Journal of Global Research in Computer Sciences
MenuISSN: 2229-371X
ISSN: 2229-371X
Payel Ghosh*1, Tapan Kumar Roy2
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| Corresponding Author: Payel Ghosh, E-mail: ghoshpayel86@yahoo.com |
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There are very common and widely used forms of solving linear and non-linear Goal Programming Problem. They are Archimedean, Lexicographic and MINMAX etc. This paper proposes a Geometric Programming method to solve a non-linear Goal Programming Problem. In particular, it demonstrates a new approach goal geometric programming in both crisp and imprecise environment. There is a numerical example and also an application of this method in two-bar truss problem. Comparison with Kuhn-Tucker conditions and crisp goal geometric programming method in the numerical example, it shows the efficiency of this method. In this paper, we have described fuzzy goal geometric programming and also implemented it on the same numerical example like crisp goal geometric programming and two bar truss problem.
INTRODUCTION |
| Linear goal programming is widely used in decision making involving linear equations and multiple conflicting goals. In this model, the best solution is achieved when the sum of weighted deviations is minimal. But there are some real life situations like production planning, location distribution, risk management, chemical process design, pollution control and other engineering design where equations may be non-linear. For such type of mathematical model goal geometric programming may be an excellent method. |
| It is worthwhile to mention that, to solve goal-programming problems, the use of multi-objective optimization technique is not new (Romero 1991[1], Steuer 1986[2]). Deb (1999)[3] has developed an approach for solving non-linear goal programming problems by multi-objective genetic algorithms. Ojha et al. (2010[4][5]) has discussed geometric programming method to solve multi-objective programming problems using weighted sum method. Luptacik (2010)[6] described elaborately about single objective geometric programming and multi-objective geometric programming. Previously Romero (1991)[1], Tamiz et al. (1995)[7] and Tamiz and Jones (1997)[8] described weighted sum method in linear goal programming problem but in this paper we have used weighted sum method in goal programming problem with non-linear equations and solved it using geometric programming technique. Further Tamiz and Jones (2010)[9] have developed goal programming problem and applied it in health care and portfolio selection. Romero (2004)[10] has described general structure of achievement function in goal programming problem. |
| When goals are not precise, then fuzzy goals are introduced. Narasiman (1980)[11] first introduced fuzzy set theory in goal programming. Further the contribution of Tiwari et.al. (1984)[12], Kim, Whang (1998)[13], Ramik (1996)[14], Li (2012) [15], Ciptomulyono (2008) [16] are mentionable. Li, Hu (2009)[17] have used weighted sum method in fuzzy multiple objective goal programming problem and further developed it. Fuzzy goal programming has very extensive applications like portfolio selection, water quality management in river basin, structural optimization etc. |
| In this paper we have applied geometric programming technique to solve a nonlinear goal programming problem. There is a comparison of results between goal geometric programming technique and nonlinear programming technique applied on a numerical example. We have also applied this technique in “two-bar truss” problem (Rao 1996)[18]. Apart from the crisp goal geometric programming, fuzzy goal geometric programming is also discussed here. Fuzzy goal geometric programming is described here by the same numerical example and also applied on the same application as in crisp goal geometric programming. |
| Body Text: |
| Formulation of Multi-Objective programming: |
| A multi-objective non-linear programming can be written as |
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| Pjo= Number of terms present in j0 th objective function. |
| Pr= Number of terms present in r th constraint. |
| Cr= Boundary value of the r th constraint. |
| In the above multi-objective programming model, there are m number of minimizing objective functions, q number of inequality type constraints and n number of strictly positive decision variables. |
| Formulation of Goal programming from Multi-Objective Programming: |
| Goal programming gives better results than ordinary multi-objective programming. To formulate the goal programming problem from the multi-objective programming problem, positive or negative deviations are minimized depending upon objective functions and constraints, e.g., for minimizing objective function positive deviation , for maximizing objective function negative deviation, for “ “ type constraint positive deviation, for “ “ type constraint negative deviation is minimized. |
| From the multi-objective programming problem (1.1), goal formulation is given below: |
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| wr>0 |
| Goal programming gives pareto optimal solutions which is already described in Romero (1991) [1]. Now we prove a result concerning the pareto optimality of the solutions of weighted non-linear goal programming problem. |
| Theorem: |
| The solution of the following weighted goal programming problem |
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| Dual form of Goal Geometric programming problem: |
| The model given by (4.1) is a normal geometric programming problem and it can be solved by using primal based algorithm for non-linear primal problem or dual programming. The model given by (4.1) can be transformed to the corresponding dual geometric programming as: Maximize d()= |
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RESULTS AND DISCUSSIONS |
| Numerical Example1: |
| A multi-objective goal programming problem |
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| The above multi-objective goal programming problem is converted into single objective goal geometric programming problem using deviations and giving the weights (priorities). The formulation is given below: |
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| Taking log on both side of (5.9) and then partially differentiating with respect to , and respectively and using the conditions of finding optimal solution we get these sets of equations |
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| Here, for different weights we get different optimum values of decision variables, deviation variables and objective functions. The logic of the theorem is also proved here since all the deviations are positive at the optimum. Hence the solutions are pareto optimal. |
| Solving (5.1) in non-linear programming (Kuhn-Tucker conditions) we get the following results: |
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| Here is the comparison of results between goal geometric programming method and non-linear programming (Kuhn-Tucker conditions) which shows that goal geometric programming gives better result than non-linear programming. We compare the values of two objective functions of two different approaches. We take the values of primal variables, 1st and 2nd objective functions for equal weights from the above table and the values which we get in non-linear programming approach. |
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| Application on “Two bar truss problem”: |
| The two bar truss is subjected to a vertical load 2P and is to be designed for minimum weight. The members have a tubular section with mean diameter d and wall thickness t and the maximum permissible stress in each member () is equal to 60,000 psi. There are two goals: Goal 1: weight should be near to 3 Goal 2: Ratio between stress and maximum permissible stress should be around 1. |
| Formulate the above goal programming problem and determine the values of mean diameter d and height h for the |
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| Therefore the multi objective nonlinear programming problem is Minimize =0.188 yd with target value 3 |
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| We solve the above model (6.2) converting it into geometric programming problem and then solving it we get the following results: |
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| From the table, it is clear that goals of decision maker (DM) satisfy here appropriately. The first goal i.e. the first objective should be near to 3 and the second goal i.e. the second objective should be around 1, is maintained here. The logic of the theorem is also proved here since all the deviations are positive at the optimum. Hence the solutions are Pareto optimal. |
| Body of Text: |
| Fuzzy Goal Programming formulations: |
| In fuzzy environment the multi-objective goal programming problem can be written as |
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| There are various kinds of membership functions such that linear, exponential, hyperbolic piecewise linear etc. The corresponding membership functions of the minimize and maximize objectives of (7.1) are |
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| Now a crisp mathematical programming is made by substituting the membership functions. A weighted sum of membership function of multi-objective fuzzy goals is taken as achievement function in this paper. There are many practical situations where DM has different requirement for each objective function in multiple objective optimization problems according to his or her preference. Then there is a big role of weight factors. Hence the crisp programming model with weighted additive objective function is as follows: |
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RESULTS AND DISCUSSIONS |
| Numerical Example 2: |
| A multi-objective fuzzy goal programming |
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| Here is the solution of the above model using geometric programming technique. |
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| Application on “Two-bar truss problem”: |
| The two bar truss is subjected to a vertical load 2P and is to be designed for minimum weight. The members have a tubular section with mean diameter d and wall thickness t and the maximum permissible stress in each member () is equal to 60,000 psi. There are two goals: |
| Goal 1: Weight should be minimized with target value 3. Decision maker gives some relaxation of target value i.e.1 and sets his opinion that this goal is „very importantâÃâ¬ÃŸ. Goal 2: Ratio between stress and maximum permissible stress should be minimized with target value 1. Here decision makerâÃâ¬ÃŸs opinion that the goal is also „very importantâÃâ¬ÃŸ and gives a relaxation 0.5 on the target value. |
| Formulate the above goal programming problem and determine the values of mean diameter d and height h for the following data: P=33,000 lb, t=0.1 in. b=30 in. =60,000 psi, density = 0.3 lb/in3. |
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| Therefore according to the first goal, weight 0.188 yd should be minimized with target value 3 and tolerance 1. |
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| Observing the results of crisp goal geometric programming on truss bar problem and fuzzy goal geometric programming on the same we can conclude that fuzzy goal geometric programming gives better result than crisp goal geometric programming. The first goal i.e. the weight should be minimized with target 3 which is „very importantâÃâ¬ÃŸ according to decision makerâÃâ¬ÃŸs choice and this target fulfills properly. Also the second goal i.e. the ratio of stress and maximum permissible stress should be minimized with target 1 which is also „very importantâÃâ¬ÃŸ according to decision makerâÃâ¬ÃŸs choice and this target also fulfills properly. |
CONCLUSIONS |
| By using Goal Geometric Programming we can solve multi-objective goal programming problem (MOGPP). This method is very useful for many real life situations where equations are non-linear. We show the efficiency of this method with weighted sum deviations where weights (priorities) can be changed as per requirement. The variation of result according to weights (priorities) also shows the perfection of this method. Comparing with non-linear Optimization (Kuhn-Tucker conditions), this method gives better result which is already described in this paper. This method has several types of applications in the field of engineering, sciences etc. Here we have applied this method in two bar truss problem and show that this method is more efficient than the previously solved process. We have also discussed this method goal geometric programming in fuzzy environment and applied this on two bar truss problem. It is clear to us that fuzzy goal geometric programming gives better result i.e. this method satisfies all our targets and requirement as per our opinion. Instead of weighted sum method one can use weighted product method, MINMAX method etc. We can apply this method in imprecise environment like intuitionistic fuzzy and interval number also. |
ACKNOWLEDGMENTS |
| ItâÃâ¬ÃŸs my privilege to thank my respected guide for his enormous support and encourage for the preparation of this research paper. The great support of my family members makes me possible to do it. |
References |
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