In the last lecture we have looked at the

life cycle of a production system and we had identified the major decisions to be taken

during the life cycle of a production system. In this particular lecture we are going to

talk about the role of models and decision making. We will have a glimpse of different

kinds of models that are used in practical decision making in various stages in the life

of a production system. The basic idea therefore is that in order to take any decision optimally

you quite often need to make a model of the situation and use this model for deriving

what appropriate decisions need to be taken. So in this particular lecture we will deal

with some of the issues. We will talk about the model. We will talk about the relevance

to decision makers of various kinds of models. We will talk about different kinds of models

which are useful and we will some examples of how models help in real life decision making. That is the intent of this particular section.

Here is a working definition of a model. Model is an abstraction of some degree of the real

life thing or process for which we want to predict performance in a most general sense. The question that arises naturally is what

are the features of models and why do we make models? One must understand that models provide

a focus on relative relevant factors and variables which means we have to understand reality.

You must be able to pick out the relevant from irrelevant. You are essentially making

models so they help us in understanding reality and picking up the right kind of factors,

using them to express the relationship that we are interested in. Then models provide

opportunity for experimentation without undue cost hazard. This is the major advantage of

using models and production management. For instance, you might be using what is known

as location models. You might want to investigate the effect of shifting the plants from Delhi

to Bombay from Bombay to Chennai in terms of costs. If you actually started doing that

you would be ruining all your assets. Model help us in making this analysis without undue

cost hazard and help, may be using the right location. The third feature of models is that they help

in prediction of real life phenomena. There is a lot of uncertainty in the real world.

The production manger has to deal with varying demands for instance; all that he has access

to, is perhaps is the past historical data. You might want to use the past historical

data to find out what is likely to be the demand for the next period of the so that

the plant production of the product might be automobiles for the next month. So in that

sense prediction is something very important for a manager. The phenomenon which is to

be predicted can be predicted by using a model. So this diagram actually summarizes the process

by which you would might make a model. What would happen is that there is the real world

which you are interested in studying or capturing and in order to study the real world, you

might have to use your judgment and experience and on the basis of these two things, you

come up with model and this model is then used for process of prediction performance

of the real life system. If the performance is okay you continue using the model. If not,

you have to revise the model. The revision of the model would generally take place either

in terms of simplifying some assumptions or introducing the additional complexities to

the model and so on. So this again refines your judgment and experience

and this is the ongoing process which helps you to make models, refine models and uses

them for the purpose of decision making because out intent here is to see how models are useful

for decision making. Of course for a model to be reliable, model validation is necessary,

this means the kinds of results that you get from the model or they are actually coming

out in conformity with what real results are and this process is known as model validation.

There are various mathematical ways of doing it but essentially we are trying to find out

whether the results from the models are good enough or not. There could be variety of models for instance,

model could be physical. Some examples of models are wind tunnel, wind tunnel and blade.

Wind tunnel is used find out the shape of the airfoil and therefore you can do various

kinds of experiments to design the airfoil in that sense it is a model. It is a physical

model or a planetarium which is the model of the universe which will talk about the

global structure or you would talk about the architect’s model of building design which

is again a physical model. The major advantage of physical model is that helps you understand

and appreciate how the whole would look together and then therefore help you in forming some

opinions or making decisions. Models could be graphic such as representation of variables

in two or three dimensional space, such as the history of demand plotted versus time

populations, food production, and traffic intensity. So all such important graphs are

graphic models which give you some ideas on how the variable of interest is actually participating. The variables could be pictorial. The models

could be pictorial and you can use visual pictures or cartoons or road signs. Road signs

are also nothing but pictorial models which tell you that you can turn right only or you

cannot turn, you cannot have a u turn or you have various other kinds of things. Those

are examples of pictorial models or a model may be a schematic which means like an organization

chart with authority relationships information flow or current flow. All these kinds of models are essentially

schematic models of the whole thing or there could be models which are mathematical essentially.

Mathematical models are those models where symbols are used to represent real life situation.

You might want to represent the pressure by the variable p, the volume by the variable

v and depending up on the situation you might say pv is equal to constant or pv to the power

zeta is equal to constant. That is in fact mathematical model that particular situation

of how gases behave. You could even set up a simulation model. A simulation model is

essentially the approximation of the real world generally carried out with the high

speed computer. You try to find out how the system would behave under varying conditions

and try to simulate those conditions and from those simulations you can draw some meaningful

conclusion. That is the intention. So these are some of the varieties of models that are

present. By and large, if you look at the variety of

models you could classify these models in 3 different categories and especially when

we talk about models for decision making, it is convenient to talk about models as iconic

or analogue or symbolic as the case may be. So this is the classification that essentially

based on structure of the model. What we are saying here is after all what an iconic model

is? When we are talking about iconic models, these are actually scaled up or scaled down

version of reality. To give an example for instance say a globe of the world is a model.

It is an iconic model of the world. It has many simplifications but it performs many

useful functions for instance you can use this model to understand how day and night

are formed. With little bit of ingenuity you can also show how various eclipses are formed

and understand how eclipses are formed so in that sense these kinds of models helps

us in this nothing but scalped up or this is scale down version there could be scalped

version reality. If you are talking about the molecular structure of let us say carbon

for instance as a tetrahedron. You want to put a carbon atom at every node of the tetrahedron.

You can have a physical model which does this and then it can help you probably calculate

a variety of atomic; inter atomic properties that you might be interested. In that stage

remember that iconic models are just scaled up or scaled down versions of reality and

they give us understanding for the major problems. With these models for decision making, it

is that they are not robust enough. You cannot keep on changing the model parameters because

you have to construct a new globe and new model of the atom every time and therefore

the usage for decision making is limited. The second kind of model class that we talk

about is an analogue model and analogue is something in which you substitute one property

of interest by another that is what it is. So you have mechanical and electrical analogues

which would try to simulate or substitute the system and all considerations. A very

interesting example of this was a model developed by the London school of economics which talks

about representing the entire economic world of different countries by a system of pistons

and cylinders with water in between. These pistons, cylinders are of different diameters

and were connected depending up on the relationship on the individual countries in the world.

The basic advantage of this was that if a particular major country like United States,

where the policy changed and it was like moving a big piston. Energy effect would be felt

in all the other countries by removing their respective small questions and so on. This

is an analogue. Let us see an example by what we mean by analogue. Then we have the symbolic model which is essentially

based on the structure where we are using mathematical symbols to represent relations.

In fact this is most commonly used in model in production system and in almost all areas

of project and production management. What we are doing here is that the quantities of

interest is represented by some kind of symbols, to establish the relationship between the

symbols, you establish your objectives your priorities and you have a mathematical model

which tries to capture the situation and some typical examples are linear programming, non

linear programming, queuing theory, inventory theory. All these are basically symbolic models.

The basic advantage of using symbolic model is that they are most robust. If anything

changes, all that changes is the value for a particular variable or a parameter and it

is very easy to run the model again with these revised changes. You do not have to make a

new globe or anything of that kind. So these models are essentially very robust. Based on the purpose we can classify models

into descriptive and prescriptive. Descriptive models are those models which merely describe

the system. This is what will happen if you do this. Typical example includes all queuing

models are essentially descriptive models. They say that this is the arrival rate and

this is the service rate and this is the queue discipline. Then the average queue length

will be such and such thing or something is the descriptive model. A linear programming

model is a prescriptive model. It tells you what should be of different quantity that

you should produce so as to maximize your profit. So the production manager gets a prescription.

He knows how much to produce each month and to maximize the profit. But both are useful

and descriptive models can often be used to choose the right decisions. If you compare

different decisions in a queuing situation you can always find out. For instance if you

could find out at a traffic light, what should be the duration that the traffic light should

be on or off? You collect distribution of the arriving cars and vehicles. Suppose you

do this exercise for let us say keeping a stoppage time of 2 minutes, 3 minutes, and

then 4 minutes in your computer, when you are doing this then for each case you would

get different amount. Let us analyze the average waiting time for the customers of waiting

cars or whatever. Based on this you would probably like to say that I would like the

car to wait more than may be 5 minutes and therefore I should have a stopping time. In

that sense descriptive model is being used for a prescriptive purpose and so now models

can also be classified based on the environment. We are talking here about the decision environment.

So broadly speaking you can speak with what is known as deterministic models, when you

are assuming a certainty for the variables or you are assuming probabilistic models where

randomness is assumed, some kind of variation is assumed for different types of variables.

Since the symbolic models are generally the most popular in production management, we

just look at a few examples of symbolic models. For instances what happens is that I said

if I have the past history, January, February, March, April, May, June. I have the actual

demand for a certain product like this and I want to utilize this information to find

out the demand for the future months. One very simple example is using a regression

model or a forecasting model. The regression type which is essentially a descriptive model

fits some function to this. It need not necessarily be only a linear function. It could be any

function and then once you fit it the function, this function then behaves as a model which

will tell you how much demand would be there for future months. Depending up on the accuracy,

model, validation process, you could relieve on this information to take decision pertaining

to the production system. Here is an example from inventory. You are all familiar with inventory and when

we talk about inventory management, the classical problem is that if I place an order for an

item, the stock level suddenly rises and then it tends to fall gradually at an average rate.

Actually the actual fall might not be smooth in this fashion but this could be this is

one of the features of approximation that we bring about in the model. Then we say the

average rate of consumption of this particular material is so much and then again the new

order is placed here and then you get the supply here. So the stock level arises and

so on. The pattern of variation of inventory typically could be modeled as the shortest

curve which we know and from this by using a bit of mathematical jugulary (which I am

not going into at the moment because I am trying to discuss the basic philosophy behind

models), you get the economic order quantity 2 into demand into the ordering cost, divided

by i which is the interest rate into c under root. You can get the optimal ordering quantity.

So it becomes the prescriptive model. It prescribes for management, what they should be ordering

to minimize the overall cost. Once you have this information you can in

fact see how the two major types of costs are involved. They are the carrying cost.

The carrying cost is the cost of keeping the inventory in stock which would be say, the

average inventory level is q/2 into i which is the interest rate into the c which is the

item cost. So you have a straight line variation for the cost here and the ordering cost per

annum is c0d/q. So this particular function would be something like this. So what really

happens is that you have two conflicting cost parameters. One increasing, one decreasing

at different rate and the total cost would be something of this nature and the EOQ is

the quantity which minimize the cost. But apart from this, the model tells you a lot

of other useful things. For instance you might not be able to operate at EOQ but you can

immediately find out if you deviate from the EOQ. What is going to be the cost penalty

that you are likely to pay? Is it better to shift in this direction or in this direction

and if I operate at a non optimal value, what is the cost I am carrying on these? These

are all important questions; managerial issues that can be answered even through a very simplified

model like the inventory cost. We can look at other examples of models which

are very relevant in production management. We can use linear programming, non-linear

programming, goal programming models of different types of production processes and we can model

product mix and scheduling by using these kinds of models. Now we will take a small example and see how

modeling for that particular product is done. Take a simple manufacturing problem. Let us

say the company produces two kinds of products. These products are simply desks and tables

and the manufacture of either desk or table requires one hour of production capacity in

the plant. So whether you produce a desk or a table, it means one hour of production capacity

in the plant and the maximum of available production capacity is only 10 hours per week. There is a limited sales capacity you can

sell at most six desks per week or eight tables per week in that sense and the gross margin

of profit is the sale of the desk is eighty rupees and from the sale of a table is 40

rupees. So if you want to model the situation, actually this is a very important prototype

occurring in industry. It is a product mixed problem. Companies always make a number of

different products so you are trying to find out what should be the optimal products. Let us suppose that the priority wise goals

of the company are the management wants to avoid any under utilization of production

capacity that is the first goal the management wants to sell as many desks and tables as

possible. But since the gross margin from the sale for a desk is twice that of a table,

there is twice as much desire to achieve sales goal for desk as for tables which is quite

obvious and the third one is management desire to minimize the overtime production of the

plant as much as possible. If you want to work with these priorities

what can happen is well incidentally you can solve this problem graphically and what we

see here is if on the x axis we have a number of desks which is x1 and on the y axis we

have a number of tables which is x2. Then what we have basically is the total sale capacity

for table is this line. You cannot sell more than 8. You cannot sell more than 6 of desks.

So you have this line here and the ideal production capacity which we have utilized, both because

each ten are available. So, one on each side and this would be just a line here with this

intercept 10 and 10 on both sides. So what you find is if you would solve this problem

as a linear programming problem, the space shown here A B D E O would be the feasible

region as far as the desks and tables are concerned you have this information itself

is useful. What can be the range of product that you can make? You cannot make for instance

so many tables which are outside this. So it gives you the feasibility range and

then of course if we work out the profits at all the corners, you find that the maximum

profit would in fact be, if we confine our sell to only the points A B D E O and exclude

c which currently is not feasible, then of course you find that the point D gives us

the maximum profit of 640. But now if you impose the priorities which we have decided

for this particular problem, you notice some interesting. What would happens is that if

we look at the examples, this line, shows you are utilizing your capacity fully and

if you go in this direction you are basically trying to over utilize your capacity. If you

come in this direction you are under utilizing your capacity. So in that sense since the

priority, said that if we do not want to under utilize our capacity, that means there is

no implicit bar on the over utilizing the capacity through over time or through sub

contracting. Whatever is the case, point c, which would

have for stated priorities, would become the best solution. At this particular stage the

first two goals are not achieved. The third goal also is not achieved and therefore since

the overtime at this stage is four hours, what can happen is that you can keep on changing

the sequence of priorities and investigate the solution changes. That is something that

you can do and in fact what we are talking about here is the variation of linear programming.

Typically known as goal programming in which we define a goal and then we talk about the

deviations from the goal and then we are looking for solutions within the certain priority. So this would be the way the goal program

would look at this particular solution and when you look at a formal definition of goal

programming for the problems, it would in fact be something like this. You have the

number of desks and the number of tables as the two variables for the problems; we define

an overtime operation if any, so if there is a d1 +, d1 — is the ideal time when the

production does not exhaust capacity. So this is the sales restriction. X1 is less than

6; x2 less than 8 or we can introduce here this which is like a slag variables. So d2

— would be the under time capacity. This is equal to 6 here. This would be 8. We have

converted these two equations into two kinds of goals using these deviational variables

d2 — and d3 –. What we can then do is we have capacity constraint

and the capacity constraints read like this. x1 + x2 must be generally equal to 10. But

we could be producing less than 10 in which this variable will come into operation or

we could be producing more than 10 in which case this variable come into operation. So

since we can operate anywhere we can have both these variables coming into play. So

this is like interpretation of this. This is the target and these are the under deviation

or over deviation in the target because you do not need to begin with it whether we would

be under deviating or over deviating from the target. So it is a very convenient device

to model such situations and then of course the objective functions would have three priorities.

Based on the priorities that we had minimized under utilization of the production capacity

this is d1 –. This was 2 times the deviation because the profits from the desks and tables

were different in the ratio of 2:1. So you have this particular goal here. You are minimizing

this deviation and the third priorities is to minimize d1 +, which is minimizing the

over capacity, over utilization of capacity here. So this solution would lead to point

c that we saw as the optimal solution. Finally this model would actually be set down

in this form and you can use the goal program goal to settle down. We notice that we have

basically deviational variable and constraints in this problem and the objective function

is in the nature of the priority and the deviational variants. That is what happens here and this

is a very common type of form which is used here. Let us look at a simple linear product

mix problem, which is product mix in the sense that if we generalize this, you are talking

of a situation where we have only two products. Now we have n products. So the n products are indexed from i=1 to

n. So this is the third variation we are talking about. The simple linear programming in 2

variables, goal programming in two variables, just to indicate to you that when you have

multiple goals and conflictive priorities we use that situation. This is the generalization

of the first model. We have m resources aij is the consumption of the jth resource per

unit production of the ith resource. bj is the availability of the jth resource in general

and pi is the profit contribution per unit of the ith product. So when you have this, the various other variables

are ui, which is the upper limit on the sale of ith product, li is the lower limit on sales

on the ith product in general and xi is the production of ith product in the planning

horizon, this is our decision variable. You want to find out how much of each quantity

to produce and typically you have this situation this model would actually look like this.

The objective is to maximize the profit so p1 is the profit per unit from each unit of

the first product, so p1 x1 + p2 x2 and so on upto pn xn is the total profit that you

earn is subject to various constraints. These constraints, if you see are each right hand

side here, is the resource availability. This is the resource consumption. This is the resource

consumption by the first product. If you are talking about labor hours, say my labor hours

are 10,000. 1 unit of the first product consumes 5 labor hours. This consumes 10 labor hours

and so on. So this would be then the consumption in terms of labor hours is less than this.

So this could be the consumption of money, this could be consumption space and so on

and you have the m resource, it is a generalized model for a product. It is a model which can

be very useful to handle the l p problem, and then of course you could have restriction

on sales. xi should be less than some upper bound, less than equal to some upper bound

and greater than equal to some lower bound. The upper bound could be the total capacity,

rather it is the demand. So it would be the total demand for this product and the lower

bound would be some minimum amount that you would have stipulated or contracted to make

for that particular product, so that is the alive. Then this would easily be solved by

nlp code. You notice that I am not talking about solution procedures here. Solution procedures

are generally simple because you have access to available codes. You can solve them easily.

What is the primary role of the production manager? It is to model a situation and set

up an appropriate model. For his factory or a situation, whatever it is, it can be solved

easily. Let me make digression to a model of the economy. As a whole you see we have

looked at various kinds of linear programming. A simple two product linear programming problem,

a simple two product goal programming problem, and a n product mixed problem in general.

Now those are all instances of models which we are actually trying to solve problems which

are generally encountered within the factory and all the examples that we are talking were

of that nature. We talked about the forecasting model developing, the demand then we talked

about the inventory model, setting production targets of, or setting ordering targets for

individual product within the factory and then we talked about the product mix which

is also the decision within the factory and so on. So in order to develop a broad aspect of the

role of models and production systems, we will look at this Leontief’s input out put

models which is actually the macro model of the national economy. After all the economy

is also a production system. It is producing different kinds of products and we will consider

a number of interacting industries within the economy. Application of this model include

integrated planning for the whole economy which means for instances whenever the planning

commission wants to make the next five year plan for your country, how does it do? The

Leontief’s input output model is the bases for doing it. So we will see exactly how this

planning can be done. You set the target for individual industries. This also comes out

and the target should be that agriculture sector should be given so much and this is

the target production for the next five years whatever it is. Resource allocation to various

sectors is also possible through this kind of a situation and price prediction and control

in the economy is also possible. You might be a little flabbergasted by saying that so

many things can be done by a simple mathematical model. Let us look at the basic of Leontief’s

input output models and see how exactly we can go about looking at this particular problem. Some of the assumptions in this model are

the major assumption that the economy consists of a number of interacting industries. Will

you agree or not? What does this mean? In real life it means for instance that if you

talk about the particular sector in the economy, let us say you talk about automobiles. For

its survival, automobiles will have to borrow from the steel sector. The steel will probably

have to be borrowed from the engineering sector and its different components and so on which

are going to be used in the automobile sector and then of course all these things are going

to be put together and there must be a demand for automobiles. So you have people from different

sectors who are actually going to use those automobiles. So the important thing really

is that in fact the key feature of Leontief’s model is to try to model the interactions

that take place between various industries. Now one assumption that is made here is that

each industry produces a single good and uses only one process of production to make this

good. For instance one might say if I divide the economy into various sectors, let us say

auto mobiles sector is one of them then for purpose of macro planning I would not say

that the auto mobile sector produces cars, jeeps, two wheelers, three wheelers and so

on which are different kind of items. There are auto mobiles, so I could club them together

I probably would say that the output of the automobile sector is in terms of general automobiles.

However if you want you might be able to say work in terms of standard passing the unit.

This says that the maruthi 800 for instance might be taken as the 0.8th standard passing

the unit. A maruthi 1000, say one standard passing the unit bigger car might be taken

as 1.2, 1.5 standard passing the unit. So that would give you a basic for aggregation

or you might work in financial units and then everything gets combined very easily and normally

in the Leontief’s input output model, we tend to work financially. This is what we mean

by saying that it produces a single good. It does not mean that automobiles do not produce

a single, but single good is automobile and each automobile could be different. So each

industry produces to satisfy the demand in all other industries apart from an exogenous

demand. Under these assumptions, we will try to see what exactly we. So let us see the interaction between various

industries in the economy. Now we are looking at the economy as a whole and we have various

industries which are interacting with each other. That means there is a two way interaction.

Something can flow from here to here, something can flow from here to there etc and there

are various possibilities and so on. So we have industry 1, 2, 3 and so on up to n and

let us look at 2 industries. Let us call this industry i and let us call this industry j

just to understand the notation which we shall be using for this particular model. So n is the total number of industries which

is there. yij is the amount of good I needed by industry j, so yij is flowing from the

ith industry to the jth industry because the jth industry during the whole year or during

the whole period, (mind you this entire exercise is for a certain period) and normal period

is 1 year. So yij is the amount of good i flowing to the industry j, in that particular

period and so on. bij is the exogenous demand of good i. So every year industry will have

some exogenous demand which is either exports or whatever the consumer consumes, that is

the exogenous demand. So this is the model that we are trying to look at in terms of

Leontief’s model. What we find here is that there are two types of things that we can

do. First of all, our intention is to find out the production of each of these individual

industries. Second objective is to find out the prices prevailing in the economy. If you

look at these two things which Leontief’s model gives, we can apply simple mass balance

equation. Mass balance equation for instance would be

the total amount xi, which industry i must produce to exactly meet the demands will be

xi=summation yij + bi. bi is the exogenous demand and yij is the amount which is going

to industry j, so this particular summation that we are talking about here would be summation

over ij. So you know for instance this should be j here, the total amount flowing from i

to all other industries plus bi should be equal to what is production. For that it is

a simple mass balance equation. This is like in electrical engineering; this would something

like Kirchhoff’s current law. Then the notion or the sort of the approach

that Leontief’s took was, he developed what are called production functions. Now what

is production function? If this is industry i and this is industry j and industry i is

supplying a quantity yij to industry j. Industry j is producing quantity xj let us say. What

does it mean? We say we relate the input yij to output xj of each industry j. So how do

we do it? I have to say that yij=aij into xj for all i j possible. All that this is

saying is in any industry there are multiple inputs from different sources. So yij i.e.,

is the amount of input required for producing a unit of j=aij in that sense of term. What

is the significance of this? You can see from this equation, yij divided by xj would be

aij and aij would have the interpretation of the number of good i need to make one unit

of good j. If it sounds a little complicated, let us

say there are numerous examples. The example is very simple. Suppose the industry makes

a cake which probably your mother does in home. So if you want to make a cake, what

are the various inputs you have for making cake? You would need chiefly maida. You would

need baking powder, you would need sugar, and you would need other inputs depending

up on the kind of flavor you want to give the cake. So the idea is to make the cake.

Here is this. So the point is if I want to make a 1 kg cake, I would probably use 750

grams of maida. The point that is being raised here is that if I want to make 1 kg cake that

means if I want 1 kg cake I need 750 grams of maida. You would probably need a pinch

of baking powder. If I make 2 kgs cake, what could we need? You would require 2 into 750

that is 1.7 kg maid and 2 pinches of baking powder and various other things. It is exactly in the same spirit that we are

relating the input to the output and what we can then say is that this if you plot the

relationship between yij and xj, we have said yij=aij. xj could be a straight line, whose

slope is aij. So aij’s are known as the input output coefficients or the technological coefficients

and here we are assuming linearity we are not assuming any economic or diseconomies

of scale. It is a static situation and the aij is constant. If it is a dynamic situation

aij would be allowed to vary that is what we are trying to say. So what is really happening

is keeping the example of the cake in mind if you want to find out or predict the total

requirements of a certain industry, it will be a fixed proportion. If I want a 2 kgs of

cakes I use 2 into 750 grams of maida. So similarly if I need to produce one ton of

something, I need this much of input. If I double my output then I would require more

of input and this aij will capture the input you must require. This is the technology coefficient. A basic production model in this particular

situation comes out like this, x1 would be equal to this particular thing which is their

plus b1by mass balance equation x2 would be again a similar equation; xn would be a similar

equation. If you put it in a matrix notation you get X=AX + B and then directly you can

solve this equation for xs. You get X=I — A inverse B. This is the basic Leontief’s

model. What it says is that you can now calculate the production quantity knowing the requirements

of the individual sectors in the economy if you know the technological coefficients, that

is the matrix a. This equation is the fundamental equation which is used for solving this particular

problem of determining the production quantities, as a function of the these exogenous demands. What does the planning commission do basically?

The planning commission finds out by to talking to different sectors as to what is likely

to be the demand for the next five years. So different people converse and generate

the vector b and once the vector b is known, and assuming that the input output coefficients

remains unchanged, you can calculate what the production should be. That is the basic

idea of the basic production model. The second aspect of this model is the prices in the

Leontief’s system. The prices in the Leontief’s system, if pj is the unit price of good j,

then aij pi would be the cost of the aij units of good i required to make one unit of good

j. Simple, and therefore the cost of goods 1, 2, 3 and so on up to n needed to make 1

unit of good j would be the summation of this. So we have summation from i equal to 1 n of

aij pi and if the value added in the industry is important because each industry adding

value It means the total cost is 20 rupees, it is making a profit of 10 rupees and selling

for 30 rupees, that is the idea. So if rj is the value added in the economy, price of

good j minus this quantity should be equal to the value added in the industry. Basically

these equations for i is equal to 1 to n will also be the n equations that we used to solve

pj’s. What happens is if you write them in matrix

notation you get I — A transpose into P=R. That is what the equation basically leads

to and therefore P can be written as I ¬– A inverse transpose multiplied with R. This

is the value added and this is the price model. So, together the price model and the production

model constitute the basic Leontief’s structure. This was the production model we studied a

short while ago in which you are calculating the production quantities as the function

of the exogenous demand b of the products. Now here you are calculating knowing the value

added. You are calculating the prices of the economy; you can calculate the prices and

the beautiful discovery of this particular model for which Leontief got the Nobel Prize.

Incidentally, here you do not have to do additional effort to calculate this. Because this is

I — A inverse which you calculate once. This is just I — A inverse, the transpose of that.

The inverse is automatically calculated. You do not have to do any matrix inversion

again and in fact it was this particular discovery that led to the difference between primal

problem, dual problem in the linear programming and it was about the same time that George

Dancy was working on the simplest method of solving the linear programming. This was in

the early fifties that Leontief discovered that this is another way of looking at duality

and therefore in this, a is the matrix of technological coefficients that we have just

seen. P is the price vector; this is the price which prevails in the economy. What would

be the cost per unit in the automobile sector? The agriculture sector in the labor sector

etc prices are determined by this model. r is the value added vector. You see what can

be done for instance the government is in planning. We can give more subsidies to a

certain section of the economy which means that they are playing with the value added

vector of that sector. If you add to the value added vector of that particular sector for

instance, the government gives conjunctions to the agriculture sector and you know how

much concession have been given. You can calculate immediately the impact of this concession

of the overall prices in the economy for the different sectors. I think it is a very useful

way of interpreting. So ultimately this was indented to give an idea of the price model

as well as the production model of Leontief which can be utilized for solving this. You

know the R vector is the value added for the economy. Now the value added for the economy

is actually governed by the competitive situation in, the market precisely. It is not it how

much value can textile value has? How much value can automobile value have for its product?

This is determined to a very large extent by those particular. Then this could be something

that they would decide to set. It is their decision. If I make a product and make it

for 100 rupees and decide to sell it for 150 rupees, I have actually set my value added.

So value added to a large extend is the amount of what the particular industry expects from

its product. You will supply the value added vector and then compute the prices. I mean

if we add the value added or keep the value added; you try to work out this particular

situation. Finally let us summarize what we have tried

to do in this class. We have tried to look at the purpose and types of models that are

there for decision making and the basic key was that all these models aid in decision

making in the production management situation. We took some examples of models. We looked

at forecasting model and lot sizing model, product mix model for graphic and LP and also

example of a macro model like a Leontief’s input output model which are insisted. It

would not be wrong to say that the steady state of production management is basically

study of models of different kinds and these models, once you carry in your bag; you apply

for real life situation. Thank you!

## Mumshad Ahmad Khan says:

thank u very much sir. i m watching ur video lectures for the last 20 days on project as well as production management. these lectures have given a new dimension to my thoughts and i m now ready to implement them into the real life of my business step by step. lectures have been very much motivating and helped me in confidence building. thanks once again sir.

## A1 INFORMATION says:

I like this

## A1 INFORMATION says:

very nice video

## kuniyil sreehari says:

thank you sir..