business mathematics_special class
TRANSCRIPT
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Kathmandu UniversityKathmandu UniversityMGTS 403: Engineering ManagementMGTS 403: Engineering Management
Decision Making
Lecture 6
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Management Science ProcessManagement Science Process
y Formulate the Problem
Construct a Mathematical Model
Test the Model
Derive a Solution from the Model
y Apply the Models Solution to the Real
System
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Types of DecisionsTypes of Decisions
y Routine Decisions delegation/rules based
Payroll processing
Reordering standard inventory
Paying suppliers
y Non-Routine Decisions
Unstructured situations
Usually more at higher level of management
Based on statistical decision making
Based on subjective decisions
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Types of DecisionTypes of Decision-- limitationlimitation
y Objective Rationality
Viewing all behavior alternatives
All consequences of choosing an alternative
Values assigned for singling out the alternative
y Bounded Rationality
Take only known factors
Time and resource constraint forever analysis
Choose a good enough criteria
Solution that satisfices rather than best
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Categories of Decision Making ToolsCategories of Decision Making Tools
Pay-off Table
State of Nature/Probability
N1 N2 Nj Nn
Alternative P1 P2 Pj Pn
A1 O11 O12 O1j O1n
A2 O21 O22 O2j O2n
Ai Ai1 Ai2 Aij Ain
Am Am1 Am2 Amj Aim
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Categories of Decision Making ToolsCategories of Decision Making Toolsy Decision Making under Certainty
Certain of the future states of nature Probabilityp1 of future N1 is 1.0 other future states has zero prob.
Choose alternative Ai that gives most favorable outcome Oij
Linear Programming, Queuing models, critical path model, Deterministic
Inventory models most widely used tools
y Decision Making under Risk Probabilities are known from experience and statistical data collection
Expected monetary value, Expected (opportunity or regret ) loss (EOL) used
y Decision Making under Uncertainty Can not know the probabilities of the outcome
Decision making is very difficult because of less information Some techniques are
x Criteron of optimism Maximax
x Criterion ofpessimism - Maximin
x Criterion of Regret (Loss) Minimax
x Criteria of Realism (Hurmiczs)
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Decision Making Under RiskDecision Making Under Risk
y ExpectedValue
y Decision Trees
y Queuing
y
Simulation
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Decision Making Under RiskDecision Making Under Risk
Expected Value criterion : Decision making Under Risk
N1: Dry Hole N2: Small Well N3: Big Well Expected Value
p1 = 0.6 p2 = 0.3 p3 = 0.1
A1: Don't drill 0 0 0 0A2: Drill alone -500,000 300,000 9,300,000 720,000
Subcontract (a3) 0 125,000 1,250,000 162,500
Decision
Alternatives
States of nature
=
= =
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Decision Making Under RiskDecision Making Under RiskExpected Value criterion : Decision making Under Risk
N1: Dry Hole N2: Small Well N3: Big Well Expected Value
p1 = 0.6 p2 = 0.3 p3 = 0.1
A1: Don't drill 0 0 0 0
A2: Drill alone -500,000 300,000 9,300,000 720,000
Subcontract (a3) 0 125,000 1,250,000 162,500
Decision
Alternatives
States of nature
=
= =
Expected Value criterion : Decision Tree
Decision Chance (Outcome) (Probability) Expected Value
node Aj node Nj Oij Pj Ei
No fire: (-200) 0.999 -199.8 +Fire (-200) 0.001 - 0.2 = -200
No fire: 0 0.999 -0 +
Fire -100,000 0.001 - 100 = -100
=
==
==
x
xx
xx
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Decision Making UncertaintyDecision Making UncertaintyCase 1: Maximax - Optimistic Criterion
High (N1) Moderate (N2) Low (N3) Nil (N4)
Expand (a1) 40,000 20,000 -30,000 -35,000 40,000
Build (a2) 65,000 30,000 -40,000 -75,000 65,000 65,000
Subcontract (a3) 25,000 10,000 -15,000 -20,000 25,000
Case 2: Maximin - Pessimistic Criterion
High (N1) Moderate (N2) Low (N3) Nil (N4)
Expand (a1) 40,000 20,000 -30,000 -35,000 -35,000
Build (a2) 65,000 30,000 -40,000 -75,000 -75,000Subcontract (a3) 25,000 10,000 -15,000 -20,000 -20,000 -20,000
States of natureDecision
Alternatives
Max.
Payoff
Max of the
max pay-off
Decision
Alternatives
States of nature Min.
Payoff
Max of the
min pay-off
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Decision Making UncertaintyDecision Making UncertaintyCase 3: Minimax - Regret Criterion
High (N1) Moderate (N2) Low (N3) Nil (N4)
Expand (a1) 25,000 10,000 15,000 15,000 25,000 25,000
Build (a2) 0 0 25,000 55,000 55,000
Subcontract (a3) 40,000 20,000 0 0 40,000
Case 4: Realism Criteron (Hurwicz Criterion)
High (N1) Moderate (N2) Low (N3) Nil (N4)
Expand (a1) 40,000 20,000 -30,000 -35,000 17,500
Build (a2) 65,000 30,000 -40,000 -75,000 23,000 23,000
Subcontract (a3) 25,000 10,000 -15,000 -20,000 11,500Cofficient of Optimism = 0.7 considered above
Hurwicz Criterion = (Max. Pay Off) + (1- )(Min. pay off)
DecisionAlternatives
States of nature Max.Regret
Mini maxRegrets
Decision
Alternatives
States of nature Hurwicz
Value
Measure of
realism
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Decision Making under CertaintyDecision Making under Certainty
y Linear ProgrammingOne of best known tools of ManagementScience
y Used to determine optimal allocation ofan organizations limited resources
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Linear Programming or LPLinear Programming or LP
y The word linear refers the existence oflinear relationship among the variables ina model developed to solve the problem
y LP is one the mathematical models whichis designed to allocate some limited orscarce resources to optimize a singlestated criterion
y Optimizing is either minimizing ormaximizing such as revenue, loss etc.
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Notations in Linear ProgrammingNotations in Linear Programming
y Linearity The relationship amongst the variables
representing different phenomenon are linearlyrelated
y Objective Function Linearly expressed function used to optimize theproblem
Mathematically
f(x1, x2, ., xn) = c1x1 +c2x2+ c3x3+ ..+cnxn
Where c1, c2 etc are constants and
x1, x2 are decision variables
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Notations in Linear ProgrammingNotations in Linear Programming
y Constraints Limitation on resources which are to be
allocated among competing activities
Resources may be raw material, human
resources, time, machinery etc. Expressed in linear equations or inequalities eg.
a11x1+ a12x2+ + a1n xn b1
a21x1+ a22x2+ + a2n xn b2
..an1x1+ an2x2+ + ann xn bn
Where a11, a12 etc are constants and
x1, x2 are decision variables
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Notations in Linear ProgrammingNotations in Linear Programmingy Non-negative Criteria
All the considering variables are assumed to benon-negative which means a decision variables,x1, x2 etc. are equal or greater than zero i.e.
.x1 0, x2 0, , xn 0
Where a11, a12 etc are constants andx1, x2 are decision variables
y Feasible Solutions Feasible region or area All those possible solutions, that can be worked
out under given constraints of aLP Problem arefeasible solution satisfy all non-negative criteria
Any solution, that optimizes the objectivefunction is the optimal feasible solution
Region or area covered by feasible solution is
referred as feasible region.
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Notations in Linear ProgrammingNotations in Linear Programmingy Formulating (or Statement) LP Problem
Generally written as:Optimizef(x1, x2, ., xn) = c1x1 +c2x2+ c3x3+ ..+cnxn
Subject to the constraints or s.t.
a11x1+ a12x2+ + a1n xn < or or = or b1
a21x1+ a22x2+ + a2n xn < or or = or b2..
an1x1+ an2x2+ + ann xn < or or = or bn
and x1 0, x2 0, , xn 0; non-negativity
y LP Problem are now solved using computer
software but the most used methods are
1.Graphical Method 2. Simplex Method
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Solutions steps for the LP ProblemsSolutions steps for the LP Problems
Identify the LPP
Formulate LPP
Convert linear constraints in
to equations
Draw Graphs
Obtain optimal solution
Evaluate the objective function
Interpret the result
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Notations in Linear ProgrammingNotations in Linear Programming1. Solve the following LP Problem graphically
Maximize f = 40x1+35x2Subject to the constraints
2x1+ 3x2 60
4x1+ 3x2 96
and x1 0, x2 0
2. Find the optimum solution using graphic methodMaximize Z = 40A+70BSubject to the constraints
3A+ 3B 365A+ 2B 50
2A+ 6B 60
A, B 0
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Linear ProgrammingLinear Programming
Objective Function
y $10 profit for selling a unit of X and $14for selling a unit of Y
Profit function:P
= 10X + 14 Ymax
im
ize
Constraints
y X requires 3 hours machining and 1 hourassembly and Y requires 2 hoursmachining and 2 hours assembly
3X + 2Y