Generalized Riser Design by Parametric Fuzzy Geometric Programming

Abstract

The optimal design of risers for castings has been the subject of numerous investigations and this paper determines the optimal dimensions when Chvorinov's rule for solidification is obeyed by both the riser and casting and solidification time is the only constraint involved .This problem can be formulated to minimize the riser volume subject to the constraint that the riser solidification time is greater than or equal to the casting solidification time. The problem is solved using the Fuzzy geometric programming technique and generalized expressions were obtained for HR , DR and VR. These expressions were applied to a cylindrical top riser, cylindrical side riser, hemispherical riser and modified hemispherical riser.

Headnote

Abstract: The optimal design of risers for castings has been the subject of numerous investigations and this paper determines the optimal dimensions when Chvorinov's rule for solidification is obeyed by both the riser and casting and solidification time is the only constraint involved .This problem can be formulated to minimize the riser volume subject to the constraint that the riser solidification time is greater than or equal to the casting solidification time. The problem is solved using the Fuzzy geometric programming technique and generalized expressions were obtained for H^sub R^ , D^sub R^ and V^sub R^ . These expressions were applied to a cylindrical top riser, cylindrical side riser, hemispherical riser and modified hemispherical riser.

Keywords: Fuzzy geometric programming, Riser design problem, Chvorinov's rule, casting modulus.

(ProQuest: ... denotes formulae omitted.)

Introduction

Geometric programming is an effective method to solve a nonlinear programming problem. It has certain advantages over the other optimization methods. Here, the advantages are that it is usually much simpler to work with the dual than primal. Degree difficulty plays an important role for solving a nonlinear programming problem by geometric programming method. Since late 1960, geometric programming has been known and used in various fields. Duffin and Petersen and Zener (1966) [12] discussed the basic theories of geometric programming. There are many references on application and the method of geometric programming. In the papers like Eckar (1980), Beightler (1979) [2,3], Zener (1971) [4], Jung and Klain (2001) developed single item inventory problems and solved by geometric programming method. Geometric programming has been applied to simple riser problems by R.C. Creese [10] using Chvorinov's rule [9]. In the last 20 yrs fuzzy geometric programming has received rapid development in the theory and application. In 2002, B.Y. Cao [6] published the first monograph of fuzzy geometric programming as applied optimization series (vol. 76), fuzzy geometric programming by Kluwer academy publishing (the present spinger), the book gives a 2.detailed exposition to theory and application of fuzzy geometric programming. The parametric fuzzy geometric programming can now be applied to a generalized riser design problem. Finally a two degrees difficulty geometric programming problem is solved.

Crisp Model

The model is to be considered such that the riser shape is completely described in terms of its height and diameter. The only restriction which will be considered is that the solidification time of the riser is greater than the solidification time of the casting. The generalized formulation would be

Minimize VR =f(DR,HR)

Subject to tR≥tC

Where the subscript R applies to the riser and C to the casting and

VR = riser volume, DR = riser diameter, HR = riser height

tR = solidification time of riser, tC = solidification time of casting.

If we can assume that the solidification times follow Chvorinov's rule in both the casting and riser such that

t =B^sub R^(V/SA)^sub R^nr

and t = B^sub C^(V/SA)^sub C^nc

Where BR = solidification constant for the riser

B^sub C^(V/SA)^sub R^= volume-to-surface area ratio for riser

nR = solidification exponent for riser

BC = solidification constant for casting

(V/SA)^sub C^ = volume-to-surface area ratio for casting

nC = solidification exponent for casting.

The volume -to-surface area ratio for casting is also called the casting modulus. The values of nR and nC vary from 1.5 to 2.5 depending upon the alloy composition and casting shape. The constraint now becomes

...

Since BC,BR , nC , nR and (V/SA)^sub C^ are constants for a given casting shape and alloy, above equation can be written as (V/SA)^sub R^ ≥ Y

Where Y = riser system modulus.

Let us now use the general relationships for the volume and surface area of the riser wherein

...

Where A, B, C and K are constants for the various shapes.

So the constraint becomes

3. Fuzzy Model

...

Subject to ...

Or, ...

Or, ...

Linear membership functions for the fuzzy objective and constraint goal are

...

According to Zimmermann

Max α

Such that ...

Which is equivalent to

Min (-á)

Such that ...

The dual geometric programming problem is

Max d(w) = ...

Such that ...

Primal-dual variable relations are

...

This is a non-linear equation in á.

4. Illustrative Example

...

5. Results

The interpretation of these results in the above table leads to some interesting findings. When comparing the three side risers, it is observed that the modified hemispherical bottom has less volume, and thus a greater yield than the other designs. The difference between the hemispherical bottom riser and modified hemispherical bottom is only 1-2 % whereas the hemispherical bottom required 16-17 % less metal than the standard cylindrical side riser. The cylindrical top riser uses 40 % less metal than the cylindrical side riser.

References

References

[1] C.B. Adams and H.F. Taylor, Fundamentals of riser behavior, Trans. AFS, 61(1953), 686- 693.

[2] C.S. Beightler and D.T. Phillips, Applied Geometric Programming, John Wiley and Sons, New York, 1976.

[3] C.S. Beightler, D.T. Phillips and D.J. Wilde, Foundation of Optimization, Prentice-Hall, New Jersy, 1979.

[4] R.E. Bellmann and L.A. Zadeh, Decision making in a fuzzy environment, Management Science, 17(4) (1970), B141-B164.

[5] H.F. Bishop, E.T. Myskowski and W.F. Pellini, A simplified method for determining riser dimensions, Trans. AFS, 63(1955), 271-281.

[6] B.Y. Cao, Solution and theory of question for a kind of fuzzy positive geometric programming, Second IFSA Congress, Tokyo, 1(1987), 205-208.

[7] B.Y. Cao, Fuzzy geometric programming (I), Fuzzy Sets and Systems, 53(1993), 135-153.

[8] B.Y. Cao, Fuzzy Geometric Programming, Kluwer Academic Publishers, Netherland, 2002.

[9] N. Chvorinov, Control of the solidification of castings by calculation, Foundry Trade Journal, 70(1939), 95-98.

[10] R.C. Creese, Optimal riser design by geometric programming, AFS Cast Metals Research Journal, 7(2) (June) (1971), 118-121.

[11] R.C. Creese, Dimensioning of riser for long freezing range alloys by geometric programming, AFS Cast Metals Research Journal, 7(4) (December) (1971), 182-185.

[12] R.J. Duffin, E.L. Peterson and C.M. Zener, Geometric Programming Theory and Application, Wiely, New York, 1967.

[13] J.B. Caine, Risering castings, AFS Transaction, 57(1948), 66-75.

[14] R.C. Creese, Generalized riser design by geometric programming, AFS Transactions, 87(1979), 661-664.

[15] R.C. Creese, An evaluation of cylindrical riser designs with insulating materials, AFS Transactions, 87(1979), 665-668.

[16] S. Liu, Posynomial geometric programming with parametric uncertainty, European Journal of Operational Research, 168(2006), 345-353.

[17] R.K. Verma, Fuzzy geometric programming with several objective functions, Fuzzy Sets and Systems, 35(1990), 115-120.