business mathematics_special class

8/3/2019 Business Mathematics_Special Class

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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


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


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)


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

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