Remove excess width

Objective

Based on price model, we get the price expression P(x,y) = f(x,y)/g(x,y) = (aX+bY)/(cX+dY). For this kind expression, there is excess width question for interval calculation. The reason for this question is that random variable is cited more one time in expression. To resolve the question, it is necessary to remove excess width in the price model.

Solution

The easiest way to handle it is to simplify the price expression such that random variable is cited only one time. It is a quick way to handle this question. But it is very restricted constraint. For most expression, it couldn’t be simplified to so kind condition.

According to the price expression, we can use another way to remove excess width. It is to use the low and high bound of interval to calculate the expression. Then from these calculated values, the result bound is determined. For two variables, there are four bounds. Every time two bounds are taken to expression to calculate the result. So total 4 values could be got. We select the minimum as the low bound of result interval, the maximum as the high bound of result interval. Let us see the example:

Supports: x = [1,2], y =[2,3] p(x,y)= (8.4x + 7.2y)/(0.04 x + 0.02y)

First: let x=1, y=2 calculate the p(x,y), we get the value 285

Second: let x=1, y=3 calculate the p(x,y), we get the value 300

Third let x=2, y=2 calculate the p(x,y), we get the value 260

Finally, let x=2, y=3 calculate the p(x,y), we get the value 274.3

So interval for p(x,y) is [260,300]

If we calculate the expression based on x and y interval directly, we can get interval [162.9, 480]. It is obvious that result interval is excess width. So we can use this method to remove excess width.

Limitation

Although the method to select the min and max value for result bound works for our case, we can see limitations for this method. When this expression is monotonous, the method can handle excess width. How about other case?

Obviously, the denominator of the expression is not including the zero, that is, zero is not result from g(x,y), otherwise this expression couldn’t be calculated.

The derivation’s direction of f(x,y) and g(x,y) couldn’t be changed in the interval of x and y. it means that f(x,y) or g(x,y) couldn’t be changed from monotonous increasing to decreasing or from monotonous decreasing to increasing. From the following figure, we can understand this constraint clearly.

In this figure, the method couldn’t be used in the interval x=[x1,x2] and y=[y1,y2] for the derived direction is changed at point A. but it works for the interval x=[x2,x3] and y=[y1,y2].

Another condition, the method also couldn’t be used when g(x,y) is close to infinite. Considering follow example. P(x,y)= f(x,y)/g(x,y) = (8.2x + 7.2y)/(y*(x-1)) for interval x =[0.5,1.5], y=[1,2]. For this case when x is close to 1, it makes the value of p(x,y) infinite. The following figure demonstrates this kind question.

Future

How to remove these limitations?