quantitative techniques
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Useful for students to have a complete and brief notes about the quantitative techniques. Also helpful for those pursuing MBA.TRANSCRIPT
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