Lecture – 22 Role of Models in Production Management

Lecture – 22 Role of Models in Production Management

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!


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