Prediction of real life phenomena
For model to be reliable, model validation is necessary
Judgement,
Experience
Simplify
Real World
Model
Revise
?
Model
Try different
simplifying
assumptions
Is performance,
prediction O.K?
No
Yes
Continue
using model
VARIETY OF MODELS
PICTORIAL
SCHEMATIC
Visual pictures, Cartoons,
Road signs
+
Organization chart with
authority relationships,
information flow, current flow
EXAMPLES OF SYMBOLIC MODELS
x
x
x
x
x
x
x
Ft = a + bt
FORECASTING
MODEL
Regression
(Descriptive)
J F M A M J
Month
J
actual Approximation, d
INVENTORY
MODEL
EOQ =2d Cordering/ iC
Time
Ordering Cost/annum = Cod /q
Carrying
Cost /annum
= q iC
2
Time
PRESCRIPTIVE MODEL
10
9
A
8
Profit
B
C
7
6
5
4
3
2
1
Sales
Representation
x1 6
x2 8
PROFIT = 80x1+ 40x2
D
O
E
1
2
x1 (desk)
3
4
5
6
Ideal
Capacity
7
8
9
Pt.
Profit
O(0,0) = 0
A(0,8) = 320
B(2,8) = 480
C(6,8) = 800
D(6,4) = 640
E(6,0) = 480
10
FOR STATED PRIORITIES :
BEST SOLUTION IS
x1 = 6, x2 = 8, d+
1 = 4
Point C above
d1 = d2 = d3 = 0
First two goals not achieved
Third goal not achieved,
Since overtime = 4 hours
TRY CHANGING SEQUENCE OF PRIORITIES AND
INVESTIGATE IF SOLUTION CHANGES.
Decision Variables
x1 = no. of desks produced /week
x2 = no. of tables produced / week
Constraints ( Goal constraints & System Constraints)
d1+= overtime operation (if any)
d1- = idle time when production does not exhaust capacity
Sales capacity
x 1 6
or
x1+ d2 - = 6
Constraints
x2 8 or
x2+ d3 - = 8
Capacity Constraint:
x1+ x2+ d1- – d2+ = 10
Objective function
P1 Minimize underutilization of production capacity (d1- )
P2 Min (2d2- + d3-)
P3 Min (d1+)
COMPLETE GP MODEL
Minimize Z = P1d1- + 2 P2d2- + P2d3- + P3d1+ subject to
(1)….
x1 + x2 + d1- d1+ = 10
(2)….
x1
+ d2=6
(3)….
x2
+ d3- = 8
Non - negativity restrictions
x1, x2, d1-, d2-, d3-, d1+  0
LP MODEL
Maximize Z = p1x1+ p2x2 + … +pnxn subject to
•
a11x1+ a12x2+ … + a1nxn < b1
•
a21x1+ a22x2+ … + a2nxn < b2
•
…
•
am1x1+ am2x2+ … + amnxn < bm
•
Li < xi < Ui (i = 1, …, n)
NOTATION
Industry 1
Industry i
Industry k
Industry 2
yij
Industry j
n = Number of
industries
yi j= Amount of
good i needed
by industry j
Industry n
bi = Exogenous
demand of
good i
bi
MASS BALANCE EQUATIONS
The total amount xi which industry i must produce to exactly
meet the demands is
xi = yij + bi ,
i = 1,…n
PRODUCTION FUNCTIONS
We must relate the inputs yij to the output xj for each industry j
Industry i
yij
Industry j
xj
aij = number of units of good i needed to make 1 unit of good j
yij = aijxj for all i,j
INPUT-OUTPUT
COEFFICIENTS
aij are known as Input-Output Coefficients or Technological Coefficients
Input of i th
industry to
industry j ,
yij
Slope = aij
• Linearity assumed
• No economies or
diseconomies
• Static (constant aij)
• Dynamic (varying aij)
Production of j th industry, xj
THE BASIC PRODUCTION
MODEL
Substituting the production function equations in the mass balance
equations, we obtain the basic production model of LEONTIEF:
x1 = a11x1+ a12x2+ … +a1nxn + b1
x2 = a21x1 + a22x2+ ... +a2nxn + b2
..
xn = an1x1 + an2x2+ … + annxn + bn
In matrix notation:
or
X = AX + B ,
X= (I-A)-1 B
PRICES IN THE LEONTIEF
SYSTEM
• pj = Unit price of good j
• aijpi = Cost of aij units of good i required to make one unit
of good j.
• The cost of goods 1, 2, …, n needed to make one unit of
good j = i=1n aij pi
• If the value added by industry j is rj ,
pj - I=1n aij pi = rj, j = 1, …, n
THE PRICE MODEL
•
•
•
•
•
•
•
•
In Matrix notation
(I-A)T P = R or
P = [(I-A)-1]T R
X= (I-A)-1 B
Price model
Production model
A is the matrix of technological coefficients
P is the price vector
R is the of value added vector
T denotes transpose & I the Identity matrix