Welcome back, we are discussing about inventory
management in the supply chain. And in the inventory management discussions
in last 2 sessions we have discussed about a very simple case of inventory management
in supply chain where we had only 2 installations in the supply chain, and we used our very
conventional EOQ model to handle those 2 installations. And we saw with the help of an example also
in the last class that how when we do the simultaneous optimisation even in the case
of just 2 installations we have some type of cost saving. The very well cost saving in a supply chain
environment. And when we do the individual optimisation
it is not possible. Now the same discussion in this class we will
take to multi echelon systems, where in a supply chain more than 2 installations are
possible. And when more than 2 installations are possible
you can have now, a supply chain where we can think of 4 stations, 1, 2, 3, 4, and as
we are moving from this side to this side we know that it can be a supplier, it can
be a manufacturer, the wholesaler, and the retailer. So, these are normally the 4 components which
are available in a supply chain, the supplier of raw material, supplier of the components,
your other type of part suppliers. Then some kind of manufacturing or assembly
of those components may take place here. And, these are the wholesalers or the dealer,
which are taking products from the manufacturer, and finally customers get their products at
the retailers end. So, this is a popular supply chain, generic
supply chain model. And, we have discussed in our first sessions
about this type of generic supply chain. Now, considering this type of generic supply
chain, and using the points which we have already discussed in earlier 2 sessions. About 2 installations of inventory management,
now here retailer requires Q4 item, the wholesalers requires Q3 item. The manufacturer require Q3 item, and supplier
has demand of Q1 item at particular time. Now, it is very well understood that the retailer
is getting it supplies from wholesaler. Wholesaler is getting it supplies from manufacturer,
and manufacturer is getting it is supply from the supplier or the vendor. Therefore, Q3 is responsible out of Q3 items
a part of that will go to retailer in the form of Q4. From Q2 items a part of that will go to wholesaler
in the form of Q3. And from Q1 a part of that will go to manufacturer
in the form of this supplier. So, therefore Q1 needs to be bigger or at
best equal to Q2, and so on for other stages also, Q2 needs to be greater than or equal
to Q3, and Q3 needs to greater or equal to be Q4. So, this type of relationship is to be there,
between various quantities which you are procuring at a particular stage. Because, a particular stage is responsible
to supply products to the next stage in the supply chain. That is one very important thing when we will
develop the model for inventory management. In this type of situation you always need
to see that Q1, Q2, Q3, Q4 or if I take it to Qn to this level. This type of relationship that the quantities
which you are procuring which you are receiving at predecessor stage, should be more or equal
to the quantities which are required at the receiving end. And now the other important thing which we
have already discuss in the 2 installation system. The holding cost, because as I moving from
1 to 4, I am doing the value addition, so this is the direction of value addition. So, therefore my holding cost which is proportional
to the these value addition h1, h2, h3, h4. So my holding cost because of increase in
value, because of more value I am adding as I am moving from left to right. So therefore holding cost will also increase
as I moving from this to this station. Now with this if I go to develop the model
for inventory management. Let us see whether we have sufficient conditions
to apply our conventional EOQ model. Now, at this point I will like to recall what
we discuss in the last 2 sessions. That to apply EOQ model the inventory consumption,
and replenishment at a particular stage must follow, if you recall you can tell me the
sawteeth pattern, sawteeth pattern must be available for applying the EOQ model of inventory
management at a particular stage. So, now will let us see whether the sawteeth
pattern is available in this particular case or not. And, then only the basic EOQ model can be
applicable for inventory management. In this particular case now, let us that at
stage number 4, the last stage where we have procuring because this knowledge we have already
from our earlier discussion. That in a 2 stage model we are procuring inventory
at other stages in relation to our last stage, so here the last stage is fourth one. So, I will procure inventory at other stages,
in relation to this fourth stage. So, what I am trying to say that Q3 should
be sum multiplier of Q4, Q3 = ni.Q4. Q2 should be sum multiplier of Q3, Q2= ni.Q3
and so on Q1 should all be sum multiplier of Q2,
Q1 = ni.Q2. So, all these inventories at other stages
are multiplier of their successive stage, and if I see, if I substitute the value of
Q3 here in this case and Q2 here in this case, I can see that I can represent Q1, Q2, Q3
all three in terms Q4. So in this case also my complete inventory
management will be done with respective quantities which I am receiving at my fourth stage. Now, at the fourth stage I am requiring this
Q4 items. Now these are the Q4 items, and I receive
Q4 items from stage 3, and I start consuming these Q4 items. Then after sometimes I will consume these,
and I have adjusted my lead time, I have adjusted my order point. In such a fashion that as soon as I am touching
this 0 level, I get a fresh supply of Q4 items. And, then I will start consuming these Q4
items, and this way my second cycle will be completed at fourth stage. Again, I will receive a new supply, and this
will go on. So, you see we have a very clear sawteeth
pattern, these vertical lines are representing the replenishment of my stocks. And these slant lines are representing my
consumption. So, at stage 4 you have a very perfect case
for applying the basic EOQ model of inventory management. Now, let us come to stage3, in a stage3 you
require Q3 items now, what we have done here, a very specific case now, what we have done,
just to make it simpler, I have taken the values of ni, ni', and ni'', equal to 2. Just for the purpose of our understanding,
all these values are 2, 2, 2 each. So Q3 is actually 2 Q4. So, all these values I have taken equals ni
to ni' equals to ni'' equals to 2. So, these things make me Q3 equals to 2 Q4. Now, I am receiving Q3 supplies here. And, out of Q3 supplies I will immediately
supplied, because I have adjusted the receiving cycle in such a manner that on day 0, when
Q3 is supplied to stage 4, the same day I receive Q3 supplies at stage 3. And out of Q3 since Q3 is 2Q4, a part of that
part of Q3 will go to stage 4. So, what is remaining with me Q3- Q4, that
is with me, and that will go with me till this particular time. And, you can see that by this time I will
be carrying this Q3 – Q4 stock in my warehouse, and I will be continuing with this, and then
at the second replenishment cycle at stage 4. When again to Q4 items will be required these
Q4 items will be shift at this point, and then I will not have any inventory for this
period. Then again I will receive a new supply of
Q3 items. But, out of that I will immediately supply
this is Q3, this point represents Q3, but out of that again I will immediately supply
that Q3, Q4 portion to the stage 4, and I will be left with Q3- Q4 here again. And, I will carry this inventory for one more
cycle of stage 4 and then this inventory will come to 0. So, with this way you will see that I have
these types of step curves. This is Q3 level, immediately Q3 comes to
Q3- Q4. I will carry this Q3- Q4, for some time, and
then my stock will come to 0 level, and then I will be having 0 stock, at stage 3 or this
much period. This much period is represented by one cycle
of stage 4. And, then again when fresh supply is required
at stage 3, I receive Q3 items here, and out of Q3 items Q3 – Q4 remains in the physical
stock at stage 3. Because, Q4 has gone to stage 4, and I will
carry with Q3- Q4 supplies, for one cycle of stage 4. And, then when the next cycle requires the
fresh supply of Q4 items, so Q4 items are available with me, and I will supply these
Q4 items to stage 4. And, the stock will again come to 0. So, I will remain 0 stock for one more cycle
of stage 4. And, then again Q3 supplies will come, and
this type of step pattern, will keep ongoing. You will see that this type of step pattern
will keep on going. Now, these lines, green line represents the
installation stock, last class also we discussed this that this is the installation stock. So on here also this is installation stock. That what is available at a particular stage
3 or stage 4. But, we already introduced a concept known
as echelon stock in our last class. So now echelon stock just to revise that echelon
stock represents the amount which you have at that particular stage at stage 3 and at
stage 4. So whatever you have, so if I talk of echelon
stock of stage 1, so whatever physical inventory you have at stage 1 + 2 +3+ 4 at a particular
time whatever inventories you have, physical inventories you have at stage1, 2, 3, 4 that
is the echelon stock for stage 1. For stage 2 what physical inventories you
have at stage 2, 3, 4. That determine the echelon stock for stage
2 and so on for stage 3 and 4. So, if I talk the echelon stock for stage
2, so you will see that here it is Q3, and here it is 0 initially, so the Q3 is the total
stock, which is available with me, and at a particular time if I talk of this particular
time, if I talk let us say a particular point here. So, at this particular point here, you see
the physical inventory at stage4 is this much. This is the physical inventory available at
stage 4. And, this is the physical inventory available
at this stage. So, I want to see the total inventories of
this + this, this point will come somewhere here. And so on, if I want to see the total inventory
at this particular point, so here it is 0 already. And, this much inventory is available at stage
4. So, this point will come here. At this point you see the total inventory
at stage 4, as well as at stage 3 is 0. So, if I combine all these points of echelon
stock I will get a line like this, this is the line which represents the echelon stock
at stage 3. The stock, which is the physical inventory
at stage 3 + stage 4. So, it is the echelon stock, in case of stage
4, since there is no stage, after stage 4, so whatever inventory is there at stage 4. That is the installation stock as well as
the echelon stock. So, in case of stage 4 this curve is making
you installation stock, as well as echelon stock. So it is same here, but in case of stage 3,
these step lines are representing the installation stock. But, this type of curve where you have a slant
line of consumption, this represents the echelon stock. And, then if you see for at any cycle, you
can have this type of curve represented by the dotted lines. This represents the echelon stock. So, you can see that we can apply now EOQ
model for stage 3 also. If we consider it is echelon stock we can
apply the EOQ model here also. And, now take this case forward you have stage
2 now, now you take the stage 2, in stage 2 you have Q2, which is 2 Q3. Q2 is 2 Q3. So, now out of Q2, whenever you procure Q2,
so to initiate we have develop the system in such a manner, that whenever you require
Q3 on time t0. The same day you have procured Q2 also. So out of Q2 half of Q2 that is Q3 will go
to stage 3. So, you are left with Q2- Q3, and you will
continue with this Q2- Q3 for this much period. The time is second requirement of Q3 comes
from stage 3, and then it will come to 0, and then again you will remain at 0 level
the same discussion which we have applied for stage 3 will be applicable to stage 2
also. So, again you can think of this type of echelon
stocks at stage 1 also. You can think of these slant lines which are
representing the consumption of echelon inventory, these slant lines represent the consumption
of echelon inventory. Echelon inventory again is the inventory available
at stage 2 + stage 3 + stage 4. So, actually this slope is coming because
of this consumption at stage 4. At stage 2 and stage 3 you have the horizontal
lines. But, the angular lines are available only
at stage 4. So, because of the angular lines at stage
4, you get the slopes in stage 2 also. And, therefore, these installation stock which
is represented by these step lines. And, the echelon stocks are represented by
these slanted lines. So, this is again showing the case, that this
slant lines make us possible the use of EOQ model, at stage 2 also. Now, come to finally a stage 1 also. The same explanation can be applied to stage
1 also where you are procuring Q1 items, in such a manner that Q1 is equals to 2 Q2. And we have adjusted the procurement cycle
of stage1 in such a manner that whenever there is a demand of Q2 items. The Q1 items come on the same day. So, on time t0, you receive Q1 items at stage
1, and out of that immediately Q2 items are shift to stage 2. So, you are left with Q1 – Q2 items. And, you will continue with Q1 - Q2 item for
one complete cycle of stage 2. And, then you will be with 0 inventory for
another complete cycle, and again by using the concept of echelon stock, you can have
this type of dotted lines to represent the echelon inventory at stage 1. So, now you see by considering the concept
of echelon stock at stage 1, stage 2, stage 3, and obviously in stage 4. Echelon and installation stocks are same. So, here we have that because of these echelon
stock concepts in these 4 stages. You can very well apply the concept of EOQ
model. So, we will see the model with the help of
echelon stocks the help of echelon stocks with at all these stages, and that will help
us in using the basic EOQ model. Here if you see that we have used the values
of ni, ni', ni'', all two. So, that will help us in very uniformly completing
the replenishment cycles. The 2 replenishment cycles of stage 1, makes
one cycle of the stage 2. 2 replenishment cycles of the stage 3 makes
one replenishment cycle of the stage 2. And, 2 replenishment cycles of stage 2 will
make one cycle of the stage 1. And, therefore this will also enable us that
I can write all these terms, all these Q1, Q2, Q3 in terms of my Q4. So, here you see that Q3 can be represented
that we have already done that. It is 2 Q2, and because 2 Q2 is 2 Q3, so I
also write Q2 equals to sorry this is Q4, here it is Q4. Q2 is equals to 2x2 Q4 that is 4 Q4. And, 2 cycles of Q2 represents one cycle of
Q1. So, Q1 is actually 2 Q2. So, here I can also write Q1 is equals to
8 Q4. So, I have represented all these Q1, Q2, Q3
in terms of Q4. Q3 is 2 Q4, Q2 is 4 Q4, and Q1 is 8 Q4. So, therefore the point is that in case of
I want to generalise this discussion. I can write Qn as my last stage of supply
chain. This is the nth stage of my supply chain. And, I can write all the stages, let us say
it is Qi in terms of my values of ni, this is how I can represent all my stages with
the help of the last stage. All my stages with the last stage, so this
is very important that here, when we are going to develop the multi echelon inventory system. In this multi echelon inventory system we
will be determining these 2 things only. We will determine the value of Q1, the value
which is required at the last stage and then for the subsequent stages. Here we have assumed for the sake of simplicity
we have assumed that all ni's are 2. So, that is why have this type of very uniform
representation. But in our other cases we will determine that
whether it is 2 or 4 or 6 or something else. And accordingly Q1, Q2, Q3 can be determined
directly determining the value of Qn only. So, now one more thing before we close this
session, one more thing will like to discuss. Now Qi can be ni into Q1. Now, Ni as we have discussed earlier also
in the case of 2 installations, should be any integer value. So, it can be 1, 2, 3, 4, 5, 6, 7 anything,
and if you remember the example which we took to understand the case of 2 installations. In that we took deliberately the value ni
as equals to 3. Q1 was 3 Q2, in that case. But, here we have a limitation and for that
limitation we will use a kind of relaxation of this problem. We will like to use a relaxation of this problem. Now, the limitation is that if for the purpose
of this problem. Here ni is 2, and here ni' is 3, and here
ni'' is 2. Let us say these are the values, 2, 3, 2. In that case it will not be possible for me
to make this type of groupings of these cycles. Because, of odd and even values of ni's I
cannot make these type of cycles of my supply chain. That all these cycles are starting and ending,
in the same manner. You have the same starting date of all the
cycles, and 2 cycles of this will make 1 cycle of stage 2. And, 2 cycles of stage 2 will make 1 cycle
of stage 1. But, if it is not so, in that case it will
be difficult for me to develop these types of relationship 2 cycle will make 1 cycle
of this. But 3 cycles of stage 2 will make 1 cycle
of stage 2 and then it will be difficult for me to have same starting and ending date. And, as a result of that either I will be
having more inventories at this stage or I will be running out of stock, for 1 complete
cycle if I want to match this type of analysis. So, for that purpose we have done a bit of
relaxation in this relationship. Where Qi equals to ni Qn we have used. Instead of that we will be writing ni as 2
to the power mi. And, mi can take any value, mi can take any
integer value 0, 1, 2, 3, 4, 5. So, what will happen because of this relaxation,
because of this additional condition we have imposed, you will get the values of ni, only
in the even values. If mi is 0, 2 to the power 0, it is 1. Then 2 to the power 1, it is 2. 2 to the power 2 it is 4. 2 to the power 3 it is 8. So, only even values will be there. 1, 2, 4, 8 like that and, that will make our
life very simpler, and therefore this type of analysis will be possible to handle the
supply chain more effectively. We can go to a bit of mathematics and Roundy
was one of the professor who suggested that if we use this type of relaxation. There will be slightly higher values of the
overall cost, but those values of higher cost will not be more than 2 %. So, that is very
well justified, because of lot of comfort we are going to receive by doing this type
of simplification. So, here we stop in today’s lecture, and
we will see the application of this type of model in our inventory management. So, in the next class we will see that now
we are clear that we are going to use this type of relaxation. And will have application of EOQ model at
all 4 stages. So, now with the help of some data, with some
practical exercise, we will see in our next session. That how do we apply this model in a multi
echelon situation. Thank you very much.
