Journal of Global Research in Computer Sciences
MenuISSN: 2229-371X
ISSN: 2229-371X
| 1*Pintu Das and 1Tapan Kumar Roy Department of applied mathematics, Indian institute of engineering science and technology, Shibpur, Howrah, West Bangal, India, 711103. |
| Corresponding Author: Pintu Das, E-mail: mepintudas@yahoo.com |
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Multi-objective geometric programming (MOGP) is a strong tool for solving a type of optimization problem. This paper develops a solution procedure to solve a multi-objective non-linear programming problem using MOGP technique based on weighted-sum method, weighted-product method and weighted min-max method .The equivalent general Multi-objective geometric programming problems are formulated to find their corresponding value of the objective functions based on duality theorem. As the numerical example Gravel- box design problem is presented to illustrate the methods.
Keywords |
| Multi-objective Geometric programming, Weighted-sum method, Weighted-product method, weighted min-max method, Gravel box. |
INTRODUCTION |
| Geometric programming (GP) is a technique to solve the special class of non linear programming problems subject to linear or non-linear constraints. The original mathematical development of this method used the arithmetic–geometric mean inequality relationship between sums and products of real numbers. In 1967 Duffin, Peterson and Zener put a foundation stone to solve wide range of engineering problems by developing basic theories of geometric programming in the book Geometric Programming [3]. Beightler and Phillips gave a full account of entire modern theory of geometric programming and numerous examples of successful applications of geometric programming to real-world problems in their book Applied Geometric Programming [1]. GP method has certain advantages. |
| The advantage is that it is easy to solve the dual problem than primal. Multi-objective geometric programming problem is a special class of non-linear programming problem with multiple objective functions. In many real-life optimization problems, multi-objectives have to be taken into account which may be related to the economical, technical, social and environment aspects of optimization problems. In multi-objective optimization , the trade –off information between different objective functions is probably the most important piece of information in a solution procedure to reach the most preferred solution .GP Liu, JB Yang, JF Whidborne gave an account with multi-objective geometric programming in their book Multi-objective Optimization and Control [5]. In this field a paper named Multi-objective geometric programming problem being cost coefficient as a continuous function with mean method by A.K. Ojha, A.K. Das has been published in the journal of computing 2010 [7].In 1992 M.P.Bishal [9] and in 1990 R.k.verma [10] has studied fuzzy programming technique to solve multi-objective geometric programming problems. In our paper we have discussed the basic concepts and principles of multi-objective optimization problem and then developed typical multi-objective methods. |
FORMULATION OF MULTIOBJECTIVE GEOMETRIC PROGRAMMING PROBLEM |
| A multi-objective geometric programming problem can be defined as |
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| Solution procedure of Multi-Objective Geometric Programming Problem based on weighted sum method (MOGPPws): |
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| Using weighted product method the multi-objective functions in (1) can be written as, |
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| So multi-objective optimization problem reduces to a single objective geometric programming problem as, |
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| Solution procedure of multi-objective geometric programming problem based on Weighted product method :(MOGPTwp): |
| The corresponding dual problem of (3) is |
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| Using Min-max method the multi-objective optimization functions in (1) can be written as, |
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| Solution procedure of multi-objective geometric programming problem based on Min-max method :(MOGPTmm): |
| The corresponding dual problem of (4) is |
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| Degrees of Difficulty: |
| Degrees of difficulty play an important role to solve the multi-objective geometric programming problems. It is defined as follows. |
| DD (degrees of difficulty)=Total number of terms –(Number of variables+1) |
| Degrees of difficulty of problem (1.1) based on weighted sum method |
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| The corresponding primal geometric programming problem based on weighted-sum method is |
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MULTI-OBJECTIVE GRAVEL BOX DESIGN PROBLEM |
| Here we have taken gravel box design problem with minor modification from [1]. A total of 800 cubic-meters of gravel is to be ferried across a river on a barrage. A box (with an open top) is to be built for this purpose. The transport cost per round trip of barrage of box is Rs .05; the cost of materials of the ends of the box are Rs20/m2. and other two sides and bottom are made from available scrap materials . Find the dimension of the box that is to be built for this purpose to minimize the transport cost and material cost. |
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| Solution procedure of the above example by Weighted sum method: |
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| The table-1 shows different optimal solutions for different weights of the problem (5) by weighted-sum method. First objective gives better optimal result when w1 increases. Similarly second objective gives better optimal result when w2 increases. |
| Solution procedure of the problem (5) by weighted product method: |
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| The table-2 shows different optimal solutions of the problem (5) by weighted-product method for different weights. If we increase the weights w1 and w2 both the optimal objective functions will increase. Here objective functions are inversely related to the weights. |
| Solution procedure of the problem (5) by weighted min-max method: |
| According to MOGPTmm |
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| The table-3 shows different optimal solutions of the problem (5) by weighted min-max method for different weights. First objective gives better optimal result when w1 increases. Similarly second objective gives better optimal result when w2 increases. . Here the objective functions are directly related to the weights. |
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| We see that DD is Minimum for weighted product method. Among three methods (weighted sum method, Weighted product method and weighted min- max method), optimal value of first objective function( x1,x2,x3) gives better result by weighted product method and optimal value of second objective function(x1,x2,x3) gives better result by weighted min-max method. For minimum total optimal variables weighted product method gives better result and for maximum total optimal variables weighted min-max method gives better result. |
CONCLUSION |
| Here we have discussed multi-objective geometric programming based on the weighted sum method, weighted product method, weighted min-max method, We have also formulated the multi-objective optimization model of the gravel-box design problem and solved this problem by multi-objective geometric programming technique based on said three methods. The different objective functions are combined into a single objective function by the above three methods. The GP technique is used to derive the optimal solutions for different preferences on objective functions. In tables 1-4 we have shown the optimal solution of our problem for different preference values of the objective functions. This multi-objective optimization model may also be solved by multi-objective geometric programming technique based on global criterion method. |
References |
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