLEMS IN OR Allocation Replacement Replacement Sequencing Sequencing Routing Routing Inventory Inventory Queuing Queuing Competitive Competitive Search Search Allocation Replacement Replacement Sequencing Sequencing Routing Routing Inventory Inventory Queuing Queuing Competitive Competitive Search Search

OR TECHINIQES Linear programming Waiting line or queuing theory Inventory control / planning Game theory Decision theory Network analysis – Program Evaluation and Review Technique – Critical Path Method (CPM) etc. Simulation Integrated production models Linear programming Waiting line or queuing theory Inventory control / planning Game theory Decision theory Network analysis – Program Evaluation and Review Technique – Critical Path Method (CPM) etc. Simulation Integrated production models

SIGNIFICANCE OF OR Provides a tool for scientific analysis Provides solution for various business problems Enables proper deployment of resources Helps in minimizing waiting and servicing costs Enables the management to decide when to buy and how much to buy? Assists in choosing an optimum strategy Renders great help in optimum resource allocation Facilitates the process of decision making Management can know the reactions of the integrated business systems. Helps a lot in the preparation of future managers. Provides a tool for scientific analysis Provides solution for various business problems Enables proper deployment of resources Helps in minimizing waiting and servicing costs Enables the management to decide when to buy and how much to buy? Assists in choosing an optimum strategy Renders great help in optimum resource allocation Facilitates the process of decision making Management can know the reactions of the integrated business systems. Helps a lot in the preparation of future managers.

LIMITATIONS OF OR The inherent limitations concerning mathematical expressions High costs are involved in the use of O.R. techniques O.R. does not take into consideration the intangible factors O.R. is only a tool of analysis and not the complete decision-making process Other limitations – Bias – Inadequate objective functions – Internal resistance – Competence – Reliability of the prepared solution The inherent limitations concerning mathematical expressions High costs are involved in the use of O.R. techniques O.R. does not take into consideration the intangible factors O.R. is only a tool of analysis and not the complete decision-making process Other limitations – Bias – Inadequate objective functions – Internal resistance – Competence – Reliability of the prepared solution

INTRODUCTION TO LINEAR PROGRAMMING Today many of the resources needed as inputs to operations are in limited supply. Operations managers must understand the impact of this situation on meeting their objectives. Linear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources. Today many of the resources needed as inputs to operations are in limited supply. Operations managers must understand the impact of this situation on meeting their objectives. Linear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources.

Linear programming We use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problems A linear program is simply another modeling abstraction (tool in your toolbox) Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (e.g. like we did for TSP) The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming, linear programming forming the foundation. We use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problems A linear program is simply another modeling abstraction (tool in your toolbox) Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (e.g. like we did for TSP) The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming, linear programming forming the foundation.

BASIC CONCEPT OF LP PROGRAM Objective function Constraints Optimization Solution of lpp. Feasible solution Optimal solution Objective function Constraints Optimization Solution of lpp. Feasible solution Optimal solution

LP PROBLEMS IN OM: PRODUCT MIX Objective To select the mix of products or services that results in maximum profits for the planning period Decision Variables How much to produce and market of each product or service for the planning period Constraints Maximum amount of each product or service demanded; Minimum amount of product or service policy will allow; Maximum amount of resources available Objective To select the mix of products or services that results in maximum profits for the planning period Decision Variables How much to produce and market of each product or service for the planning period Constraints Maximum amount of each product or service demanded; Minimum amount of product or service policy will allow; Maximum amount of resources available

 Objective function: the linear functions which is to be optimized i.e maximized or minimized this may be expressed in linear expression.  Solution of Lpp: The set of all the values of the variable x1,x2……xn which satisy the constraints is called the solution of Lpp.  Feasible solution: The set of all the values of the variable x1,x2……xn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp.  Objective function: the linear functions which is to be optimized i.e maximized or minimized this may be expressed in linear expression.  Solution of Lpp: The set of all the values of the variable x1,x2……xn which satisy the constraints is called the solution of Lpp.  Feasible solution: The set of all the values of the variable x1,x2……xn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp.

Recognizing LP Problems Characteristics of LP Problems in OM A well-defined single objective must be stated. There must be alternative courses of action. The total achievement of the objective must be constrained by scarce resources or other restraints. The objective and each of the constraints must be expressed as linear mathematical functions. Characteristics of LP Problems in OM A well-defined single objective must be stated. There must be alternative courses of action. The total achievement of the objective must be constrained by scarce resources or other restraints. The objective and each of the constraints must be expressed as linear mathematical functions.

Steps in Formulating LP Problems Define the objective. (min or max) Define the decision variables. (positive) Write the mathematical function for the objective. Write a 1- or 2-word description of each constraint. Write the right-hand side (RHS) of each constraint. Write for each constraint. Write the decision variables on LHS of each constraint. Write the coefficient for each decision variable in each constraint. Define the objective. (min or max) Define the decision variables. (positive) Write the mathematical function for the objective. Write a 1- or 2-word description of each constraint. Write the right-hand side (RHS) of each constraint. Write for each constraint. Write the decision variables on LHS of each constraint. Write the coefficient for each decision variable in each constraint.

Linear Programming An optimization problem is said to be a linear program if it satisfied the following properties: There is a unique objective function. Whenever a decision variable appears in either the objective function or one of the constraint functions, it must appear only as a power term with an exponent of 1, possibly multiplied by a constant. An optimization problem is said to be a linear program if it satisfied the following properties: There is a unique objective function. Whenever a decision variable appears in either the objective function or one of the constraint functions, it must appear only as a power term with an exponent of 1, possibly multiplied by a constant.

LP Problems in General Units of each term in a constraint must be the same as the RHS Units of each term in the objective function must be the same as Z Units between constraints do not have to be the same LP problem can have a mixture of constraint types Units of each term in a constraint must be the same as the RHS Units of each term in the objective function must be the same as Z Units between constraints do not have to be the same LP problem can have a mixture of constraint types

No term in the objective function or in any of the constraints can contain products of the decision variables. The coefficients of the decision variables in the objective function and each constraint are constant. The decision variables are permitted to assume fractional as well as integer values No term in the objective function or in any of the constraints can contain products of the decision variables. The coefficients of the decision variables in the objective function and each constraint are constant. The decision variables are permitted to assume fractional as well as integer values

Examples of lpp We are already familiar with the graphical representation of equations and inequations. here we describe the application of linear equations and inequations in solving different kinds of problems. The examples are stated below. Example 1: Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum. Step1: we have to choose the positive two numbers. Let the 2 positive numbers be x and y. this x and y are decision variables.

Step 2: our objective is to minimize the product x,y Let z=xy we have to maximize z Step3: we have the following conditions on the variables as x and y. step 4: x+y>=15 x-y<=7 x,y>0 as the linear constraints.the mathemetical constraint of this equation is to maximize the objective function z=xy.

PROBLEMS 1.A producer wants to maximise revenues producing two goods x 1and x2 in the market. Market prices of goods are 10 and 5 respectively. Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000. Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is How much should this firm produce x 1and x 2 in order to maximise the total revenue. Max R =10x 1 + 5x 2 Subject to: Skilled labour constraint: 25 x1 +10x 2 <=1000 Unskilled labour constraint: 20x 1 +50x 2 <=1500 Non-negativity constraints: x 1,x 2 >=0 1.A producer wants to maximise revenues producing two goods x 1and x2 in the market. Market prices of goods are 10 and 5 respectively. Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000. Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is How much should this firm produce x 1and x 2 in order to maximise the total revenue. Max R =10x 1 + 5x 2 Subject to: Skilled labour constraint: 25 x1 +10x 2 <=1000 Unskilled labour constraint: 20x 1 +50x 2 <=1500 Non-negativity constraints: x 1,x 2 >=0

SOLUTION Max R =10x 1 + 5x 2 Subject to: Skilled labour constraint: 25 x1 +10x 2 <=1000 Unskilled labour constraint: 20x 1 +50x 2 <=1500 Non-negativity constraints: x 1,x 2 >=0 Max R =10x 1 + 5x 2 Subject to: Skilled labour constraint: 25 x1 +10x 2 <=1000 Unskilled labour constraint: 20x 1 +50x 2 <=1500 Non-negativity constraints: x 1,x 2 >=0

Pounds of each alloy needed per frame Aluminum Alloy Steel Alloy Deluxe 2 3 deluxe Professional 4 2 Pounds of each alloy needed per frame Aluminum Alloy Steel Alloy Deluxe 2 3 deluxe Professional 4 2 Example: LP Formulation

Define the objective Maximize total weekly profit Define the decision variables x 1 = number of Deluxe frames produced weekly x 2 = number of Professional frames produced weekly Write the mathematical objective function  Max Z = 10x x 2 Define the objective Maximize total weekly profit Define the decision variables x 1 = number of Deluxe frames produced weekly x 2 = number of Professional frames produced weekly Write the mathematical objective function  Max Z = 10x x 2

Write a one- or two-word description of each constraint Aluminum available Steel available Write the right-hand side of each constraint Write for each constraint < 100 < 80 Write a one- or two-word description of each constraint Aluminum available Steel available Write the right-hand side of each constraint Write for each constraint < 100 < 80

Write all the decision variables on the left-hand side of each constraint x 1 x 2 < 100 x 1 x 2 < 80 Write the coefficient for each decision in each constraint + 2x 1 + 4x 2 < x 1 + 2x 2 < 80 Write all the decision variables on the left-hand side of each constraint x 1 x 2 < 100 x 1 x 2 < 80 Write the coefficient for each decision in each constraint + 2x 1 + 4x 2 < x 1 + 2x 2 < 80

LP in Final Form Max Z = 10x x 2 Subject To 2x 1 + 4x 2 < 100 ( aluminum constraint) 3x 1 + 2x 2 < 80 ( steel constraint) x 1, x 2 > 0 (non-negativity constraints LP in Final Form Max Z = 10x x 2 Subject To 2x 1 + 4x 2 < 100 ( aluminum constraint) 3x 1 + 2x 2 < 80 ( steel constraint) x 1, x 2 > 0 (non-negativity constraints

Example:graphical method

x 20 0 Example:graphical method

x y (5,5) Example:graphical method

x y (5,5)

x (5,5) Example:graphical method

x Example:graphical method

x (5,5) Example:graphical method

x y (5,5)

THE SIMPLEX METHOD

So far we find an optimal point by searching among feasible intersection points. The search can be improved by starting with an initial feasible point and moving to a “better” solution until an optimal one is found. The simplex method incorporates both optimality and feasibility tests to find the optimal solution(s) if one exists So far we find an optimal point by searching among feasible intersection points. The search can be improved by starting with an initial feasible point and moving to a “better” solution until an optimal one is found. The simplex method incorporates both optimality and feasibility tests to find the optimal solution(s) if one exists

An optimality test shows whether an intersection point corresponds to a value of the objective function better than the best value found so far. A feasibility test determines whether the proposed intersection point is feasible. The decision and slack variables are separated into two nonoverlapping sets, which we call the independent and dependent sets An optimality test shows whether an intersection point corresponds to a value of the objective function better than the best value found so far. A feasibility test determines whether the proposed intersection point is feasible. The decision and slack variables are separated into two nonoverlapping sets, which we call the independent and dependent sets

THE SIMPLEX METHOD Transform Linear Program into a system of linear equations using slack variables: Transform Linear Program into a system of linear equations using slack variables:

THE SIMPLEX METHOD

Start from the vertex (x=0, y=0) Move to the next vertex that increases profit as much as possible. Start from the vertex (x=0, y=0) Move to the next vertex that increases profit as much as possible. THE SIMPLEX METHOD

At (0,0), P = 0 Increasing x can increase P the most (x coefficient has larger magnitude than the y coefficient) Compute check ratios to find pivot row (smallest ratio) At (0,0), P = 0 Increasing x can increase P the most (x coefficient has larger magnitude than the y coefficient) Compute check ratios to find pivot row (smallest ratio)

Basic Idea: Start from a vertex (x=0, y=0) Move to next vertex that increases profit as much as possible Basic Idea: Start from a vertex (x=0, y=0) Move to next vertex that increases profit as much as possible At (0,0), P = 0 Increasing x can increase P the most (x coefficient has larger magnitude than the y coefficient) Compute check ratios to findpivot row (smallest ratio) Pivot around the element inboth pivot column and row At (0,0), P = 0 Increasing x can increase P the most (x coefficient has larger magnitude than the y coefficient) Compute check ratios to findpivot row (smallest ratio) Pivot around the element inboth pivot column and row

x y s1 s2 P RHS Pivoting means solve for that variable, Then substitute into the other equations Pivoting means solve for that variable, Then substitute into the other equations

x y s1 s2 P RHS Pivoting means solve for that variable, Then substitute into the other equations Pivoting means solve for that variable, Then substitute into the other equations

x y s1 s2 P RHS Pivoting means solve for that variable, Then substitute into the other equations Pivoting means solve for that variable, Then substitute into the other equations

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