Inverse Optimization for Linear Fractional Programming

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1 444 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: International Journal of Physical and Mathematical Sciences journal homepage: Inverse Optimization for Linear Fractional Programming Sanjay Jain 1 *, Nitin Arya 2 1 *Department of Mathematical Sciences, Government College, Ajmer Affiliated to M. D. S. University, Ajmer , INDIA 2 Department of Mathematics, Government Engineering College, Jhalawar Research Scholar, Bhagwant University, Ajmer, INDIA 1* drjainsanjay@gmail.com, 2 nitin.arya1234@gmail.com RECEIVED DATE ( ) Abstract: In this paper, we have proposed an inverse optimization model for linear fractional programming (LFP) problem. In our proposed model, the parameters of the objective function are adjusted as little as possible (under L 1 norm), so that the given feasible solution become optimal. We formulate the inverse linear fractional programming (ILFP) problem as a linear programming problem having large number of variables, which can be solved by many existing methods or optimization software such as: TORA, EXCEL SOLVER etc. The method has been illustrated by a numerical example also. Keywords: Inverse optimization, Linear Fractional Programming, Complementary slackness. 1. Introduction The purpose of this paper is to present a brief taxonomy of an important group of programming problems which occur in certain branches of applied mathematics. A rough heading for these problems is Mathematical Programming Problem. Here special emphasis is given on inverse fractional programming problem. Recently, there has been interest in inverse optimization problems in the mathematical programming community and a variety of inverse optimization problems have been studied by researchers namely, minimum cost flow problem, NP complete problems, minimum spanning tree problem, shortest path problem, linear programming problem, inverse quadratic programming problem etc. Inverse optimization is a relatively new area of research and study of inverse optimization is useful in many branches. In mathematical programming problem, the cost, weights or penalties are not known with precision, but from the past experience or practice, it is plausible to call a solution x 0 or objective value z 0. In inverse problem, we generally assume a solution and to make it an optimal one (if possible), the parameter values associated with the decision variables in objective function or constraints are adjusted as little as possible (under L 1, L 2 or L norm). In numerical analysis, errors for solving the system Ax b are measured in terms of inverse optimization. Some interesting scenario in which solving the inverse problem is faster than solving for the optimal solution: An example is matroid intersection for representable matroids. For some algorithms, the solutions converge to the optimal in an inverse manner: An example is cost scaling algorithms for minimum cost flows. Burton and Toint [1] have first been considered inverse shortest path problem arising in seismic tomography used in predicting the movement of earthquakes. Since then a lot of work has been done on inverse optimization and plenty of inverse problems and their applications have found. Zhang and Liu [2] have first been calculating some inverse linear programming problem and further investigated inverse linear programming problems in [3]. Huang and Liu [4] studied on the inverse problem of linear programming problem and its applications. Ahuja and Orlin [5] provide various references in the area of inverse optimization and compile several applications in network flow problems with unit weight and develop combinatorial proofs of correctness. Amin and Emrouznejad [6] have considered applications of inverse problem. Zhang and Zhang [7,8] worked on inverse quadratic programming problems. Wang [9] has given the cutting plane algorithm for inverse integer programming problem and Hladik [10] have first been considered inverse problem for generalized linear fractional programming. Jain and Arya [11] provide some references in the field of inverse linear fractional programming problem. The linear fractional programming problem seeks to optimize the objective function of non negative variables of quotient form with linear functions in numerator and denominator subject to a set of linear and homogeneous constraints. Bajanilov [12] compiled the literature of Linear Fractional Programming: Theory, Methods, Applications and Software in the form of book. Charnes Cooper [13],

2 445 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: Kantiswarup [14], Naganathan and Sakthivel [15], Chadha [16], Jain and Mangal [17, 19], Chadha and Chadha [18], Jain, Mangal and Parihar [20] and many researchers gave different methods for solving linear fractional programming problem. In the following section, we describe in brief how inverse optimization is applied on LFP. In the proposed method, first we consider a feasible solution x 0 and objective value z 0 and using the dual problem and primal dual relation (complementary slackness conditions), we adjust the cost vectors c to c and d to d in such a way that the given solution become optimal for the modified problem. Some standard transformations and L 1 norm are used to reformulate the inverse problem as a linear programming problem. Thus obtained linear programming problem have a large number of variables, and is solvable by any existing methods or optimization software such as: TORA, EXCEL SOLVER etc. 2. Problem Formulation In this section, we study the inverse linear fractional programming problem. We will consider the inverse version of the following linear fractional programming problem, which we shall subsequently refer to as primal problem: Maximize for all and J Where I and J denote the index set of decision variables and constraints respectively. In inverse problem, we fix a feasible solution x 0 and objective value z* and for making it optimal, we perturb the cost vectors c to and d to in such a way that and are minimum, where is some selected L 1 norm defined as. The inverse linear fractional programming (ILFP) problem under L 1 norm can be stated as follows: Minimize, for all, for all Which is a linear programming problem having large number of variables. The notations used in this problem are defined in the next section. 3. The Method The standard linear fractional programming problem is given as follows: Maximize for all and J (1) where,, I and J denote the index set of decision variables and constraints respectively. Dual of the above linear fractional programming problem proposed in [18] is the following linear program: Minimize z for all i I

3 446 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: for all i I (2) Where y i is the dual variable associated with ith constraint in (1) and v j is the dual variable associated with each variable x j. If u is the slack variable associated with each constrains in primal problem (1) then the complementary slackness condition for linear fractional programming states that: ( x*,u*) solve (1) and (y*,z*,v*) solve (2) iff v* x* 0 and u* y* 0 In other way it can be written as: If If then then If x 0 is any feasible solution of primal problem and B is the index set of binding constraints in (1) with respect to x 0 (that is ), let L and F denote the index set of variables defined as and, then using these notations the complementary slackness conditions can be restate as: y i 0 for all i v j 0 We want to make x 0 an optimal solution of equation (1) by perturbing the cost vectors c to and d to, so replacing the cost vectors c to and d to and using the complementary slackness conditions in equation (2) gives the following characteristics of the cost vectors: (3) Now we also want to make the value of objective function (z) z* for the feasible solution x 0, so substituting z z* in (3) along with the objective function of primal problem (1) gives the following characteristics of the new cost vectors The inverse problem is to perturb the cost vectors c to and d to so that the feasible solution x 0 become an optimum solution to the modified problem with the objective value z* and also and are minimum, where is some (4) selected L 1 norm defined as.the inverse problem under L 1 norm can be stated as follows: Minimize

4 447 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: The above problem is not a linear programming problem but it can be converted to a linear programming problem using a standard transformation. We know that minimizing and are equivalent to minimizing and (5) respectively, subject to the conditions: ; and ; (6) Using these transformations in (5) the inverse linear fractional programming problem can be restate as follows: Minimize, for all (7) Clearly it is a linear programming problem which is solvable by any existing method but due to complexity of the problem it can be solve by optimization software like TORA, EXCEL SOLVER etc. 4. Numerical Example Let the linear fraction programming problem be and is the optimal solution with the objective function value. Let is a feasible solution of above LFP problem and we want to make x 0 an optimal with the objective function value that both the constraints are binding with respect to feasible solution x 0 and also. It can be seen therefore by the complementary slackness conditions and. The inverse linear fractional programming problem is as follows: ; j 1, 2

5 448 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: and for i 1,2, for j 1,2 Substituting the values a 11 1, a 12 1, a 21 3, a 22 1, b 1 4, b 2 6, c 1 2, c 2 3, c 0 1, d 1 1, d 2 1, d 0 4, and z* 1,625 in above equations and simplifying gives: Minimize ) and Optimal solution of the inverse problem using optimization software TORA

6 449 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: We get y , q , q 2 0, Therefore, the modified cost using (7) is. All other cost coefficients will remain same. Using the modified costs the given LFP problem becomes We observe that the solution of the modified problem is x 1 1, x 2 3 and z and 5. Particular Case: If we substitute for all and in original problem, it reduces to linear programming problem and our proposed method reduces for inverse linear programming problem which is studied earlier by [5]. 6. Conclusion Inverse optimization is an important area in both academic research and practical applications. Using the inverse optimization this paper suggested an inversed based methodology for the solution of linear fractional programming problem. An illustration observation used to demonstrate the advantage of the new approach. REFERENCES [1] Burton, D. and Toint, Ph. L. (1992). On an instance of shortest paths problem, Mathematical Programming, 53,

7 450 International Journal of Physical and Mathematical Sciences Vol 4, No 1 (2013) ISSN: [2] Zhang, J., & Liu, Z. (1996). Calculating some inverse linear programming problems. Journal of Computational and Applied Mathematics, 72, [3] Zhang, J., & Liu, Z. (1999). A further study on inverse linear programming problems. Journal of Computational and Applied Mathematics, 106, [4] Huang, S., & Liu, Z. (1999). On the inverse problem of linear programming and its application to minimum weight perfect k matching. European Journal of Operational Research, 112, [5] Ahuja, R. K. & Orlin, J. B. (2001). Inverse Optimization. Operation Research, 49, [6] Amin, G. R., & Emrouznejad, A. (2007). Inverse Forecasting: A New approach for predictive modeling. Computers & Industrial Engineering. doi: /j.cie [7] Zhang, J. and Zhang, Li. (2009). Solving a class of inverse QP problems by a smoothing Newton method. Journal of Computational Mathematics, 27(6), [8] Zhang, J. and Zhang, Li. (2010). An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems. Applied Math. & Opt., 61(1), [9] Wang, Li. (2010). Cutting plane algorithm for the inverse mixed integer programming problem. Operation Research Letters, 37, [10] Hladik, M. (2010).Generalized linear fractional programming under interval uncertainty. Eur. J. Oper. Res., 205(1), [11] Jain, S. and Arya, N. (2012). Inverse linear fractional Programming: A new approach. Journal of Computer and Mathematical Sciences, 3(5), [12] Bajanilov, Erik B. (2003). Linear Fractional Programming: Theory, Methods, Applications and Software. Kluwer Academic Press. [13] Charnes, A. and Cooper, W. W. (1962). Programming with linear fractional functionals, Naval Research Log. Quart, 9, [14] Swarup, K. (1965) Linear Fractional Functionals Programming, Operations Research, 13, [15] Naganathan, E. R. and Sakthivel, S. (1998). A ratio multiplex algorithm to solve linear fractional programming problems. Opsearch, 35, [16] Chadha, S. S. (1999) A Linear Fractional Program with homogeneous Constraint, Opsearch, 36, [17] Jain, S. and Mangal, A. (2004). Modified Fourier elimination technique for fractional programming problem. Acharya Nagaarjuna International Journal of Mathematics and Information Technology, 1, [18] Chadha, S. S. and Chadha, V. (2007). Linear Fractional Programming and Duality. Operation Research, 49: [19] Jain, S. and Mangal, A. (2008). Extended Gauss elimination technique for integer solution of linear fractional programming. Journal of Indian Mathematical Society, 75, [20] Jain, S. and Mangal, A. and Parihar, P. (2011). Solution of fuzzy linear fractional programming problem. Opsearch, 48,