Quantitative Techniques

Quantitative Quantitative TechniquesTechniques

Deepthy Sai Manikandan

Topics:Topics: Linear ProgrammingLinear Programming Transportation ProblemTransportation Problem Assignment problemAssignment problem Queuing TheoryQueuing Theory Decision TheoryDecision Theory Inventory ManagementInventory Management SimulationSimulation Network AnalysisNetwork Analysis

LINEAR LINEAR PROGRAMMINGPROGRAMMING

Linear ProgrammingLinear Programming It is a mathematical technique for It is a mathematical technique for

optimum allocation of scarce or optimum allocation of scarce or limited resources to several limited resources to several competing activities on the basis competing activities on the basis of given criterion of optimality, of given criterion of optimality, which can be either performance, which can be either performance, ROI, cost, utility, time, distance ROI, cost, utility, time, distance etc.etc.

StepsSteps Define decision variablesDefine decision variables Formulate the objective functionFormulate the objective function Formulate the constraintsFormulate the constraints Mention the non-negativity Mention the non-negativity

criteriacriteria

Components & Components & AssumptionsAssumptions ObjectiveObjective Decision VariableDecision Variable ConstraintConstraint ParametersParameters Non-negativityNon-negativity

ProportionalityProportionality AddivityAddivity DivisibilityDivisibility CertainityCertainity

Problem:Problem:

An animal feed company must An animal feed company must produce at least 200 kgs of a mixture produce at least 200 kgs of a mixture consisting of ingredients x1 and x2 consisting of ingredients x1 and x2 daily. x1 costs Rs.3 per kg. and x2 daily. x1 costs Rs.3 per kg. and x2 Rs.8 per kg. No more than 80 kg. of Rs.8 per kg. No more than 80 kg. of x1 can be used and at least 60 kg. of x1 can be used and at least 60 kg. of x2 must be used. Formulate a x2 must be used. Formulate a mathematical model to the problem.mathematical model to the problem.

Solution:Solution:

Minimize Z = 3x1 + 8x2Minimize Z = 3x1 + 8x2Subject to x1 + x2 >= 200Subject to x1 + x2 >= 200

x1 <= 80x1 <= 80 x2 >= 60x2 >= 60

X1 >= 0 , x2 >= 0X1 >= 0 , x2 >= 0

Graphical SolutionGraphical Solution Formulate the problemFormulate the problem Convert all inequalities to equationsConvert all inequalities to equations Plot the graph of all inequalitiesPlot the graph of all inequalities Find out the feasilble regionFind out the feasilble region Find out the corner pointsFind out the corner points Substitute the objective functionSubstitute the objective function Arrive at the solution Arrive at the solution

Problem:Problem: Maximize Z = 60x1+50x2Maximize Z = 60x1+50x2 subject to 4x1+10x2 <= 100subject to 4x1+10x2 <= 100 2x1+1x2 <= 222x1+1x2 <= 22 3x1+3x2 <= 393x1+3x2 <= 39

x1,x2 >= 0x1,x2 >= 0

Solution :Solution : 4x1+10x2=1004x1+10x2=100 (0,10)(25,0)(0,10)(25,0) 2x1+x2=22 2x1+x2=22 (0,22)(0,22)

(11,0)(11,0) 3x1+3x2=39 3x1+3x2=39 (0,13)(13,0)(0,13)(13,0)

0

x2

x1

10

13

22

11

13 25

E

C

BA

D

A (0,0) = 60*0+50*0 = 0A (0,0) = 60*0+50*0 = 0B (11,0) = 60*11+50*0 = 660B (11,0) = 60*11+50*0 = 660C (9,4) = 60*9+50*4 = 740C (9,4) = 60*9+50*4 = 740D (5,8) = 60*5+50*8 = 700D (5,8) = 60*5+50*8 = 700E (0,10) = 60*0+50*10 = 500E (0,10) = 60*0+50*10 = 500

Max Z is at C (9,4) and Z = 740Max Z is at C (9,4) and Z = 740

Z = 60x1 + 50x2Z = 60x1 + 50x2

TRANSPORTATION TRANSPORTATION PROBLEMPROBLEM

Transportation Transportation ProblemProblem A special kind of optimisation A special kind of optimisation

problem in which goods are problem in which goods are transported from a set of sources to transported from a set of sources to a set of destinations subject to the a set of destinations subject to the supply and demand constraints. supply and demand constraints. The main objective is to minimize The main objective is to minimize the total cost of transportation. the total cost of transportation.

Initial Basic Feasible Initial Basic Feasible SolutionSolution North West Corner MethodNorth West Corner Method Least Cost MethodLeast Cost Method Vogel’s Approximation MethodVogel’s Approximation Method

The solution is said to be feasible The solution is said to be feasible when one gets (m+n-1) when one gets (m+n-1) allotments.allotments.

Assignment ProblemAssignment Problem It is a problem of assigning It is a problem of assigning

various people, machines and so various people, machines and so on in such a way that the total on in such a way that the total cost involved is minimized or the cost involved is minimized or the total value is maximized.total value is maximized.

QUEUING THEORYQUEUING THEORY

Queuing TheoryQueuing Theory A flow of customers from A flow of customers from

finite/infinite population towards finite/infinite population towards the service facility forms a queue the service facility forms a queue due to lack of capacity to serve due to lack of capacity to serve them all at a time. them all at a time.

Input Input Output OutputServer

MeasuresMeasures Traffic intensityTraffic intensity Average system lengthAverage system length Average queue lengthAverage queue length Average waiting time in queueAverage waiting time in queue Average waiting time in systemAverage waiting time in system Probability of queue lengthProbability of queue length

Queuing & cost Queuing & cost behaviorbehavior

Cost of service

Cost of waiting

Total cost

DECISION THEORYDECISION THEORY

Decision TheoryDecision Theory

The decision making environmentThe decision making environment Under certainityUnder certainity Under uncertainityUnder uncertainity Under riskUnder risk

Decision making under Decision making under uncertainityuncertainity Laplace CriterionLaplace Criterion Maxmin CriterionMaxmin Criterion Minmax CriterionMinmax Criterion Maxmax Criterion Maxmax Criterion Minmin CriterionMinmin Criterion Salvage CriterionSalvage Criterion Hurwicz CriterionHurwicz Criterion

Inventory Inventory managementmanagement Inventory is vital to the sucessful Inventory is vital to the sucessful

functioning of manufacturing and functioning of manufacturing and retailing organisations. They may retailing organisations. They may be raw materials, work-in-be raw materials, work-in-progress, spare progress, spare parts/consumables and finished parts/consumables and finished goods. goods.

ModelsModels

Deterministic Inventory ModelDeterministic Inventory Model Inventory Model with Price breaksInventory Model with Price breaks Probabilistic Inventory ModelProbabilistic Inventory Model

Basic EOQ ModelBasic EOQ Model

Slope=0 Total cost

Carrying cost

Ordering cost

Minimum total cost

Optimal order qty

SIMULATIONSIMULATION

SimulationSimulation It involves developing a model of It involves developing a model of

some real phenomenon and then some real phenomenon and then performing experiments on the performing experiments on the model evolved. It is descriptive in model evolved. It is descriptive in nature and not an optimizing nature and not an optimizing model. model.

ProcessProcess

Definition of the problemDefinition of the problem Construction of an appropriate Construction of an appropriate

modelmodel Experimentation with the modelExperimentation with the model Evaluation of the results of Evaluation of the results of

simulationsimulation

NETWORK ANALYSISNETWORK ANALYSISPERTPERTCPMCPM

A project is a series of activities A project is a series of activities directed to the accomplishment directed to the accomplishment of a desired objective.of a desired objective.

PERTPERT CPMCPM

Network Analysis / Network Analysis / Project ManagementProject Management

CPM-Critical Path CPM-Critical Path MethodMethod Activities are shown as a network Activities are shown as a network

of precedence relationship using of precedence relationship using Activity-On-Arrow (A-O-A) network Activity-On-Arrow (A-O-A) network construction.construction.

There is single stimate of activity There is single stimate of activity timetime

Deterministic activity timeDeterministic activity time

Project Evaluation & Project Evaluation & Review TechniqueReview Technique Activities are shown as a network Activities are shown as a network

of precedence relationships using of precedence relationships using A-O-A network construction.A-O-A network construction.

Multiple time estimatesMultiple time estimates Probabilistic activity timeProbabilistic activity time

CrashingCrashing Crashing is shortening the activity Crashing is shortening the activity

duration by employing more duration by employing more resources.resources.

cost slope = Cc – Cn/ Tn - Tccost slope = Cc – Cn/ Tn - Tc