Published on July 10, 2014
QUANTATIVE TECHNIQUES II: QUANTATIVE TECHNIQUES II PowerPoint Presentation: REAL – LIFE PROBLEM QUANTATIVE ANALYSIS QUALITATIVE ANALYSIS PROBLEM ANALYSIS FINAL DECISION DECISION - MAKING PROCESS HISTORY OF OR: HISTORY OF OR OR – During World War – II – need to manage scarce resources. OR – result of research on military operations - cause it involved strategic and tactical problems. The economic and industrial boom after World War II – Mechanization, automation, decentralization of operation and division of management functions. Features of OR Approach: Features of OR Approach Interdisciplinary Approach Methodological Approach Wholistic Approach Objective Approach Definition: Definition OR is the application of the methods of science to complex problems in the direction and management of large system of men, machines, materials and money in industry, business, government and defense. The distinctive approach is to develop a scientific model of the system incorporating measurements of factors such as chance and risk, with which to predict and compare the outcomes of alternative decisions, strategies or controls. The purpose is to help management in determining its policy & actions scientifically -- Operation Research Society - UK PowerPoint Presentation: OR is concerned with scientifically deciding how to best design and operate man-machine systems usually requiring the allocation of scarce resources. -- Operation Research Society, America PHASES IN OR: PHASES IN OR JUDGEMENT PHASE. RESEARCH PHASE. ACTION PHASE. JUDGEMENT PHASE: JUDGEMENT PHASE Identification of real-life problem. Selection of an appropriate objective & the values of various variable related to this objective Application of the appropriate scale of measurement Formulation of appropriate model of the problem, abstracting the essential information to obtain the decision’s makers goals. RESEARCH PHASE: RESEARCH PHASE Observation & data collection for better understanding of the problem Formulation of hypothesis and model Experimentation to test the hypothesis on basis of additional data. Analysis of the available information & verification of the hypothesis using pre-established measure of desirability Prediction of various result from the hypothesis Generalization of the result & consideration of alternative methods. ACTION PHASE: ACTION PHASE Making recommendation for implementing the decision by an individual who is an the position to implement result Advantages: Advantages Structured approach to problems Critical approach to problem solving. Shortcomings: Shortcomings Solutions are derived by making simplified assumptions so the solutions have limitations. Sometime Models do not represent the realistic situations in which decisions must be made. Decision maker is not fully aware of the limitations of the model. Many real world problems just cannot have an OR solution. Opportunities: Opportunities It compels the decision maker to quite explicit about the objective, assumption and his perspective to the constraints. Variables which influence decisions are considered. Gaps in data required to support solutions to a problem. Management of time because models can be solved by a computer. Application & Scope of OR: Application & Scope of OR Finance & Accounting Marketing Purchasing, procurement and Exploration Production Management Manufacturing Maintenance and Project scheduling Personnel Management MIS & General Management Government. LINEAR PROGRAMMING MODEL: LINEAR PROGRAMMING MODEL Linear Programming Model Formulation: Linear Programming Model Formulation Decision – making environment model formulation is important because it represents the essence of business decision problems. Formulation is the process of converting the verbal description and numerical data into mathematical expressions which represents the relevant relationship among decision factors, objectives and restriction on the use of resources. PowerPoint Presentation: Linear programming (LP) is the OR technique for economic allocation of ‘scare’ or ‘limited’ resources such as man, machine, capital, space to several competing activities such as product, service, new technique, project on the basis of a given criterion of optimality. Criterion of optimality is either performance, return on investment,, profit, cost, utility, time, distance, etc.. PowerPoint Presentation: Linear programming – ‘Linear’ refers to linear relationship among the variables in a model. ‘Programming’ is modeling & solving a problem mathematically that involves the economic allocation of limited resources by choosing a particular course of action or strategy among various alternative strategies to achieve the desired objective STRUCTURE OF LP MODEL: STRUCTURE OF LP MODEL The general structure of LP Model consist of three components. Decision Variable The Objective function The constraints DECISION VARIABLE: DECISION VARIABLE Activities are needed to evaluate various alternatives (course of action) for arriving at the optimal value of objective function. Evaluation of various alternatives is guided by the nature of objective function and availability of resources. Activities are denoted by x 1 ,x 2 ,x 3 ….,x n . The value of these activities represents the extent to which each of these is performed Example: In a product mix manufacturing, the management use LP to decide the how many units of each of the product to manufacture by using its limited resources such as personnel, machinery, money, material etc PowerPoint Presentation: These activities are also known as decision variables because they are under the decision-makers control Decision variables are interrelated in terms of consumption of limited resources which are required for simultaneous solutions. Decisions variables are continuous, controllable and non-negative. x 1 >=0,x 2 >=0,x 3 >=0….,x n >=0 THE OBJECTIVE FUNCTION: THE OBJECTIVE FUNCTION LP model is a mathematical representation of objective in terms of a measurable quantity such as profits, cost, revenue, distance etc., The general form is Optimize (Max or Min) Z is measure of performance variable which is a function of x 1 ,x 2 ,x 3 ….,x n , quantities c 1 ,c 2 ,c 3 ….,c n are the parameters that represent the contribution of a unit of the respective variables x 1 ,x 2 ,x 3 ….,x n to measure of performance Z. THE CONSTRAINTS: THE CONSTRAINTS The limitations on the use of resources, e.g. man, machine, raw material, space, money etc., that limits the degree to which objective can be achieved. These constraints must be expressed as linear equalities or inequalities in terms of decision variables. The solution of an LP model must satisfy these constraints. ASSUMPTIONS OF LP: ASSUMPTIONS OF LP Certainty All the model parameters such as availability of resources or cost contribution of a unit of decision variable and consumption of resources by a unit of decision variable must be known and may be constant. These may be random variable represented by a known distribution or may tend to change, then the problem can be solved by a stochastic LP model or parametric programming PowerPoint Presentation: Divisibility (Continuity) The solution values of decision variables and resources are assumed to have either whole number (integer) or mixed numbers ( integer & fraction). IF integer variables are desired, e.g. machines, employees etc., the integer programming method may be applied to get the desired result PowerPoint Presentation: Additivity The value of the objective function for the given values of Decision variable and the total sum of resources used, must be equal to the sum of the contribution (profit or cost) earned from each decision variable and the sum of the resources used by each decision variable, respectively. PowerPoint Presentation: Linearity (Proportionality) All relationships in the LP model must be linear, i.e. both Objective function and constraints. For any decision variable j, the amount of resource i used and its contribution to the cost in the objective function must be proportional to its amount. ADVANTAGES OF LP: ADVANTAGES OF LP Helps in attaining the optimum use of productive resources. It also indicates how decision-maker can employ his productive factors effectively by selecting & allocating these resources. Quality of decision is improved because the user is more objective & less subjective. LP techniques provide possible and practical solution since there might be other constraints operating outside the problem which must be taken into account. It highlights bottlenecks in the production process It helps in re-evaluation of basic plan for changing conditions. If conditions change when the plan is partly carried out, remainder of the plan can be adjusted for best results. LIMITATIONS: LIMITATIONS Linear programming treats all relationships among decision variables as linear. There is no guarantee that we get integer valued solutions. It does not take into consideration the effect of time and uncertainty. Large-scale problems can be solved with LP technique with assistance of computers Parameters in the model are assumed to be constant. It deals with single objective Applications: Applications Production Management Financial Management Marketing Management Personnel Management Military Application Agriculture Application General Mathematical LP MODEL: General Mathematical LP MODEL The general LP model with n decision variables and m constraints is: Find the values of decision variable x 1 ,x 2 ,x 3 ….,x n so as to Optimize( Max or Min) Subjected to the linear constraints, a m1 x 1 + a m2 x 2 +………+a mn x n ( x 1 ,x 2 ,x 3 ….,x n 0 a 11 x 1 + a 12 x 2 +………+a 1n x n ( a 21 x 1 + a 22 x 2 +………+a 2n x n ( ) b2 ) bm ) b1 . . . . . ( Non – negativity condition) Constraints STEPS IN LP MODEL FORMULATION: STEPS IN LP MODEL FORMULATION Identify the decision variables. Identify the problem data. Formulate the constraints. Formulate the objective Function Identify the decision variables: Identify the decision variables Express each constraints in words. See whether the constraints is of the form ≥ (at least ,as large as) or of the form ≤ (no larger than) or = (exactly equal to). Then express the objective function verbally. Above steps should allow to verbally identify the decision variable. If there are several decision alternatives, then identify the decision variables by asking “what decision must be made in order to optimize the objective function?” Identify the problem data: Identify the problem data For solving a problem, actual values has to be provided for the decision variables identified. These quantities constitute the problem data. Identify the problem data Express the constraints verbally in terms of requirements and availability of each resource. Convert the verbal expression of the constraints imposed by the resources availability as linear equality or inequality in terms of the decision variables defined in step 1 Formulate the objective Function : Formulate the objective Function Identify if the objective function is to be minimized or maximized. Then Express it verbally such as, maximize total profit/ min cost and then convert it into a linear mathematical expression in terms of decision variables multiplied by their profit or cost contribution.