Session 1 Intro&LPP

7/30/2019 Session 1 Intro&LPP

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Quantitative Methods for Decision Making

Course Credit: 3Objective :This course is designed to expose thestudents to various managerial decision

making that emerge in the business and tohelp them understand how tostructure/formulate the decision problems aswell as to obtain optimum practical solutions

for the same.


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Topics that we will cover (refer courseoutline)

Linear Programming:-Formulations- Graphical Solution- Interpretation of the solution- The Simplex Method of solving Linear Programming

problem-Duality and Sensitivity Analysis.

Transportation and Assignment Problems

Replacement Problem Decision Theory Game Theory Simulation


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Operations Research

Origin During World War-II when theBritish Military asked Scientist to analyzemilitary problems

The applications of mathematics andscientific methods to military applicationswas called as Operations Research

Today it is also called as ManagementScience


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Analysis Phase of Decision-Making ProcessQualitative Analysis based largely on the managers judgment and

experience includes the managers intuitive feel for the

problem is more of an art than a science

Analysis and Decision Making


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Analysis Phase of Decision-Making ProcessQuantitative Analysis analyst will concentrate on the quantitative facts

or data associated with the problem analyst will develop mathematical expressions

that describe the objectives, constraints, andother relationships that exist in the problem

analyst will use one or more quantitativemethods to make a recommendation

Quantitative Analysis and Decision Making


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Quantitative Analysis and

Decision Making Potential Reasons for a Quantitative

Analysis Approach to Decision Making The problem is complex. The problem is very important.

The problem is new. The problem is repetitive.


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Quantitative Analysis

Quantitative Analysis Process Model Development Data Preparation Model Solution Report Generation


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Model Development

Models are representations of real objects or situations Three forms of models are:

Iconic models - physical replicas (scalar representations) of real objects Analog models - physical in form, but do not

physically resemble the object being modeled Mathematical models - represent real world

problems through a system of mathematicalformulas and expressions based on key

assumptions, estimates, or statistical analyses


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Advantages of Models

Generally, experimenting with models(compared to experimenting with the realsituation): requires less time is less expensive involves less risk

The more closely the model represents thereal situation, the accurate the conclusionsand predictions will be.


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Mathematical Models Objective Function a mathematical expressionthat describes the problems objective, such as

maximizing profit or minimizing cost Constraints a set of restrictions or limitations,

such as production capacities Uncontrollable Inputs environmental factors

that are not under the control of the decision

maker Decision Variables controllable inputs; decisionalternatives specified by the decision maker,such as the number of units of Product X to

produce


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Mathematical Models

Deterministic Model if all uncontrollable inputs to themodel are known and cannot varyStochastic (or Probabilistic) Model if any uncontrollableare uncertain and subject to variationStochastic models are often more difficult to analyze.


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Mathematical Models

Cost/benefit considerations must be made in selectingan appropriate mathematical model.Frequently a less complicated (and perhaps lessprecise) model is more appropriate than a morecomplex and accurate one due to cost and ease of

solution considerations.


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Transforming Model Inputs into Output

Uncontrollable Inputs(Environmental Factors)

ControllableInputs

(DecisionVariables)

Output(Projected

Results)

MathematicalModel


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Linear Programming

First conceived by Dantzig in 1947 in hisresearch paper Programming in Linear Structure

Koopmans Coined the Term Linear Programming in 1948

Simplex Method was published in 1949 byDantzig


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Linear Programming Formulation

Lets take a problem (Product Mix Problem):

A firm is engaged in producing 2 types of products A & B.

Each unit of product A requires 2kgs of RM and 4 LH.Each unit of product B requires 3kgs of RM and 3 LH of same type.The firm has an availability of 60kgs of RM and 96LH in a week.One unit of product A & B yields Rs. 40 and Rs. 35 as profit

respectively.Formulate this problem as LPP to determine as to how many units of each of the products to be produced /week so that the firm can earnmaximum profit.


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Terminology

Decision Variable Objective Function (that we are supposed

to be optimized) Constraints Non-negativity restrictions

The process of developing algebraicrepresentation (mathematical model) isfinally called as Problem Formulation


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Problem Formulation has followingsteps:

Identify the Decision Variable Writing the Objective Function

Writing the Constraints Writing the NN ConstraintsIn the given example the Objective Function

and the Constraints are linear- so it is alinear programming model


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So for a linear programming problem has- A linear Objective Function

- Linear Constraints- All the decision variables are non negative


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Assumptions of LP model

Proportionality or linearity Additivity Continuity or divisibility Certainty

Finite choices


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Linear Programming Formulation

Lets take another problem (Product MixProblem)-Minimization Case:

A agricultural research institute suggested to a farmer to spread out at least4800 kgs of phosphate fertilizer not less than 7200 kgs of a specialnitrogen fertilizer to raise productivity. Two sources of obtaining these-Mixture A & B. Both of these are available in in bags of 100 kgs andcosting Rs. 40 and 24 respectively.

Mixture A contains phosphate fertilizer and nitrogen fertilizer with a ratio of 20:80 while mixture B contains with a ratio of 50:50.

Formulate this problem as LPP to determine as to how many bags of eachtype the farmer should buy in order to obtain the required fertilizer atminimum cost.

Session 1 Intro&LPP

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