In today’s lecture we study inventory models. Inventory essentially deals with materials the procurement of materials. Materials management is to how we store the material for subsequent use in every manufacturing organization or
manufacturing situation. We buy material and these material are usually bought as raw material. Sometimes these raw material undergo manufacturing and become semi finished of finished products. Some of the bought out items go directly into the assembly of the product. Earlier we have seen aggregate production planning where the output of the aggregate planning is telling
us how much of production capacity is available for every period. Later we would also be studying disaggregation models, where the production capacity that
is available in each period is, now allocated to the various products that are made. So,
at the end of disaggregation we would know what are the products that are going to be
made. In order to make these products we need to buy material. So, we are now going to look at how we buy material. So, inventory essentially answers two very important questions. If there is an item or material that we are going to buy there are two important decisions that
are associated with the purchase of the material. And these two questions are how much to order and when to order. So, obviously when the production planning is complete and the allocation of time to the various products are made one has an idea of how much or how many products of a certain type we are going to produce. So, the final product, whatever we are going
to produce of the final product of each of the products. Now, requires certain raw materials or bought out items in a certain number. So, if the
demand of the various products are known. Now, this demand can be suitably multiplied to find out the requirement of each of these items or material that goes into the final
product. Once we know the annual demand of each of the item that goes into the final
product we need to make decisions on what quantity we are going to order every time
we place an order and when exactly we are going order. So, inventory control or inventory management essentially deals with these two aspects.
Let us also spend some time on the classification of inventory models. Broadly two types of
classification. One is called single period inventory models and multiple period inventory models. Single period inventory models assume that the planning horizon is a single period and the decision is made only once. Multi period models assume that there are several planning periods or multiple periods. And decisions
are made more than once. Sometimes these are also called static and dynamic inventory problems. Now, the important thing that drives the inventory management is the actual demand for the item. The moment we know that demand then only we can answer these two questions how much to order every time we order and when to order. So, now depending on the nature of the demand we can have broadly three classifications one is called deterministic demand the other is probabilistic demand and within the probabilistic demand we have two classifications. One is called inventory under risk the other is called inventory under uncertainty. Deterministic
means the annual demand is known and known with certainty. Risk would mean that the demand is known, but the demand is not deterministic it follows a certain distribution and the
entire distribution is known. If the distribution of demand is known then it is called problems under risk. uncertainity, If it is assumed that the demand is not deterministic the distribution of demand is not known but, certain important parameters such as mean and variance of the demand are known, then it is said to be inventory models under uncertainty. And in this lecture series on operations and supply chain management we will be looking at multiple period deterministic models and multiple period with risk. We would also try and look at some one or two single period models with nondeterministic demand. So, we will first take the case of the deterministic demand and then look at
some models where the demand is known and deterministic. Then we will move into probabilistic inventory models particularly covering situations of inventory under risk. Then as mentioned
in inventory problems we basically answer these two questions which are how much to order and when to order. One of the reasons we have to answer these two questions is the material that we buy or the items that we buy for manufacturing
are special items which somebody else is producing and selling to the organization - that is
manufacturing this product. So, the vendor or the person or the supplier who actually
provides this material also requires some time to make this and then sell it.
Therefore, it is necessary to fix on a certain order quantity which is optimal such that
the person who makes it and sells it has the time to do. So, organizations do not for example buy items the same item everyday traditionally they would. But, a slightly larger quantity than what was required every day. So, the question of how much to buy or how much to order came about more recently organizations have moved towards just in time manufacturing. And just in time purchasing where organizations tend to buy exactly what is demanded or they buy for the demand of a very short period of time. Unlike in the older models where
you actually end up fixing an order quantity which is optimized which may result in the
quantity being the demand of a larger time period. We will first look at some traditional
models where we actually try and optimize and try and find out what is the quantity
that we have to buy. Usually this thing how much to order is called the order quantity and is denoted by the letter Q and when to order is called reorder level
and is denoted by r. Now, the second question is also important because when we place an order to the supplier the supplier has to make that item and then give it. So, there
is a time that the supplier needs to make it and the variable called when to order is
chosen such that when we place the order. And by the time the order arrives we should
have enough material with us to meet the demand during the period between placing an order
and arrival of the order. Now, this period between placing an order
and arrival of the order is called the lead time. Very broadly defined, lead time is the
time between placing an order and receiving it. Lead time also actually has several components. Sometimes we look at identifying that an order has to be placed and placing an order between these two certain time elapses and then the order is placed and the order is met or supplied. And then by the time the item is supplied and the time the item is put to use certain
amount of time elapses. Sometimes we consider all of them as the lead time, but for the
purpose of our discussion we would assume that the lead time is the time between placing an order and receiving it. So, the when to order question becomes important because we need to place an order such that by the time the material arrives or for the
period taken between placing the order. And the arrival of the material which is the lead
time we should have sufficient stock to meet the demand for the item during the lead time. So, generally reorder level or the point at which we place the order is equal to the demand during lead time. So, that that quantity which is called the reorder level we should be able to meet the demand during lead time which is called lead time demand. When the demand is deterministic and the lead time is also deterministic and known with
certainty, then the lead time demand can be computed as the product of lead time and daily demand. But if one of them is nondeterministic, then we have to look at expected value of
the lead time demand and then try and see whether the reorder level is about the expected value of the lead time demand or sometimes more than that. We will see those aspects
as we move along as we move towards probabilistic inventory models later. Now, we will restrict ourselves to deterministic inventory models where the demand is known with certainty we go back to the other question how much to order quantity there one other
reason. That we do not order the same item everyday for consumption is that there are
other costs associated with inventory. So, obviously when items are bought there is a price or which is called the cost of the item. So, when we buy a single item we there is
a unit cost associated with that item. So, if we order a quantity Q and if the unit
cost is C then Q into C is the actual worth or cost of the item that we have to incur
when we buy that item. If we assume that the annual demand is known and this capital d
then D into c is the worth of the item or the money spent on buying the item per year under the assumption of course. That the cost of the item does not change during the planning period we normally assume that C remains as C. And it does not increase or there is no
effect of inflation on the cost of the item and so on. But, in addition to the cost of the item there are few other costs that we incur. Now those are three other types of
costs that we incur. One is called order cost or ordering cost
which is denoted by the notation C naught or C 0 or C o order cost which is incurred
every time an order is placed for an item. Now, this cost is different from the actual
cost of the item this does not depend or is independent of the quantity that is being
ordered. But, every time there is a decision to place an order or make an order a certain order cost is incurred order cost has several components some of which I will list here
now. First one is the cost of people or payroll money given to the people who work in the
purchasing areas their salaries have to be factored in. And they become part of the ordering
cost because they order several items day by day.
So, they there is a part of the order cost that goes into meeting the salaries of the
people who are involved. Second is there is a transportation cost for the items to come
from the supplier to the organization which has been ordered which is included in with
cost of order. We can assume that there is a onetime transportation cost for bringing
the items from the supplier point to the organization third is there could be inspection costs because
these items that come in have to be inspected. Sometimes we could have hundred percent inspection.
Sometimes there could be random sampling. But, in some form these incoming goods are
inspected and there is a cost associated with the inspection. Next is however undesirable,
we could have a situation where at the end of the inspection there is a reject. So, the
rejected items have to go back and then have to be replaced, so there is a delay there
is a time element as well as there is a cost element.
So, there could be cost of reject and rework with respect to the supplier side. Fifth one
and the perhaps another important one is called the cost of follow up. For example, an order
is placed and sometimes there could be delays and then the retime is estimated and known.
And just about the time we are going to reach the lead time which means the item has to
come then there is a lot of follow up activity going on. And some of these follow up activities
could even be senior people from this organization visiting the supplier and making sure that
the critical items that are required come in time.
So, there is a cost associated with follow up. Follow up could be over the phone it could
be over other forms which are expensive it would also mean a visit of a person going
to this place and then may be inspecting it then and there and bringing it here. So, there
are these costs that are associated with the cost of ordering. Of course, there is any
delay then each of these delays is going to into result in a higher cost.
So, all these are the components of what is called order cost or ordering cost and this
cost does not depend on – it is assumed that it does not depend on the quantity that
is ordered. For example, if you look at these components one could say that at least this
component could be could depend on the quantity that is ordered. But, then right now we assume
that the order cost is not dependent on the quantity that is ordered and is fixed for
every order. Now, the other important cost that comes - that
the organization has to incur after the item arrives is called inventory holding cost or
carrying cost - holding or carrying cost. Now, this is denoted by C c inventory carrying
cost of carrying inventory which is C c. This also has many components. Now the material
arrives and it arrives in a certain quantity which is Q which we have to find out.
And then as I mentioned when we do the optimization Q, let us say, is a little more than the daily
demand which means that the material that comes is going to be used for a certain number
of days. And then an order is going to be placed the next order is going to arrive and
it is going to be used. So, when this quantity Q comes it has to be saved it has to be stored
it has to be held in control and there is a cost associated with that, now that cost
is our inventory holding cost or inventory carrying cost.
So, the first cost that we can think of is the cost of space to keep these items. So,
they have to be kept in some warehouse or some storage area and stored and kept safe
and secure and the moment we want safety and security. We need people who will guard these
items from theft and keep it safe and secured. So, the salary associated with - salary of
the people who are associated with storing the inventory is another component.
Sometimes along with space you could also have power and other accessories which are
required. So, there will be cost of power that is there in the storage area and other
accessories. Sometimes some of these items may require very specific environment such
as a dust proof environment or sometimes certain temperature which has to be maintained for
some of these items. These are not very common, but not uncommon
either that there could be some items which require a certain temperature conditions,
dust proof conditions and so on. So, we need cost to do certain other things such as other
special requirements. Now, the next thing that can happen is suppose we order a very
large quantity Q. And before we consume this entire quantity which would say run into several
days of consumption for some reason either the technology changes or there is a better
product that comes in and so on and this item that we have bought can become obsolete and
unusable. So, if such a situation occurs there could
be cost of obsolescence. So, I write this as cost of obsolete items again however undesirable
it is, there are times when the items get stolen. And there is a certain pilferage cost
that the organization has to occur particularly when they are metals tend to be easily stolen
if they are not properly watched. So, however small and undesirable it is there
could be a cost of pilferage. Now, over and above all these, is an extremely important
cost which is a very integral and a large component of the cost of carrying. Now, when
we buy this item say Q quantity of Q of a particular item which costs C then Q into
C is the cost of the item and the organization has to spend this money in buying it.
Now, the organization is going to recover the money much later after all these items
are put to use the product is manufactured the product is sold and then the money is
recovered. So, when the organization spends this Q into C to, but this item the organization
actually borrows this money Q into C from some source. So, for the borrowed money the
organization has to pay interest which is called the cost of capital.
So, the most dominant cost of holding the inventory is the cost of capital though I
have written it as the sixth item or the last item this is indeed the most important of
the costs. So, this is also defined as some kind of interest rate that the organization
spends which is given by small I and therefore this C c if we consider all these costs. Then
this C c is the cost of holding the item it comes in. So, many rupees per unit per year
or per period many times it is customary and if we actually workout the calculations for
the various components one would see that this takes a very large proportion of the
C c and the rest of them are actually very small compared to the contribution of this.
So, in a situation where this dominates the rest of them it is also customary to write
C c as i into C where i is the interest percent per year and C is the unit cost of the item.
So, carrying cost can be written in two ways one is it is called C c cost of carrying where
all these costs are included. And there are situations where cost C c is written as i
into C where the interest on the borrowed capital dominates and is taken and the rest
of them are assumed to be negligible compared to the interest rate.
Now there is a fourth cost. Let me write the fourth cost here and the
fourth cost is called cost of back order or cost of shortage. Now, let us assume that
we are buying a certain item and based on the availability of this item. Let us say
that we have scheduled the production of a certain product or a component let us say
on a Friday. And let us assume that this item is expected to come on Thursday. So, that
on Friday morning when the production happens the material is available for production.
And let us assume for some reason we are not getting on Thursday or Friday and when the
Friday morning when the production is about to begin the item has not come it is not available.
Therefore, we will not be able to produce it on Friday say now if the due date for the
item or for the product that is to be made on Friday, which means we will not be able
to deliver it to the customer on Friday. So, there will be a delay now that delay is
ordinarily called as shortage. We have not been able to meet the due date or not able
to meet the customer requirement for want of material. Now, this can be handled in two
ways one is to say that well I am not able to deliver it to the customer on Friday. At
the same time I will carry the unmet demand to the next day which is say Saturday produce
it and give it. Now, such a situation is called back order
where the unmet demand is carried to the next period produced and then the demand is met.
The other is called shortage or also called lost sale which means if I am not able to
meet a certain demand on Friday I have lost it and I will not try to recover it by back
ordering it. Very loosely both of them are called shortage. Shortage has to be specified
whether we are looking at a back ordering situation or on a last sale situation usually
when it is not specified it is assumed that it is back ordering situation. So, there is
a cost associated with this if the material does not come in time, now such a cost is
given by the notation C s cost of shortage. Now, this also has several components the
first component particularly if it is a lost sale is the loss of profit associated with
the sale because the sale is not complete because for want of material. So, there is
a loss of profit associated with that one could even go to the extent of adding something
like loss of opportunity. For example through the sale I might get further business which
is lost because I have not met that deadline or due date.
If it is a back ordering which means I carry the demand to the next period I need some
more capacity in the next period to meet the additional demand. So, cost of that capacity.
So, cost additional capacity to make this item in either case whether it is a back order
or a shortage I do not have the material. So, I would not be able to produce, so either
I have to do some rescheduling and produce some other things for which material is available.
So, if we do that then there is a cost of reschedule if we are unable to produce some
other item during the time that we have specified for this item. Then the resource is not utilized
for this period is idle for this period. So, cost of idleness of resource or underutilization
of resource. Now, let us go back to the situation where
we have not been able to produce it on Friday. But, we try and give it on the next day we
produce it on the next day, now let us assume for a moment that we had we had actually decided
scheduled it for Friday. Now, let us assume that if we were able to make it on Friday
then let us say we would have used a truck to take the finished goods to the customer.
Now, there is a back order we are not able to produce and meet it on Friday. Let us say
we are able to do it only on Saturday, but then we do not want that delay to be carried
to the customer we do not want it to go much later. So, one of the ways we can actually
speed it up is by instead of sending it by road or by truck one could air lift the item
and send it to the customer. So, that would that would result in increased freight.
If we choose to take faster modes of transport which are expensive. So, there could be an
increased freight if in order to meet the due date we try and speed it up by sending
it by air instead of sending it by road. Now, we have seen six different aspects of C s,
but there is one most important one just as cost of capital is the most important one
in C c there is very important component of this which is called loss of customer good
will. Now, if there is a delay when the customer
is not going to be happy with the performance of the manufacturing organization. And if
this delay continues then definitely there is a point at which the organization would
lose the customer or the organization starts getting worried. That they are going to lose
the customer and there is a goodwill which is extremely important for the survival of
any organization now that gets hurt and there is a loss of good will.
So, organizations are very concerned about the loss of goodwill that could happen because
of delays and because of material not being available. So, the cost associated with loosing
customer goodwill is an integral part of C s. The only difficulty there is it is extremely
hard to give a numerical value to this one cannot say that it might cost goodwill so,
many rupees per day delayed. Every organization knows that this cost is high and every organization
wishes to treat this cost as high. So, that we do not compromise on that or we do not
take that for granted at the same time almost no organization can accurately estimate what
is the cost of loss of customer goodwill. So, these are all the components that go into
the inventory, now four important costs the unit cost of the item that is bought the ordering
cost the carrying cost. And the back order or shortage cost which are described using
notation C for the cost of the item C naught or C o which is the ordering cost C c or i
into C which is the carrying cost and C s which is the shortage cost.
Now, the other three costs are different and they are additional costs they are over and
above the actual cost of the item. So, when an organization maintains - buys maintains
inventory the organization actually incurs all these four costs depending on certain
situations. So, the inventory management which talks about how much to order which is Q which
is the first question that we will look at we will, now have to try and optimize and
keep all the four costs to the minimum. So, the objective will be to try and minimize
the sum of all the associated costs. All the four costs which are there depending on the
setting or depending on the assumptions the decision variable is how much to order which
is Q. So, with this we will try and look at some very basic inventory models and then
extend these inventory models to slightly more advanced situations. So, the first inventory model that we will
look at which we call as model one is called single item continuous demand
and instantaneous replenishment. Now, let
us assume that the annual demand for this item
is D per year and is known. Now, let Q be
the ordering quantity that we decide to order every time we place an order. So, let us look
at how the inventory for this item behaves with respect to time. So, let us draw a simple graph which is like
this is time and let us assume that we have just placed an order for Q. And we have just
instantly received it instantaneous replenishment would mean that as soon as you place the order
the item arrives which also means that lead time is 0. So, the moment the order is placed
the items are received. So, let us assume that we have received Q
and we are consuming or using this Q. Now, when we say continuous demand we assume that
the demand is there the same amount of demand is there at every instance of time. Therefore,
as time goes on the demand D is met is there for every instance of time it is met and the
stock of the item comes from Q. It is consumed and comes till it reaches 0, now this slope
is D which is actually the demand for the item to put it in simple terms more understandable
terms. It will be like saying if the annual demand is ten thousand and let us say if the
company works for two hundred and fifty days. Then the daily demand is forty if the company
works for eight hours a day then the hourly demand is five if the hourly demand is five.
The demand per minute is five by sixty and demand per second is that divided by sixty
and so on and the demand is the same. That is that is how we should look at this continuous
demand which means at every instance of time there is a demand. And since that demand is
same which is D per year, now this will come as a straight line from Q to 0.
Now, the moment this stock position reaches zero because we have instantaneous replenishment
and we also assume no shortage. We do not want to model shortage situations no shortage
since we do not want to model shortage as soon as the demand reaches as soon as the
inventory position reaches zero. I will call this as stock or inventory position on the
y axis time on the x axis. So, as soon as the inventory position reaches zero I place
an order I place an order for another Q and there is an instantaneous replenishment of
that. So, as soon as I place an order here my stock position goes up to this Q.
Now once again the cycle continues and then I start consuming this till I reach this.
Then again I place an order and then I start consuming. So, this is how the inventory position
will behave please note that it is the same Q which is here as well as here and. So, one
now let us call time period as capital T, now let us derive a very simple formula to
find out Q, now every time we place an order we order a quantity Q. So, number of orders
per year
will be D by Q because every time we place an order we order for Q number of orders per
year is d by Q total order cost is equal to D by Q into C naught.
Now, C naught is this order cost. So, every time we place an order we incur a C naught
and this C naught has a unit of rupees per order. So, D by Q is the number of orders
per year C naught is rupees per order. So, D by Q into C naught will have a unit of rupees
per year spent in order cost, so this cost we have considered. Now, we have to consider
this cost holding cost or inventory cost now holding cost or inventory cost is defined
as C c and this has a unit called rupees per unit per year.
So, either year, or the same time period that we are taking about since we are looking at
year here we would also define this as year rupees per unit per year. So, this is the
cost of holding one unit rupees per unit per year. So, what is the average inventory that
we have at any point in time multiplied by C c will give us the holding cost per year
and the average inventory should be in terms of number of units that we have.
Now, there are two ways of calculating the average inventory. So, average inventory
is like saying if I start this cycle with
Q then it comes down it comes down comes down to zero at time t. So, in a particular cycle
the beginning inventory is Q the ending inventory is zero the average inventory is Q by 2 because
you actually have a line here that is one way of saying that the average inventory is
Q by 2. Other way is to say that to take any period of time find out the total inventory
that you have divide it by time to get the average inventory.
So, if you take one particular cycle the total inventory that we have held during this cycle
is the area of this triangle which is half into base into height, so half into Q into
T. Now, that much inventory is held for a period T, so half into Q into T divided by
T which gives us half into Q. So, average inventory either way is Q by 2 units average
inventory is the kind of inventory that we have at any point in time averaged over a
reasonable value. Now, the cost of holding this inventory is called total holding cost
is equal to Q by 2 into C c cost of holding this also has rupees per year because this
is rupees per unit per year this is units. So, it becomes rupees per year the units get
cancelled and you get rupees per year. Now, we look at this cost which is back order
or shortage cost one of the assumptions is that there is no shortage. So, we do not incur
shortage in this particular model which means the inventory position does not become negative.
So, there is no shortage assumption in this model therefore, we will not consider back
order or shortage cost in our total cost. Now, there is this cost of the item, so cost
of the item is equal to D into C D is the annual demand C is the unit cost of the item.
So, D into C rupees per year, so total cost total annual cost T C which is the sum of
the order cost the holding cost as well as the cost of the item is given by d by Q C
naught plus Q by 2 C c plus D into C. Now, in this total cost expression as I said there
are three costs cost of the item cost of ordering cost of holding the inventory. There is no
shortage or back order cost when we want to find the variable as Q rest are all the things
are known they are known parameters. So, only thing that we need to do is to find out Q,
now this total cost is a non-linear function of Q because you find Q in the denominator
here you find Q in the numerator here. It is a single variable non-linear function,
so differentiating with respect to Q and setting it to zero will give the first derivative
equal to zero will give the optimum value of Q. Whether it is maximized or minimized
will depend on the second derivative. Now, our objective is to minimize this T C, so
setting it first derivative to zero would give us d T C by d Q equal to zero will give
us minus D by Q square C naught plus C c by 2 equal to 0. The third term is D into C which
is a constant it does not contain Q. Therefore, its derivative will be 0, so a first derivative
equal to 0 would give us minus D by Q square C naught plus C c by 2 equal to 0 which on
simplification would give us Q equal to root of 2 D C naught by C c.
Now, actually speaking Q is a square root of something, so Q should be plus value or
minus value mathematically. But, since Q is the quantity that is ordered it cannot take
a negative value, therefore it is customary to write Q equal to root of 2 D C naught by
C c. And write only the positive value we do not write plus minus root of 2 D C naught
by C c only the positive value is taken; now we have to find out whether at this value
of Q the original function is a maximum or a minimum. Now, we are interested in minimizing
the total cost, so a second derivative will be this is minus D by Q square C naught. This
will become a constant, now there is a negative term and a Q square here.
So, on further differentiation we will get a positive term here and Q cubed will come
in the denominator. So, for a positive value of root over 2 D C naught by C c the second
derivative will be positive indicating that this quantity is actually the minimum quantity.
Now, this quantity Q is called the economic order quantity which also are called E O Q for short. Now,
the formula that we have derived is called the E O Q formula for a single item and this
formula is also called the Wilsons E O Q formula attributed to Wilson who derived it somewhere
I think in the early forties or thirties so on. Sometimes this formula is also credited
the credit goes to Harris it is also called Harris inventory formula. And the literature
talks about the contributions of both Wilson and Harris who supposedly have independently
derived or thought about the use of such a formula, to find out the economic order quantity.
Or the order quantity that minimizes the sum of the ordering cost as well as in the carrying
cost. So, out of the four costs as I said we did
not consider the back order or shortage cost we also observe that the three remaining costs
that we have got this is the order cost component this is the inventory holding cost component.
This is the item cost component we now realise that the item cost actually does not play
a part in the economic order quantity. So, it is also a customary to leave out the item
cost and derive Q simply by considering the order cost. As well as the carrying cost alone
and not including the item cost because item cost being a constant is not going to contribute
to the derivation of the economic order quantity. So, there are only two costs which are optimized
which are the order cost as well as the carrying cost.
So, in this formula of Q equal to plus root of 2 D C naught by C c. D is known which is
the annual demand. C naught is known which is rupees per order C c is rupees per unit
per year what is very important is that D is annual demand which is demand per year.
Then C c should be defined as rupees per unit per year if D is defined as month then C c
should also be defined as, so many rupees per unit per month. So, D and C c should have
the same time period in their unit of measurement. So, if D C naught and C c are known then the
economic order quantity can be calculated. And then we can substitute this Q into the
total cost function to find out the minimum total cost that the organization would incur.
In the next lecture we will consider numerical examples and we explore a little further on
inventory models.
