Lecture – 36 Aggregate Production Planning: Basic Concepts

Lecture – 36 Aggregate Production Planning: Basic Concepts

In the last two lectures we have been talking
about forecasting and various methods of forecasting. As you all know the forecast of demand is
a very important input to the planning exercise and in today’s lecture we are going to be
talking about utilizing demand forecasting for purposes of aggregate production planning. Aggregate production planning is the exercise
of developing an overall plan for a company which specifies how resources of the company
are going to be committed overall the next six months to one year. So the typical planning horizon that we are
talking about is a planning horizon up to about one year and in this particular lecture
we are going to be talking about some of the basic concepts of aggregate production planning
and we will see how aggregate production planning is actually useful for a company. This is the basic definition of aggregate
production planning. So if you see aggregate production planning
is basically concerned with planning overall production. The emphasis is on overall production of all
products combined and this overall production could be in tones of steel liters of paint
etc and we are talking about a planning horizon for generally next 3 to 6 months or even extending
up to a year for a given forecast demand schedule. So the basic idea here is that the demand
for the various products of a firm could be fluctuated and if the demands are fluctuating
how do we commit our resources to meet this fluctuation in the demand? That is the basic intention. I think a word about how we call it aggregate
production planning and how is aggregate production planning different from production planning
within the context of a factory. You see for a planning horizon for the next
six months or one year, the company is basically interested in knowing how the overall performance
of the company shall be as the demand keeps on fluctuating. So a company might be manufacturing let us
say hundreds of products. The demand for one product may be going up. The demand for the other product might be
coming down and as far as the hundreds of products are concerned each demand would exhibit
its own kind of behavior. So if the company tried to track the performance
of these individual products it could probably go mad in tracking the demands of hundreds
of products which are there and since the company is interested only in the overall
growth and the overall commitment of resources for the next one year, it is therefore interested
in aggregating the demands of the individual products and talking about the aggregate production
plan and rather than talking about individual production plans. So if you are talking about a steel plant,
for instance a steel plant could be manufacturing sheets scalp, various types of ingots of different
sizes, rail sections strips in a variety of sections. So you would have a large variety but when
the managing director of the steel plant wants to know how the steel plant would be performing
over the next year he would probably be concerned about the total production in terms of tones
of steel of all types of products put together. So tones of steel would be a convenient aggregate
measure of the performance of the company and you could then aggregate the demand of
all the products in terms of the total tones of steel and use this as an aggregate measure
of the requirements of the company. Similarly if you are talking about a paint
manufacturing company, a paint manufacturing company produces paints in different colors,
different sizes and for different applications and for each of those paints. The demand would be different but if you want
an overall idea of how the paint manufacturing company is performing, you could probably
aggregate the total requirements of paints and the total liters of paint requirement
for the whole company as a whole. This concept could be extended to even an
automobile manufacturer. For instance if you talk about an automobile
manufacturer, let us take the typical example of Maruti udyog, the company produces different
types of cars. It produces a Maruti 800. It produces a zen, it produces a van it produces
a gypsy it produces a versa and so on. It produces different types of cars and the
demands of these cars will be different over the planning horizon of the next one year
but if the company is trying to find out what would be its growth level for the next one
year, it could probably try to mix up the demand of all these cars. How would it mix up the demand? It could probably have to think in terms of
a unit like standard passenger cars. So may be they might say that a Maruti 800
is like a 0.8 of a standard car, a Maruti zen is like one standard car. Esteem is like 1.2 passenger units and develops
some units like this and then ultimately measures the performance of the company in terms of
standard car units something like this and then it would have an aggregated variable
which would be able to deal with this. So the point that we are trying to say is
apart from this aspect of focusing on the overall growth, the second major advantage
of using aggregating or aggregating the demand is that normally if you have 50 products and
you make their forecast and then you try to work with individual forecast each forecast
will be subject to some errors and if you try to combine these forecasts the aggregate
demand figure would be subject to much less error because the positive and negative errors
will tend to randomly cross each other out and therefore there is greater accuracy in
obtaining the aggregate demand forecast then the individual demand forecast. So this is another reason for dealing with
aggregate production planning. So the whole process in aggregate production
planning is essentially to determine the aggregate forecast. First you do a process of aggregation, i.e.,
do aggregate production planning. Once you have done it, you will have to disaggregate
to obtain the individual pattern of demands or rather production levels for individual
products in that sense. So I think this perception of aggregate production
planning must be kept in mind. Basically what are we trying to do in aggregate
production planning is we are working with a forecast of demand and we are trying to
work out the capacities cost and commitments that you are making. So you know that this is the capacity of plant
so much in January, so much in February, so much in March because you might have committed
capacity to other products. So the capacity availability in different
periods is known. So you know the various types of costs that
are incurred and you are therefore trying to find out how these capacities should be
committed to the aggregate demand over a certain period of time and this is what we call an
aggregate production plan. From the aggregate production plan we are
then able to work out the work force requirement over the planning period and we are also able
to determine the machine time, allocation over the planning horizon. So basically this is the purpose of aggregate
production planning. n trying to determine the commitments of the
capacity to various products, we have already indicated that the basic problem in aggregate
production planning that we are trying to solve is the problem of meting a fluctuating
demand. If you were sure for instance that the demand
in each month is going to be hundred units and this will continue add infinitum then
production planning will be relatively simple affair but that is not what happens in practice. In practice demands keep on fluctuating. You have to forecast these demands and therefore
let us be clear about the options that management has to meet. Fluctuating demand in any business, how can
it meet a fluctuating demand? Probably the first method is one that is used
quite commonly used in industry is to build inventories in slack periods in anticipation
of higher demands later in the planning horizon. So you can keep on building inventories when
there are no sales in the hope that in some future you will be able to sell off the inventories
that you have made. This could be a good strategy provided the
demands actually show up in the future but if you have been able to if you have not been
able to forecast the demands properly then this could be a risky strategy. You probably are aware that in the Maruti
factory in Gurgaon at one stage, they had such a large stock of cars that they had to
procure additional land to just store these finished cars outside and so much so that
the cars were actually getting damaged in the adverse weather conditions in which they
were being stored. So this strategy of using inventories in slack
periods in anticipation of higher demands may work in practice but it is not without
its problems. The other option could be that you could carry
back orders or tolerate lost sales during peak periods. Do you know the difference between a back
order and lost sales? A back order is something when the customer
comes to a shop and you don’t have the item in stock and he places an order with you and
he is willing to wait. So whenever you are able to supply you give
in the order. So that is called back ordering. A lost sale would be when a customer comes
to your shop and he finds that you do not have the brand of shampoo that you need and
therefore he just walks away to some other shop and that is a lost sale. So both lost sales and back orders are actually
situations which occur when there is a shortage. When you do not have enough stock available
of whatever item is demanded so whenever the stock level is positive you will inventories. Whenever the stock level is negative you either
carry back order or tolerate lost sales in that sense. The third option that management may have
to meet a fluctuating demand is to use over time in peak periods under time in slack periods
to vary output while holding work force and facilities constant. This is a very common strategy. This is the kind of strategy that students
typically tend to use. When there is a peak period, that is five
assignments had to be submitted, they work over time at night and get these assignments
ready and otherwise when there is a slack period, they don’t. I mean they use under time in slack period. They don’t work and that is the way they vary
their output and while holding the work force and facilities constant. I think another point that needs to be mentioned
in the context of aggregate production planning is that normally we are not talking about
the option of increasing or machinery or other kind of infrastructure within your plant. For instance if there is an increased demand
for something, one option might be to set up another factory. We are not talking about those capital intensive
options in this particular situation because our planning horizon is in typically one year
ahead. So we are talking about those options in which
generally the facilities are constant though of course work force can be varied in general. The other option available to the manager
for dealing with fluctuating demand is you can vary the capacity by changing the size
of work force through hiring and firing. It is a very common strategy these days. You probably heard or read a couple of days
to go about Cadbury’s laying off something like 20,000 workers all over the world in
their various plants. So they are adopting this strategy of changing
the size of the work force through either hiring or firing. So most of the companies fire their workforce
and when there is a need you can always hire the work force. So changing the capacity by hiring and firing
is a very common strategy. It will be contract labor. For instance most farmers would engage additional
labor when they require them for purposes of reaping the harvest and then when they
do not need them they would fire them. This is exactly that but hiring and firing
as a typical term as a normal in normal parlance is the legacy of the American culture. The American culture people are very fond
of hiring and firing. You find that an executive is doing very well. He is given a promotion today but when he
comes to work, the next Monday he finds a pink envelope on his desk which says thank
you Mr. Ralph you are fired today because we do not need you and here is our cheque
for whatever number of days you have served us. This kind of culture prevails more in the
united states not in India as much though of course, with the advent of multinationals
and so on this kind of culture is also coming into being the hire and fire culture. But normally when you talk about employment
in the government of India then if you are hired once, you are fired only when you retire. That kind of a situation but then this is
an option available to management for changing capacity by changing the size of the work
force through hiring and firing. I would like to mention here that hiring and
firing is not without its costs. When you hire people you have costs of hiring. That is the process of selecting people, conducting
an interview, training people before they actually become productive. So these costs are the costs of hiring and
similarly when you fire people, there is a cost associated with firing as well. There is loss of customer good will and the
loss of customer good will would mean that the person whom you are fired will probably
spread the message in other organizations. Do not ever join this company because they
hardly keep you for two weeks and then fire you. So that reputation travels. So there are costs associated with hiring
and firing and one of the major purposes in aggregate production planning is actually
considering these various costs which are associated with these various options and
try to select a strategy which is the optimum strategy in terms of meeting the fluctuating
demand. You can vary the capacity through changes
in plant and equipment but generally as I said this is the long term option and is not
considered when you are talking about aggregate production planning. So the point really is that each option that
we have considered here involves cost. The cost may be tangible or intangible cost
and the aim in aggregate production planning is to choose the best option considering the
various costs. It is essentially an optimization problem
with these types of costs, fluctuating costs and the kinds of constraints that we will
just try to mention. So before we talk about the problem involved
let us try to identify the costs which are there in typical aggregate production planning. One of the major costs involved is procurement
cost. So whenever you are producing you have a typical
option of either buying the material or producing the material in either cases procurement is
to be made. You can either buy the material for hundred
rupees or then process it within your factory or you can procure the item may be for 500
rupees whatever it is. So costs of procurement have to be taken into
consideration and in fact all these costs could be fluctuating or varying with the time
period over the planning horizon. Then you have costs of production. Cost of producing a piece in the first period
might be 100 rupees. In the next period it might be 110 rupees
and then it might be constant at 110 rupees a piece and then it might go up again in the
sixth period to something else. Inventory holding costs the inventory holding
cost is that cost by which if you produce something in a certain period whatever is
not consumed will have to be held for that particular period and that is the inventory. There are costs of storage costs of spoilage
cost of pilfering etc. These are basically all aspects of inventory
holding costs and typically one of the most common components of inventory holding cost
is the loss of interest on the capital that you have invested. For instance if I stock something worth one
thousand rupees, if i had kept that one thousand rupees in the bank or given it to somebody
else, I would have been able to earn some interest on it by holding it in inventory
in the form of a car or a whatever product I have. I have actually foregone any returns on that
particular product. So what we mean by saying that there are always
inventory holding costs, Shortage losses associated with back orders and lost sales. So depending upon the situation that is if
you are not able to supply an order because you don’t have the quantity ready with you
then you incur a shortage and the shortage losses could be in the nature of back orders. So in a back order, the cost would be in terms
of rupees per unit time. Because you are satisfying the demand at a
later point of time but for a loss sale it is a one time. So it will be like so many rupees per units
lost. The element of time would not be there in
lost sales. So that would be the unit of a back ordering
cost and lost sales. Then we talk about costs of increasing and
decreasing the work force. These are typically the hiring and firing
costs that we were talking about. If you increase the work force the cost of
employing additional people training them and bringing them up to the mark so that they
can contribute effectively to your process. That is the cost of increasing the workforce. Cost of decreasing the work force could be
the cost of ideal capacity or it could also be the cost of loss of good will. So these types of costs are involved. Then we have the cost of overtime and under
time. Cost of changing production rates. What can happen is that there could be set
ups. There could be opportunity; losses etc and
these are the costs of changing the production rate. For instance if you are producing at a certain
level in the month of January and you decide to change the production rate in february
it would either mean that you have to change some set ups, make some new adjustments in
your plant and that is the kind of set up cost that you have here and or there could
be some kind of opportunity losses as a consequence of these kinds decisions. Let us now take a simple example to show how
aggregate production planning could be done. Suppose that we have data on demand for the
next let us say one year. So during the first period the expected demand
is 100 and I also have a cumulative demand figure which tells me what is the cumulative
demand up to the end of that period. So, in the second period the demand is180. The cumulative demand is now 280 and so on. In the third period the demand is 220. So this added to 280 will give me 500. Now the significance of this cumulative demand
is that it tells us that, up to the end of the third period the total demand for all
the periods is 500 units. Similarly the total demand up to the end of
the eighth period is 1500 units and this is how the demand has been fluctuating over the
various periods. So this information would be available to
us from a demand forecast. We have been talking about demand forecasting
in the previous two lectures. So basically the output of an exercise of
this kind would be that we would have access to information like what is the expand demand
over the next period. What we are seeing here is that there are
these are four week periods, so there will be thirteen four week periods in a year 52
weeks. So we have the demand figures available for
all the four week periods. So these are like four week periods in that
sense. The cumulative demand is 2600 for all these
thirteen four week periods and that means for the entire year. Now what we would like to do is to find out
that in the face of this uncertain demand what would it be what should be the policy
for the company to produce so that you can meet this kind of demand and we will approach
this problem from an intuitive perspective and we will see how we can solve this particular
question. We will use a graphical procedure and incidentally
this graphical procedure is the most common one with practical managers because they find
it simple and it does not involve any complicated mathematics and obviously the prize is in
terms of the fact that this leads to only a heuristic solution. It does not guarantee an optimal solution
but let us say it has certain advantages to offer. The first thing that you do in the procedure
is that you plot the cumulative demand. So the solid line which you see here is a
plot of the actual cumulative demand that we had in the table. So you plot the cumulative demand which is
shown here and you go up to the thirteenth period and you have this demand. Now let us consider two possibilities. We call them plan one and plan two. After all what we need is, what is the production
plan? A production plan is nothing but a path going
from this point to this particular point here and any line which goes from here to here
is actually a production plan. Since infinite number of production lines
can be drawn from zero to this terminal point therefore there are infinite numbers of production
plans. But normally what is the ideal thing for a
production manager? He would like to vary the quantity that he
produces every year. So the total quantity that is to be produced
in thirteen period’s is2600. So if we assume that the constant production
level is to be maintained at 2600 divided by thirteen, we should maintain a constant
production level of 200 per period and we would have a constant level of production. If you plot this plan this plan is actually
shown by this top dotted line, it is a straight line from here to here and that is what it
is. Do you notice something about this plan? One is the constant production. If you plan at a constant production rate,
you compare the actual demand forecast with this line. You find that the actual demand forecast for
most of the time except somewhere here is below this line. What does it show? It shows that the cumulative production is
greater than the cumulative demand which means if this particular line, that is the production
line is above the cumulative demand line we would have costs of carrying inventories. So one thing that you can immediately see
from this plot is that if we follow this policy of having a constant production level of 200
per units, in most periods we will be carrying inventories except in some periods where there
could be a shortage. This amount could be calculated as we will
do subsequently. So if the cost of carrying inventory is not
too high this should be a good plan because you are most of the times not incurring any
back orders or shortages. When you are carrying inventories and this
is what it is. Of course one might say that another possible
plan may be that you follow this demand line closely which means you produce exactly as
per demand. That could be a production policy. So if you produce exactly as per plan what
will happen is that your cumulative production will exactly follow this and you will neither
be carrying inventories nor there shortages. But the kinds of cost that you will involve
are cost of changing production rates every period. So there would be a situation where you would
not have to carry any inventories. You would not have to incur any shortages
but you will have to change production every period because you are following the demand. You are chasing the demand. It is actually a chase model and when you
are chasing the demand every time you are changing the period there is a cost involved. This cost could be through other hiring and
firing or it could be by varying production rates. So whatever it is in various ways, you would
have a situation like this. So this policy although good in terms of minimizing
the costs of carrying and back orders it would not be so good from the point of view of changing
the production rates because you will have cost of changing production rates will be
pretty high. Practically you will be changing the production
rate every month and this would mean havoc for the production manager. You would not like to do this kind of thing. So you might argue that can we not have an
intermediate plan and the intermediate plan could be that rather than changing this production
rate every period, we have minimum number of changes and then see. For instance by looking at this curve you
might find for instance that it is going up and there here. You could have a policy that let us say up
to the fifth month I maintain a constant production rate. So in these months I am having inventories
and in this period I am carrying shortages here. So the total production here the cumulative
demand up to this level is750. So 750 divided by 5 is 150. So the policy is that you carry on the varying
production level at a rate of 150 per period during the periods from one to five. So you get this lower dotted line up to here. Then you can look at this and say up to here
from this point from the fifth period to the eleventh period. You again calculate what the cumulative demand
here is and then see, divided this by the total number of periods and you get 250 per
period is what you produce during the period from 6 to 11. So you have production plan is something like
this and in the subsequent periods twelve and thirteen which are the last periods, you
again know that this is what you have to produce. So you are producing 175 per period during
the period from twelve to thirteen. So what is the advantage? The advantage is this is like a compromised
plan. It shows that there will be some holding inventory
some shortages but the number of changes in production are limited because during the
entire planning horizon first five periods we are changing, keeping the production level
at 150. Then we are changing it to 250 during this
period and then we are changing it to175. So one slope next slope next slope. Only these three slopes are involved and it
involves starting with this production period and then only two changes. One changes here another change here. In the earlier plan that we were talking about
where you are changing the production level each time, that would be more expensive. So the basic advantage of the graphical procedure
is that it helps us to in fact by looking at the demand profile suggest what could be
the possible alternative plans and this is the feature that practical managers love most. What they do is they can put down their plan
on it and then basically it can be evaluated. So if you put down a couple of plans on this
particular line you can then evaluate them and pick up the best one. Evaluation is generally done in a convenient
tabular form like this. We are talking about plan one which is constant
production. So production is constant in all the periods
and because of the varying we have the demand. So if you have produced hundred the demand
was hundred, so the inventory at the end of this period is going to be hundred and the
back order is zero and the capacity changes plus 20. Assuming that the capacity was set for something
else the capacity changes so much, the overtime is zero and the subcontracting is also zero
and you compile this table. Obviously what you notice from the plan is
the first plan is that there are back order costs generally. There is only one capacity change cost. There is no overtime cost, no subcontracting
cost at least in the first eight months but there are mostly costs of holding inventories. So depending upon the relative cost of this
you can determine what the total cost is. Capacity change means that if I am operating
at hundred units today and if I change this capacity to 120 in the next period there is
going to be some cost associated with that capacity change. That is what it is. So this plan for instance it does incur some
back order costs as I said in the tenth, eleventh and twelfth periods. So what you have to basically do is you know
these various cost parameters? You have to talk about the cost of production,
the cost of inventory, the cost of back order, the cost of capacity change, the cost of overtime,
the cost of subcontract and we now know each of these multiply with the appropriate figures,
you will get a here. So you will know the total rupee value in
terms of the various types of costs involved for this particular plan. We can do an exactly similar exercise for
the second plan. What was the second plan? We were keeping the production constant up
to the fifth period up to150. Then it was 250 and thereafter it was 175
in the last two periods. So you can calculate similarly the inventory,
the back orders, the capacity changes, the over times and the subcontracting which is
necessary. Of course this is all based on a certain set
of assumptions. We will talk about them in a short while and
ultimately even for this plan you will have this. So the inventory costs here are zero and back
order is there. Capacity change means reduction in capacity
over time costs, subcontracting all these costs are there. So if you know the relative values of these
particular costs, you can find out this particular component. This is the basic frame work that you can
follow for analyzing a plan. So you can compare plan one with plan two
with plan three, find out the overall costs and then take a plan which is the best for
your use. So this is how the graphical method will actually
be a way of choosing the best plan out of the ones that you have been trying to that
you have listed. So if you have not listed the optimal plan
obviously you will not get the optimal plan. It is a heuristic procedure but it is good
in the sense that it tries to consider all the costs involved. Now in the carrying out those computations
some assumptions were made which are quite commonly followed. All shortages were assumed to be backlogged
which means that we did not have any loss sales. If there was a shortage of 20, we said that
it is backlogged. Regular time capacity was taken to be 200
units per period. The maximum overtime capacity was taken to
be 20 percent of the regular capacity. So, 20 percent would mean that the maximum
overtime was 40 units per period. Overtime is generally preferable to subcontracting
and that was an assumption. Normally you would tend to produce something
on regular time followed by something on over time and if you cannot make it on overtime
then you will subcontract. That is the idea. So the relative preference was regular time
overtime and subcontract. Assumed initial inventory was zero. The initial regular time production capacity
was 180. So these were the figures that were assumed
that last year the company was operating at an initial capacity of 180. So that this here if you want to produce 200,
we have to change the production capacity to 200, so the production capacity change
costs are + 20 and that is what you found in the table. So you will always have a set of situations
something like this and you can use the graphical procedure which is the simplest method of
aggregate production planning to actually identify the best plan of action for your
situation. However the biggest limitation with the graphical
procedure is that it does not give us an optimal solution. It does not give us the solution which has
the minimal cost and therefore we have procedures which in fact are dependent to a very large
extent on the nature of costs and depending upon the nature of costs we can choose an
appropriate procedure. For instance if you have a situation where
the production cost in a particular period say Xt is the production and period and Ct
Xt is the production cost. If this is linear that means it passes through
the origin. It is like saying that then slope of this
line is the cost of per piece. So if it is whatever is the cost like this
it is a linear production cost, similarly the inventory cost. The inventory cost could be of two kinds,
it could be a situation where the inventory is positive which is the holding cost. The situation when the inventory is negative
is the shortage cost. What is positive inventory? It is your holding stocks. What is negative inventory? You have items on back order. So it will be one of the two situations. Either I can supply immediately 20 cars, if
I have more, stocked with me. So it is a situation where the inventory level
is high and if there is any demand less than this I can supply it off hand but in this
case there is a shortage. So I back order. If I somebody places an order for 20 cars
and I do not have 20 cars in stock I will say okay, the next two months I will manufacture
these 20 cars and supply them to you. That is back ordering cost but back order
is also at a cost. It is in terms of loss of goodwill whatever
it is. So there are these costs which are determined
by the slopes. You can look at it this way. If it is allocated some additional land, if
he is not stocking cars on it, he can probably give it on rent and on revenue on that particular
land which it is forfeiting by putting some cars on stock. It is a potential loss of an opportunity which
is a cost. That is what it is. So it would always be a situation of this
nature. So if you have linear cost of this kind that
is production costs are linear and the inventory costs are also linear, then you can use linear
programming for solving this problem. You can use a simplified transportation model
or you can use a transportation model for solving this particular problem. I am just trying to give you an overview of
the various solution procedures. In the next lecture we shall go into the details
of how these models can be applied for determining the optimal solution but the point here is
that these solution procedures that we have identified here are valid only if the cost
structure is linear for both production costs and for inventory holing and shortest path. Then you can use any of these procedures. On the other hand if costs are convex this
is now a case when all the costs are convex. What do you mean by convex a cost, that is
the production cost is something like this. What does it mean? If you produce the first unit the cost is
less, as you keep on producing more units the cost keeps on increasing. When do you think this would be a relevant
thing to happen? This would happen for instance when initially
you have lower costs and subsequently beyond a certain production level the costs are lower. So if you are talking about regular time production
over time production and subcontracting the cost slope would be progressively increasing
piecewise linear in that sense right So those costs are there for convex. In a convex cost if you join any two points
on this cost function, the line segment lies wholly above this particular cost. That is how you define a convex cost function. Similarly you will find that the inventory
costs are also convex. This is convex and in a situation like this
where the costs of production cost of inventory are convex again you can use a transportation
model but after piece wise linearization, piece wise linearization is quite common. Regular time, over time subcontracting the
cost will increase per unit or another very classic example of convex cost is a model
due to holt Modigliani muth and simon for people and they have developed what are known
as linear decisions rules or LDR’s and they are typically valid for convex costs of this
kind and in fact what has what they have assumed is that this is based on a case study in a
paint manufacturing company and for that paint manufacturing company they have approximated
all costs as quadratic costs of this type. This is a quadratic function and when you
differentiate quadratic function you get a linear function. The solution that you get is a linear decision
rule. That is how the term comes about. We will have a quick glance at this particular
method. If the costs are concave and piece wise concave
this is the most complicated category of costs. Mathematically you are saying that the cost
is first increasing and then flattening out. What happens is this is like economies of
scale the more you produce the lesser is the cost per unit. Similarly for inventory costs you have this
behavior and for concave and piecewise concave costs we can use dynamic programming for determining
a solution or we can use a network flow solution which is based on the shortest path in a network
developed by Wangner and Whitin. In fact it is Wagner and Whitin. It should be Wagner and Whitin. To
give you an idea of these linear decision rules let’s quickly run through this example. This was a study done by Holt Modigliani,
Muth and Simon in the in 1955 on a large paint manufacturing unit. What they did was they defined these variables. Wt is the work force level in period t; t
going from one to capital T, Xt is the aggregate production level in period and then we have
It is the actual production net inventory at the end of period and It star is the ideal
aggregate net inventory at the end of period t. What is done in this particular model is that
various types of costs are modeled. How are they are modeled? What is the regular time payroll cost? It will have a fixed component and a variable
component like this. This is the workforce Wt number of workers
in period t. So this cost is modeled by C1 Wt + C13 which
is a constant. So C1 and C13 is constants. Similarly workforce change cost. Workforce change cost is something like this. This is for shortage and this is for holding
inventory. So you have Wt — Wt –1 is the increase or
decrease in workforce. So this is the hiring cost. This is the firing cost. They approximate this function by this quadratic
component and this quadratic component is written as C2Wt — Wt 1 — C11whole square This is how the workforce change cost convex
costs. Then coming to the over time cost, over time
cost will also be that, beyond this level the cost goes up. This approximated by this particular function
and it is Xt — C4Wt whole square multiplied with C3 + C5 Xt — C6 X into Wt + C12 into
Xt Wt. It is nothing but a function of all Xt and
Wt. The Xt — Wt whole square, Xt term Wt term
and the product of the two terms, all the components are there with various parameters. Now these parameters will have to be determined
from the actual cost of the company. Similarly invented in related cost which of
this nature are approximated by function which is quadratic of this type and then you have
C1 It — It star whole square which is equal to C7 into It — C8– C9 into Dt whereas Dt
is the demand for the T th period whole square and these are all the costs and this is for
just to give you an idea that once you sum up all these costs and you take partial derivatives
with respect to W1 star and X1star which are your variables. You will get lineary functions of Dt W0 I0. What really happens is that W1star W X1star
are now becoming out linear functions of Dt W0 I0 in the previous period. Similarly W2 X2 will be linear functions of
W1 X1and so on. Those are the linear decision rules that this
particular model is actually trying to use. There are a number of other methods to handle
aggregate planning, aggregate production planning. These are some of the models which are available
in the literature for solving aggregate production problems this type. So a very common formulation is the linear
programming formulation. So if the costs are linear, you can easily
write down a LP to represent the costs and the various constraints. What would be the constraints? The constraints would simply be inventory
balance equations. I have so much, I produce so much. I sell so much and therefore at the end of
the period, I have so much. This kind of a constraint will have to be
written for each period. That is the inventory balance equation conservation
of mass equations. Then any other constraints that you may have
and the costs that we have discussed will constitute your objective function. This is the typical LP model formulation. Search decision rules was one category of
procedures which were actually determined by Taubert in the sense that you talked about
one type of rule, another type of rule that you are using and you can then talk about
a combination of these rules. It is like rule one into alpha + rule two
into one — alpha and you are trying to basically search with what particular value of alpha
you would have a rule which would give you an appropriate kind of solution. These have not been produced to not come out
very popular in aggregate production planning. Goal programming formulation is an extension
of linear programming and it would be useful when you are dealing with multiple goals. Parametric production planning is a procedure
developed by Jones in which basically some parameters are introduced which can change
over certain periods and through the process of optimization you are actually determining
the values of those parameters. That is the basic idea and a management coefficients
model is an approach in which you basically try to talk to different managers and find
out what is the optimum strategy. You then try to find out something like a
weighted average of the methods that are suggested or the strategies that are suggested by different
managers which might happen. For instance if you gave the example of the
graphical procedure we had two plans. So one could be the possibility of product
one manager one. Second one could be the idea of manager two. So what you can do is if you thought of may
be 75 percent confidence in this manager and 25 percent confidence in the other manager. You could take the two plans and find out
what would be the production quantity in individual period something like that. That would be the approach that you would
follow to determine this. There are a variety of methods for solving
the aggregate production planning problem. To conclude, let us summarize what we have
tried to see in this particular lecture. We have seen aggregate production planning
is relevant for fluctuating demands for medium term horizons typically in the range of 6
months to 1 year. I think that is one key factor that you have
to understand is that basically it is the horizon of 6 months to 1 year that we are
trying to see and we are talking about the fluctuating demand of the company and we are
trying to see how the resources should be allocated to meet that fluctuating demand. This is the problem. This is the problem that we have been trying
to see. A simple graphical procedure that generates
good solutions by examining the demand pattern was present. It was simple in the sense that you could
pose a procedure and then just find out what is the cost of that procedure. That was the idea and there could be various
ways to meet a fluctuating demand. It could be through hiring firing through
storing inventories through shortages through back orders etc. We saw that there were a variety of ways. In fact the very fact that there are a variety
of ways that is why the optimization problem is so intense. You have to find out what is the best way
and finally we had taken an example of solution procedures including the linear decision rules
where basically the costs were convex and because they were modeled by a quadratic function,
when you differentiated the cost function you actually landed up with linear decision
rules and this was one approach. Now in the next lecture we shall be talking
about some specific procedures that I will use for optimizing the production plan. That is finding out what is the best way to
determine the production plan. Thank you!


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