1. Prediction of real life phenomena For model to be reliable, model validation is necessary Judgement, Experience Model Simplify Real World Is performance, prediction O.K? Revise Model Continue using model No Yes ? Try different simplifying assumptions

2. PICTORIAL Visual pictures, Cartoons, Road signs SCHEMATIC Organization chart with authority relationships, information flow, current flow + VARIETY OF MODELS

3. JFMAMJJ x x x x x x x Ft = a + bt Month Demand FORECASTING MODEL Regression (Descriptive) EXAMPLES OF SYMBOLIC MODELS

4. INVENTORY MODEL EOQ =  2d C ordering / iC Time Inventory Approximation, d actual

5. PRESCRIPTIVE MODEL Time Annual Costs Ordering Cost/annum = C o d /q Carrying Cost /annum = q  iC 2

6. 12356789104 1 2 3 4 5 7 8 9 6 A B C D E Ideal Capacity Profit 800 640 320 480 Sales Representation x 1 (desk) x 2 (tables) x 1  6 x 2  8 PROFIT = 80x 1 + 40x 2 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 O

7. FOR STATED PRIORITIES : BEST SOLUTION IS x 1 = 6, x 2 = 8, d 1 = 4 d 1 = d 2 = d 3 = 0 First two goals not achieved Third goal not achieved, Since overtime = 4 hours TRY CHANGING SEQUENCE OF PRIORITIES AND INVESTIGATE IF SOLUTION CHANGES. + - -- Point C above

8. Decision Variables x 1 = no. of desks produced /week x 2 = no. of tables produced / week Constraints ( Goal constraints & System Constraints) d 1 + = overtime operation (if any) d 1 - = idle time when production does not exhaust capacity Sales capacity x 1  6 or x 1 + d 2 - = 6 Constraints x 2  8or x 2 + d 3 - = 8

9. Capacity Constraint: x1+ x2+ d1 - – d2+ = 10 Objective function P1 Minimize underutilization of production capacity (d 1 - ) P 2 Min (2d 2 - + d 3 - ) P 3 Min (d 1 + )

10. COMPLETE GP MODEL Minimize Z = P 1 d 1 - + 2 P 2 d 2 - + P 2 d 3 - + P 3 d 1 + subject to (1)….x 1 + x 2 + d 1 - - d 1 + = 10 (2)….x 1 + d 2 - = 6 (3)…. x 2 + d 3 - = 8 Non - negativity restrictions x 1, x 2, d 1 -, d 2 -, d 3 -, d 1 +  0

11. LP MODEL Maximize Z = p 1 x 1 + p 2 x 2 + … +p n x n subject to a 11 x 1 + a 12 x 2 + … + a 1n x n < b 1 a 21 x 1 + a 22 x 2 + … + a 2n x n < b 2 … a m1 x 1 + a m2 x 2 + … + a mn x n < b m L i < x i < U i (i = 1, …, n)

12. NOTATION Industry 1 Industry j Industry 2 Industry i Industry n Industry k y ij n = Number of industries y i j = Amount of good i needed by industry j b i = Exogenous demand of good i bibi

13. MASS BALANCE EQUATIONS The total amount x i which industry i must produce to exactly meet the demands is x i =  y ij + b i, i = 1,…n

14. PRODUCTION FUNCTIONS We must relate the inputs y ij to the output x j for each industry j Industry j Industry i y ij xjxj a ij = number of units of good i needed to make 1 unit of good j y ij = a ij x j for all i,j

15. INPUT-OUTPUT COEFFICIENTS a ij are known as Input-Output Coefficients or Technological Coefficients Production of j th industry, x j Input of i th industry to industry j, y ij Slope = a ij Linearity assumed No economies or diseconomies Static (constant a ij ) Dynamic (varying a ij )

16. THE BASIC PRODUCTION MODEL Substituting the production function equations in the mass balance equations, we obtain the basic production model of LEONTIEF: x 1 = a 11 x 1 + a 12 x 2 + … +a 1n x n + b 1 x 2 = a 21 x 1 + a 22 x 2 +... +a 2n x n + b 2.. x n = a n1 x 1 + a n2 x 2 + … + a nn x n + b n In matrix notation: X = AX + B, or X= (I-A) -1 B

17. PRICES IN THE LEONTIEF SYSTEM p j = Unit price of good j a ij p i = Cost of a ij 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 a ij p i If the value added by industry j is r j, p j -  I=1 n a ij p i = r j, j = 1, …, n

18. 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