The New Theorems to Measure the Distance among Triangular Fuzzy Numbers

In the reference with the article of “Fuzzy Method for Decision Making: A Case of Asset Pricing Model”, you can find the important pionts about the final part of above article which described new definitions and laws of the distance measurement among fuzzy numbers.

 The purpose of this article is to demonstrate the new theorems for measuring the distance among triangular fuzzy numbers where these theorems have been inferred from the method modified by Hsieh and Chen (1999).

Before releasing these theorems, let me divide the debate of the distance among triangular fuzzy numbers into two types as follows:

1)      All triangular fuzzy numbers have been limited into interval [x1, x2] in which “xm” referred to membership function µ(x) is the same for all fuzzy numbers and equal to 1.   For instance, if we consider the fuzzy numbers of “A”, “B”, “C” and “D”, we will have:

A = (x1, xm, x2)

B = ((x1< x < xm), xm, (xm < x < x2))

C = ((x1< x < xm), xm, (xm < x < x2))

D = ((x1< x < xm), xm, (xm < x < x2))

Fig (1) shows us the concept of type (1):

2)      In this case, there is not the limitation for independent variable of “x” and also “x” assigned to membership function µ(x) = 1 is not the same for all fuzzy numbers in which fuzzy numbers A, B, C and D can be considered as follows:

      A = (x1, x2, x3)

B = (x4, x5, x6)

C = (x7, x8, x9)

D = (x10, x11, x12)

Fig (2) illustrates the concept of type (2):

 

The method Modified by Hsieh and Chen (1999)

According to Fu (2006), there are many approaches to measure the distance between two triangular fuzzy number such as Bortolan and Degani (1985), Liou and Wang (1992) and Heilpern (1997) who applied a geometrical distance measuring and his method was modified by Hsieh and Chen (1999). This modified method is described as follows:
If we have two triangular fuzzy numbers below cited:

 

                   

Then, we can calculate the distance by below function:

The New theorems for measuring the distance among triangular fuzzy numbers

In the reference with above function, new theorems can be listed as follows:

Theorem (1)

Assume, we have three triangular fuzzy numbers A, B and C where:

Theorem (2)

According to theorem (1), we can define theorem (2) as follows:

Theorem (3)

If A = (a1, a2, a3) and B = ((n + a1), (n + a2), (n + a3)) are triangular fuzzy numbers so that “n” is real numbers (n ϵ R), Then we can define the distance between A and B as follows:

d (A, B) = n

Theorem (4)

If triangular fuzzy number “Ж” is the sum all triangular fuzzy numbers A, B, C, D, E…

We have:

Ж = A + B+ C + D + E+…    and    Д = B+ C + D + E+…   

Then we will have:

Д = Ж – A

Therefore, in the reference with Theorem (1), we can define below mentioned:

d ((Ж – A), Ж) = d (A, 2A)

Or

d (Д, Ж) = d (A, 2A)

To be continued…….

Reference

- Fu, Guangtao. (2006). A fuzzy optimization method for multicriteria decision making:

An application to reservoir flood control operation. Expert Systems with Applications, 34, 145- 149.

Newton's Law of Cooling and Fuzzy Method for Decision Making

Following to article of “Fuzzy Method for Decision Making:  A Case of Asset Pricing Model”, the purpose of this article is more practice of this method by presenting a case in physics which is about Newton’s law of cooling. At the first, a case has been explained then reviewing Newton’s law of cooling and finally the application of method mentioned in above article to analyze the case is demonstrated.  Of course, there are many cases to analyze by this method such as the circuit electric of R – L in which we have:  i = V/ R [1- (e^ (-Rt/L))] in the field of electrical engineering, Benjamin Franklin (1706-1790)’s devise about composite interest rate in finance, Compressor Performance Control to track the distance between the operating point and set point, Head Generated by the Circulation Pumps to find out the operating point which is the intersection of pump head and circuit losses in mechanical engineering and so on.

I have chosen Newton’s law of cooling because it is the general case for all engineers.

Case Study: Predictions in temperature transferring 

Mr. X is willing to decrease the temperature of a pot filled of water boiled at 100 degrees C. He puts this pot into a sink full of water at 5 degrees C in which water temperature into sink will be stayed the constant at 5 degrees C during the period of the time. He would like to predict the temperature of water into pot after one hour. How?

We as well as know that there are three mechanisms to transfer the temperature which are Conduction, Convection and Radiation. This case is referred to only Conduction.

Definitely he should use Newton’s law of cooling.

Newton’s law of cooling

In early 1701, Isaac Newton observed that the rate of heat loss a hot area during the period of the time has directly the relationship with the difference temperature between this area and adjacent area where the temperature of adjacent area (surrounding) will be the constant during the period of the time. Where we have:

T0 = the temperature of hot area in time zero

Ta = the temperature of adjacent area (the constant)

T (t) = the temperature of hot area at time “t”

T (t) > Ta, T = T (t)

We can replace above proportional relationship to a differential equation, if we include a heat transfer coefficient (k) which is positive. Since we have: (T – Ta) > 0, the trend of the curve will fall down. Therefore, we should consider a negative mark for this differential equation as follows:

Solution of differential equation:

We have:

In the reference with data included in the case study, the rate of heat loss will be as follows:

As we can see, Mr. “X” will be able to predict the temperature of water into pot after one hour, if he finds out a constant amount for “k”.

One the best ways to obtain “k” is to utilize Fuzzy set theory.

Fuzzy Method for Decision Making

Here, I am willing to apply fuzzy method mentioned in my previous article to calculate “k” step by step as follows:

Ø  On the spreadsheet of excel, we enter a algorithm to calculate T (t)

Ø  We use of the formula of µ = T (t) / T0 to change rate of temperature loss to Fuzzy membership function assigned to “t”

Ø  If we have the heat loss by falling down the trend ((T – Ta) > 0), it can be also considered vice versa. It is means, if the temperature of pot water is 5 degrees C and the temperature of sink water is 100 degrees C (the constant), we will have the increase heat into pot where the trend of the curve will go up ((T – Ta) < 0). Therefore, the differential equation will be the same and we can do above two steps for this situation (Please see below algorithms on my spreadsheet of excel).

Ø  The most important step is to find out the domain for rectangular k – t. As the first try, I start by method of try and error in which I reach a constant “µ” or where the rate of changes for “µ” will be negligible.

Ø  By Using of “µ” discovered from previous try, I utilize PLUG order in excel to limit “µ” to zero in which it is named α – cut.

Ø  As the second try, I use from the table of two way sensitivity analysis for rectangular k – t where the domain has been extracted from previous step.

(Please see Appendix I)

If we review Appendix I, we will see that the range of “k” is between 0.0106 and 0.1 (per min) while the range of “t” is between 1 (min) and 45 (min)

Ø  As the final try, I repeat to make the table of two way sensitivity analysis for rectangular k – t in accordance with above ranges. (Please see Appendix II and III)

Appendix II is for status of warming to cooling and Appendix III is for status of cooling to warming.

The final result has been presented in below diagram:

As we can see, the most density has been included into triangular ABC. Therefore, gravity centre of this triangular (ABC) is the answer for “k” where the coordination of gravity centre is:

t = 13.5 (min)

α – cut = 0.54

According to Appendix II or III, we can see that the answer for “k” is approximately 0.055 per min. Thus we have:

 

In the result, the temperature of pot water will be about 8.5 degrees C after one hour.

Now, you can do an experimental test at your home to prove this method.

Perhaps, we say that the solution of this case is very easy because Mr. X can measure the temperature of pot water after next 5 minute then he can solve above function to get “k”. Yes, it is true but we should consider that there are some adjacent areas into the space or even into the earth with different temperatures where we have not access to measure the temperatures during the period of the time.

 

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this method as a template for further practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call me on my cellphone: +989109250225. Please be informed these spreadsheets are not free of charge.”

Appendix I

 

Appendix II

Appendix III

Fuzzy Method for Decision Making: A Case of Asset Pricing Model

There are always many ways to solve the problem but we should only choose the best way among them to take an action. How can we find the optimized choice? The process of optimization the ways for solving of the problem is defined as decision making in which the output of this process will be an action. Nowadays, we need to use the process of decision making in all sciences such as Strategic Management, Financial Management, Engineering, Medicine and especially Political science. When we refer to this process, we can often see that there is the lack of information or vague values for the parameters which are affecting on the case something like uncertainty on data and so on that it is named as Fuzzy environment. Fuzzy Set theory gives us the opportunity to grade all alternatives (ways) where we should abandon the approach of 0, 1 in our mind. In fact, this method is one the best tools to solve the paradoxes.

The purpose of this article is to apply intersection of fuzzy sets to obtain the results (alternatives) in the frame of a new fuzzy set as the process of decision making where the alternatives are a crisp set and maximum membership function of this new fuzzy set gives us the best alternative (best way) to take an action. Then a new method of fuzzy number intersections has been demonstrated. A case study of asset pricing model is given to perceive purposed method. Finally, some laws and a new formula about distance measurement among fuzzy numbers have been listed.

Methodology

According to Bellman and Zadeh (1970), if two fuzzy sets “Д” and “И” have the intersection, there is a new fuzzy set “Ж” with membership function µ Ж (x) in which “Ж” is named the decision as follows:

Ж = Д ∩ И = {(x, µ Ж (x) |x ϵ [a1, a2], µ Ж (x) ϵ [0, α ≤ 1]}

Where interval [a1, a2] is a crisp set which is named the alternatives consequently for fuzzy set “Ж”, we have:

µ Ж (x) = min (µ Д (x), µ И (x))      and      x ϵ [a1, a2]         

Here is a schematic figure of above definition:

Fig (1)

We definitely apply the maximum degree of membership function of set “Ж” that it gives us the maximum “x” among alternatives. Therefore, we have:

x max  = {x |x ϵ [a1, a2], µ Ж (x) = max (min (µ Д (x), µ И (x)))}

Above schematic diagram shows us the same left and right sides of continuous piecewise – quadratic fuzzy number. We can also use triangular and trapezoidal fuzzy numbers to demonstrate the concept of this method as follows:

Fig (2)

Case Study: Asset Pricing Model

Mr. X (or company X) is willing to launch a finance plan as follows:

According to his economic study, the parameters such as money supply, monetary base, money multiplier, interest rate, inflation rate, Demand – Supply curves and so on show a fuzzy environment. In fact, there is a stagnated market to transact some assets in which you cannot easily liquidate them to get cash money while the attractive markets including stock markets, corporate bonds have a high volume of transactions. He says: “Where money goes?” He starts his finance plan by the schedule in the period of 6 months. According to his projection, If he sells his second car to invest in corporate bonds and some stock markets, the internal rate of return (IRR) will be about 26% after 6 months whereas the increase valuation of his second car plus the costs of transportation and so on will show his IRR equal to 18% during the period of next 6 months. Therefore, the sale of his second car and investment on stock markets and corporate bonds will be ok. But how can he sell his second car in the stagnated market? He has to advertise his second car on websites or newspaper which will have the cost. How many times should he advertise his second car? It is clear, he should pay the cost for each time advertisement. Several times of advertisement will have several times of making payment. How can he optimize his cost of advertisement? He has to find the best price (the best way or the best alternative) for advertising where the increase of calls by buyers will proceed to make a deal. In a fuzzy environment, he has considered to apply fuzzy method to find the best price for advertising step by step as follows:

Ø  He review second car prices which are the same model with his car and they have advertised on websites or newspaper during the period of the last week.

Ø  To calculate the average of the prices collected from Internet and newspaper.

Ø  He will look for some agents who will certainly buy his second car by cash but very chipper than the prices advertised on Internet or newspaper.

Ø  To calculate the average of the prices quoted by the agents.

Ø  To get the maximum price advertised on Internet or newspaper

He arranges the set of alternatives (crisp set) for right and left side of fuzzy number as follows:

Right side of fuzzy number:

Average of prices quoted by the agents = $ 5000

Maximum price advertised on Internet or newspaper = $ 10000

Referring to Figure (2), we have:

Д = {(x, µ Д (x)) | (x ϵ [0, 5000], µ Д (x) = 1), (x ϵ [5000, 10000], 0 ≤ µ Д (x) ≤ 1)}

Left side of fuzzy number:

The average of the prices collected from Internet and newspaper = $ 6400

Maximum price advertised on Internet or newspaper = $ 10000

In the reference with Figure (2), we have:

И = {(x, µ И (x)) | (x ϵ [6400, 10000], 0 ≤ µ И (x) ≤ 1), (x ≥ 10000, µ И (x) = 1)}

Finally, we will have fuzzy set “Ж” as follows:

Ж = Д ∩ И = {(x, µ Ж (x) |x ϵ [6400, 10000], µ Ж (x) ϵ [0, α ≤ 1]}

Therefore, crisp set of alternatives is interval [6400, 10000]. Figure (3) shows us that the best price of the first try to advertise is equal to $7900 with confidence level of 0.42 (α = 0.42).

Fig (3)

 

New Method of Fuzzy Numbers Intersection

The most crucial problem to use above method is to take the mistake about minimum prices. The most of time, we have doubt about minimum prices quoted by buyers. On the other hand, considering the average of the prices collected from Internet or newspaper as minimum price of left side fuzzy number is not really logical and accurate price in which this is also the same for right side fuzzy number. What can we do?

One of the best ways is, to define the range for minimum prices instead of a fixed average price. Then we will apply two ways sensitivity analysis for the range of minimum prices and “x” which is the price variations from 0 to maximum prices. In this case, we will have an area of intersections by fuzzy numbers on our diagram. The best price of first try to advertise will be calculated by two methods as follows:

Ø  Applying Finite Element Analysis (FEA) on spreadsheet of excel

Ø  Calculating the Gravity Centre of the area

In this article, I will utilize the second method which is to calculate the gravity centre of the intersection area.

Note (1): “The professional people who are interested in applying Finite Element Analysis (FEA) on spreadsheet of excel, can use below link which is an article of “Mechanical and Electromechanical Systems

Simple Thermal Heat Transfer Finite Element Analysis in Excel 2007” by Tom Scarince on April 1st, 2010:

http://www.thsengineering.com/2010/04/thermal-heat-transfer-finite-element-analysis-excel/”

Now, let me start this new method by two examples as follows:

Example (1)

Mr. X (or company X) believes that the range of minimum prices quoted by buyers is into interval [$5000, $10000]. He also considers the range of the prices collected from Internet or newspaper into interval [$5000, $10000]. The results are as follows:

 Right side of fuzzy number:

The range of minimum prices quoted by the agents = [$5000, $10000]

Maximum price advertised on Internet or newspaper = $ 10000

Left side of fuzzy number:

The range of the prices collected from Internet and newspaper = [$5000, $10000)]

Maximum price advertised on Internet or newspaper = $ 10000

According to the method defined for fuzzy number intersection, the final results can be presented in Figure (4):

Fig (4)

If we calculate the gravity centre of area (ABCD), the best price of the first try to advertise will be equal to $8883.33 with confidence level of 0.52 (α = 0.52).

Example (2)

Mr. X (or company X) discovers that the range of minimum prices quoted by buyers is into interval [$4000, $6000]. He also gets the range of the prices collected from Internet or newspaper between $6000 to $8000 and the maximum price is still equal to $10000. The results are as follows:

 Right side of fuzzy number:

The range of minimum prices quoted by the agents = [$4000, $6000]

Maximum price advertised on Internet or newspaper = $ 10000

Left side of fuzzy number:

The range of the prices collected from Internet and newspaper = [$6000, $8000)]

Maximum price advertised on Internet or newspaper = $ 10000

According to the method defined for fuzzy number intersection, the final results can be presented in Figure (5):

Fig (5)

If we calculate the gravity centre of area (ABCD), the best price of the first try to advertise will be equal to $8183.33 with confidence level of 0.4 (α = 0.4).

This method can be also used in business, finance, strategic management, political science, engineering, and medicine and so on. For instance, in financial management, we can replace dividend, time series analysis of share prices, IRR and NPV of the projects instead of second car price.

Case Questions:

Ø  What is the impact of sale a second car by using of above fuzzy method on overall market? Can we say this transaction will be considered as the index for new situation of market and new minimum price for this fuzzy model where new fuzzy numbers will go ahead?  

Ø  Assume, Mr. X (or company X) is willing to sell 50 second cars simultaneously. What is the impact of sale simultaneously 50 second cars by using of above fuzzy method on overall market? Can we say these transactions will be considered as the index for new situation of market and new minimum prices for this fuzzy model where new fuzzy numbers will go ahead? 

Ø  How many second cars should be sold to move overall market? Should we add the application of Game theory to this fuzzy model to answer this question?

Ø  If Mr. X (or company X) sells his second car by $8000, he has to buy the same model car by $11500 after 6 months. Will his investment cover his lost?

 For answering to above questions, we should try to learn new definitions and laws of distance measurement among fuzzy numbers. Let us review some basic rules in this case as follows:

If we have triangular fuzzy numbers Д, И and Ж:

(1)   d (Д, Д) = 0

(2)   d (Д, И) = d (И, Д)

(3)   If d (Д, И) < d (Д, Ж), then И is closer to Д.

(4)   The New Inequality Theorem in Fuzzy Numbers (refer to link: http://emfps.blogspot.nl/2012/02/fuzzy-delphi-method-to-design-strategic_06.html)

Here, I am willing to demonstrate two new laws or formulas as follows:

Ø  If we have triangular fuzzy number Д = (a, b, c) and interval [x1, x2] where:

a ≤ x1≤ b  and   b ≤ x2 ≤ c and  ∆x = x2 - x1

By using of geometrical theorems, we can easily prove below formula:

 µ = 1 – [∆x / (c – a)]                   0 ≤ µ≤ 1

In the result, we have:

Ø  In the reference with above formula, if we have two triangular fuzzy numbers

 Д = (a1, b1, c1) with membership function µ Д (x) and И = (a2, b1, c2) with membership function µ И (x) and below conditions:

a1 ≤ a2 ≤ b1   and     b1 ≤ c2 ≤ c1  

Then, we can get below formula:

(c2 - a2) / (c1 - a1) = 1 – µ Д (x)

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this method as a template for further practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call me on my cellphone: +989109250225. Please be informed these spreadsheets are not free of charge.”

To be continued……

References:

- Bojadziev, George., & Bojadziev, Maria (2007). FUZZY LOGIC FOR BUSINESS, FINANCE, AND MANAGEMENT (2^nd ed.). London:  World Scientific Publishing Co. Pte. Ltd.

- Fu, Guangtao. (2006). A fuzzy optimization method for multicriteria decision making:

An application to reservoir flood control operation. Expert Systems with Applications, 34, 145- 149.