**Free Disposal Hull (FDH) approach — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/free-disposal-hull-fdh-approach/

Introduction to Free Disposal Hull (FDH) approach as an alternative to DEA models, focusing on convexity and technical efficiency estimation.

## Key Takeaways

- FDH is a non-convex alternative to DEA that can better model real-world production with indivisible inputs and outputs.
- Convexity assumption in DEA can be unrealistic and lead to hypothetical peers that complicate interpretation.
- FDH uses free disposability to construct the production possibility set without requiring convex combinations.
- FDH’s linear programming model differs by restricting lambda variables to binary values, simplifying interpretation.
- FDH efficiency scores can be computed with simple tools like Excel or MATLAB for small datasets.

## What the video covers

- The video introduces the Free Disposal Hull (FDH) approach as a subset and alternative to the conventional Data Envelopment Analysis (DEA) model.
- FDH is easier to estimate manually even with multiple inputs and outputs compared to standard DEA models.
- The session revisits the convexity assumption in DEA and explains its theoretical and practical limitations.
- Convexity assumes that any convex combination of feasible input-output bundles is also feasible, which may not hold true in real-world scenarios with indivisible or discrete inputs.
- The video discusses foundational criticisms of convexity, such as ignoring increasing returns to scale and marginal productivity effects.
- FDH addresses these limitations by relaxing the convexity assumption and better handling indivisible inputs and outputs.
- The concept of disposability (free disposability of inputs and outputs) is explained as a key property in constructing the production possibility set without convexity.
- The video includes a diagrammatic representation of free disposability and technical efficiency estimation under FDH.
- Linear programming formulations for FDH are introduced, highlighting differences from DEA, such as lambda variables restricted to 0 or 1.
- The video concludes with practical notes on implementing FDH efficiency estimation using MATLAB and Excel.

## Chapters

1. 00:00 Introduction and overview of FDH and DEA models
2. 01:36 Manual estimation and advantages of FDH
3. 02:38 Diagrammatic representation of free disposability and efficiency
4. 04:21 Convexity assumption and its mathematical formulation
5. 06:11 Criticism of convexity in real-world applications
6. 07:44 Foundational microeconomic criticisms of convexity
7. 08:49 Limitations of DEA and introduction to FDH as a solution
8. 12:31 Constructing production possibility sets without convexity
9. 15:37 Efficiency estimation and algorithmic approach in FDH
10. 18:28 Linear programming formulation differences between FDH and DEA

Answers

## Questions about this video

What is the main difference between FDH and DEA models?

FDH relaxes the convexity assumption present in DEA, allowing for modeling of production with indivisible inputs and outputs, making it more realistic in certain contexts.

Why is the convexity assumption criticized in DEA?

Convexity assumes that any weighted average of feasible input-output points is also feasible, which may not hold true for discrete or indivisible inputs like labor, leading to unrealistic or hypothetical peers.

How does FDH estimate technical efficiency differently from DEA?

FDH uses a linear programming model where the lambda variables take only binary values (0 or 1), avoiding convex combinations and simplifying the interpretation of efficiency scores.

## Full Transcript — Download SRT & Markdown

00:05

Speaker A

[music] [music] Hi, welcome back to the course Applied Production Analysis using MATLAB. In the last session, we saw one advanced model.

00:25

Speaker A

Uh, we can claim that as a DEA model. Basically, uh, that was the super efficiency model.

00:32

Speaker A

In this case, we consider one, uh, different model in literature mainly because you don't see much of the study using this approach. Few people consider it as an advanced model, but to me, it's not an advanced model. It is

00:54

Speaker A

basically a subset of the basic DEA model, and sometimes people call this as an MDSDA model also, mainly coming from the fact that the formulation remains more or less the same, but from a, uh, very rooted theoretical point of view, this is basically a model

01:15

Speaker A

less advanced as compared to DEA model. Someone gives you a data point if it is more number of inputs and output, multiple input and multiple output, it will be very difficult for you to plot and get a DEA efficiency score. But if I

01:36

Speaker A

give you a data set of, so say, four input and five output, still, uh, manually by putting a bit of effort, you will be able to estimate this model. That is the beauty of this model. Frankly speaking, um,

01:56

Speaker A

in the first instance, it looks like a very complicated framework, uh, because of which you don't see much of the application of FDH or whatever.

02:05

Speaker A

But from a, uh, estimation point of view, it is going to be even easier than our, uh, standard DEA model.

02:16

Speaker A

In the second instance, from a practical or an applied point of view, this model may become more realistic as compared to our earlier DEA models.

02:29

Speaker A

So that is coming from the first thing, the convexity assumption. We'll revisit the convexity assumption in this session.

02:38

Speaker A

We try to diagrammatically represent the free disposability, uh, and estimate the technical efficiency or get an idea about the technical efficiency of decision-making unit, and at the end, I would like to give you the linear programming problem involved

02:53

Speaker A

in the free disposability. So we'll revisit the concept of convexity. Basically, uh, among the set of assumptions that we are having, all observed input-output bundles are visible, disposability of input, disposability of output, convexity, and additionally, if you are assuming CRS,

03:15

Speaker A

that radial expansion and contraction, VRS or the not, the convexity became the most solid or most uncomplicated assumption over there.

03:33

Speaker A

So we have something called convexity. It was an assumption of our DEA. We kept it as, say, fourth assumption or whatever in the same order what we follow. So here, say, if x a and y a is

03:59

Speaker A

feasible, say, let's put x a and y a is

04:21

Speaker A

feasible and disposability we represent by expressing that the output-input, input-output bundle is a part of the technology set, and alternatively, you have x b, y b as another input-output combination that is also feasible.

04:50

Speaker A

So now, by convexity, lambda x a plus 1 minus lambda x b, that is becoming the input bundle that we are having, and the same weightage lambda y a plus 1 minus lambda y b, this is going to be the

05:14

Speaker A

point that you can conceptualize by following the convexity property. And if this one, these two points are feasible, which is part of the technology set by formulation, this convex combination should also become feasible under DEA.

05:30

Speaker A

This is basically the data point. We were having these two points, and we are connecting these two points, and our technology set becomes convex and piecewise linear. The piecewise linear we are getting from the fact that two points are feasible. Any point in

05:39

Speaker A

between also you can represent in terms of a convexity property, and that will also become feasible.

06:11

Speaker A

But can convexity be controversial in reality? Sometimes it can happen. Say you have A using, say, six units of input, B using seven units of input, and we are considering that as labor. The convex combination, like say lambda A

06:28

Speaker A

and lambda B, you are taking value 0.5 such a way that it adds to one, and in the convex combination that you are going to get, it is going to be 6.5.

06:52

Speaker A

Does it make any sense? Especially when you are talking about labor, and we are asking a firm to use 6.5 labor, which is going to be controversial in this context. So any context where you have an indivisible technology or

07:07

Speaker A

the input values are discrete values, convex combination does not make any sense. Alternatively, say we had output bundle here. Say it is producing 10, and it is producing 12.

07:24

Speaker A

And now we are talking about a convex combination which uses 6.5, and it will become 11 as the output with the, uh, lambda A, lambda B formulation that we are doing.

07:44

Speaker A

Can this also be criticized from an application or a real-life or very foundational microeconomic point of view? Yes, it can be, because here moving from six to seven, the firm is, uh, operating at a larger scale here, not a

07:54

Speaker A

sudden jump, but if 60 to 70, it would have been a sudden, like a huge, uh, jump in terms of the scale of operation.

08:05

Speaker A

So, and also our micro theory says that there can be increase in returns to scale or the marginal productivity of labor may improve when there is an improvement in the scale of operation,

08:23

Speaker A

expansion in the scale of operation. But the moment you consider convex combination and get a more of a pseudo data, which is basically the hypothetical data that you are constructing using the convex combination according to the weight being assigned by the linear

08:36

Speaker A

programming problem, it does not take into account the marginal productivity and the increasing marginal productivity nature of the conventional production framework.

08:49

Speaker A

So these are the very foundational criticisms that we are having in the context of, uh, convexity, and the moment you have convexity as a property within or convexity as an assumption that you embed into a DEA

09:12

Speaker A

model, it will be open to this network system also. Going beyond, sometimes it becomes very complicated for us to suggest any peers, like say if firm C is there, and we are asking for 0.5 of firm A, 0.5 of firm B to become

09:31

Speaker A

technically efficient, it becomes very complicated. It becomes even confusing for the firm to see what to do. It's like, uh, parents asking, um, someone doing good in math and another one doing good in, say, uh, science, you're asking them to take a

09:45

Speaker A

convex combination of that and asking, okay, you do like, uh, in between this much of, uh, person one and person two, but it is going to be even complicated in the DEA use case.

10:05

Speaker A

So to point another limitation, the conventional DEA model may give you a peer which is going to be a hypothetical peer or hypothetical, uh, reference point with multiple peers, which is going to be difficult for our, uh, policy point of

10:48

Speaker A

view. So now, as a solution for these two points, not particularly these two points, FDH may solve even problems beyond convexity and, uh, the indivisibility nature. Here we see particularly these two problems as the main problem.

11:06

Speaker A

So what does free disposability hold? Basically, when we are doing the DEA model, we had it in the back end. The first instance what we, uh, did empirically, it was constructing a free disposability set for inputs, and that becomes the input

11:19

Speaker A

free disposability of input, and then same for input disposability of output. That was the way we extended our, uh, individual data points to construct the entire, uh, production possibility set.

11:43

Speaker A

So what does this, uh, disposability hold us? So we can take the case of one input, one output case. So here, say this was the output and input that we are having, and these are the six data points that we

12:03

Speaker A

are considering for our symbol match. So now going ahead, we need to construct the production possibility from here with disposability of input and disposability of output. But we are not imposing any convexity property, and we are considering, uh, the disposability case

12:31

Speaker A

over here. For that, consider this point since this point is feasible means, say, x a, y a, any point beyond this should also be feasible. That means if firm A is producing y a unit of output by using x a unit of input, any point

12:47

Speaker A

beyond that should also be possible. Firm should be another firm's. So the same firm can use input more than that x a and should be able to produce the same y a amount of output.

13:08

Speaker A

Similarly for this observation, we can get th

13:26

Speaker A

So once you have these free disposal uh points or we can call it as technology set TFDH.

13:40

Speaker A

This is basically the set of all x and y. And here we are considering x0 y 0 to be part of our technology free disposability.

14:04

Speaker A

That means it is technically feasible any point here we had x1 up to x 6. So once we have these uh points, this is going to be the ender point that is basically a combination of all x and y such a way

14:24

Speaker A

that that x that you are considering should be greater than or equal to x0 and the y that you are considering here should be less than or equal to y 0.

14:41

Speaker A

This is basically the very foundational uh way to explain the technology set here and this thing will become um it becomes real values with only positive values and we can write it as P + 1 P + Q here

15:00

Speaker A

it is going to be 1 + 1 it is going to be a two-dimensional context.

15:06

Speaker A

So once we have this technology set in their technology set we can define the boundary of the technology set as the predisposable frontier. So this is going to be this is going to be the FDH from D only

15:37

Speaker A

FDH from over here and here since these observations are lying on the frontier all of them will get a technical efficiency under FDH whether it is input oriented or output oriented as one as against that since this point it is

15:56

Speaker A

lying away from the from if you consider output oriented case it may get a technical efficiency less than one but close to one but if you consider input oriented it will be compared again this observation and it is going to be

16:09

Speaker A

a point far less than one in the context of input oriented technical efficiency the philosophy of projecting remain the same as compared But here as again a hypothetical observation. So here if we had a DA model our model would have looked like this.

16:33

Speaker A

We by convexity they may connect these two points also. So in that case the projection should have been here.

16:42

Speaker A

But here in this case of free disposability all the projection is somewhere here. That means the technical efficiency that you are estimating using FDH will be greater than or equal to technical efficiency under DEM whether it is CRS or VRS.

17:08

Speaker A

Okay. So what is what are the other complication that you are going to get in this context. So here it is an algorithm that search for uh whether other firms are dominating a particular firm in terms of their input

17:29

Speaker A

usage or output uh creation. And then since we are not considering any uh convex combinations into the framework, the FDH framework is likely to give more observation with value one and also it is going to be very sensitive to outliers than the uh

17:50

Speaker A

sensitivity of our DEA model for our outliers. That is one uh complication. We'll discuss it. uh we'll see that uh how to estimate and how how what are the problem that we get from an empirical application.

18:07

Speaker A

Now we see how to formulate this problem FDH the LP formulation there is nothing new that we are going to introduce we'll be using the same framework what we had for our DEA model but here these values we cannot have a convex

18:34

Speaker A

values convex combination So lambda J.25 was possible in the context of our uh DA model. If it was VRS, the other lambdas will take care of the remaining 75.

18:53

Speaker A

But in case of FD this lambda has to be zero either zero or one. So here you can say in this case this will become the observation and say this was a b c and in this case of c l

19:13

Speaker A

lambda b will become one and lambda a or d if we had a d here become zero. This was the context. But if it was a DEA model, we would have get lambda A not equal to Z and lambda B also not

19:32

Speaker A

equal to zero. When you are estimating the technical efficiency of form C to make it more clear, we follow an output oriented VRS case or O VS. For that we have to maximize fee which is called as f star

19:55

Speaker A

maximum value that we are going to get at the end. We can have say uh one input one output case we'll consider first sum of lambda j yj should be greater than or equal to f of yk and we are considering dmu

20:19

Speaker A

k for the estimation of technical efficiency of under fdh sum of lambda j xj J should be less than or equal to X K. We have sum of lambda J should be equal to 1. These are very familiar to

20:40

Speaker A

you what we had it in the context of DEA model. But as against that lambda J will take only either zero or value one. That is the framework that we have it in MDH as compared to our DA

21:01

Speaker A

model. Here you can generalize this model. Say if you have R say Y RJ XRK and R 1 up to say M. We can have x i j and x i k where i one up to say n where

21:33

Speaker A

m is the number of outputs and n is the number of inputs From an estimation point of view, the lambda js it can take either zero or one. Basically, these are the integer values.

21:57

Speaker A

And at the same time, we have a sum of lambda is equal to one. So this program has this constraint lambda j can take only uh value zero or one and sum of lambda has to be equal to one. So this makes a

22:14

Speaker A

case that uh this has to be solved by using a mixed integer programming. If some of you want to approach this problem from an operation research point of view, I think uh the mixed it it it will be solved by using

22:38

Speaker A

integer programming program. But here uh in our case we try to do it from a data point of view by seeing in the first instance we see whether any firms are dominating firm K in terms of their output generation.

22:57

Speaker A

If s we go and see whether uh this firm that dominating the firm K in terms of output whether they are dominating the same firm in context of inputs and then based on this information and we get a new set

23:14

Speaker A

of firm which dominates firm K in terms of its output and as well as in terms of input and then take the ratio of that and we get an idea uh whether this firm K is efficient or if not efficient what

23:29

Speaker A

is the inefficiency score that you can assign or what is the level of efficiency score that you can consider.

23:36

Speaker A

So here this is the of lambda = 1 lambda is equal to 1 so and so and once you get f 1 by f star say we get lambda j star and fear as the solution and 1x star can be technical efficiency

24:00

Speaker A

under fd. Okay, so just to summarize uh this was a big error but I would like to clarify the sum of lambda equal to 1. This is similar to our convexity constraint. And just to summarize in this case, we

24:16

Speaker A

revisited our idea of convexity in the context of DEA model and how or why convexity does not make in the real life scenario. Especially when you are having indivisible uh inputs where your inputs can take only integer value it become complicated

24:35

Speaker A

for a convex combination and the referral point become a divisible value non- integer value or continuous value.

24:47

Speaker A

Then another constraint or another limitation of conventional DEA it does not take into account the uh dynamics involved in the marginal productivity and it simply takes a convex combination. sometime that ignore the margin of productivity uh or increasing

25:04

Speaker A

margin of productivity or increasing returns to scale context as a uh result or just to overcome these limitation we proposed our predisposability model and then diagrammatic manner this was the uh predisposability hull that you can consider construct and the philosophy of

25:23

Speaker A

efficiency estimation remain the same as in the case of DEA And the technology can be defined this manner. And here we started with the physible point. Sometime you can conceptualize disposability for each data points also. Here we are defining

25:42

Speaker A

the technology set for the entire uh data point. So and so then we saw how to show we saw how to estimate this from a u linear programming problem point of view. But it's not a linear program. is

25:57

Speaker A

basically mixed integer programming. Here the same constraints input and output constraint that we are having in the context of uh DEA but only constraint the additional constraint that we are having basically the sum of lambda J can take only value zero or one

26:15

Speaker A

in this context. So in the next session we will try to estimate predisposability of a very simple data. Unfortunately I couldn't find any inbuilt package.

26:25

Speaker A

Initially I was thinking that inbuilt package is not coming from the fact that the model might be so complicated and it involves mixed integer programming so and so but the model is so simple uh by simply modifying your data and getting a

26:38

Speaker A

reference point. You can even estimate the model using a excel. So we'll see how to program that in our uh matlab framework and see how to estimate the free disposable efficiency scores of small data set that we are having. Thank you. [music]

26:59

Speaker A

[music]

Topics: Free Disposal Hull FDH Data Envelopment Analysis DEA convexity assumption technical efficiency production possibility set linear programming applied production analysis MATLAB

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