Wednesday, April 24, 2013

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



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 (2nd 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.