Mixed 0-1 Linear Programming for an Absolute. Value Linear Fractional Programming with Interval. Coefficients in the Objective Function

Transcription

1 Applied Mathematical Sciences, Vol. 7, 2013, no. 73, HIKARI Ltd, Mixed 0-1 Linear Programming for an Absolute Value Linear Fractional Programming with Interval Coefficients in the Objective Function Mojtaba Borza 1, Azmin Sham Rambely 2 and Mansour Saraj 3 1,2 School of Mathematical Sciences, Faculty of Science & Technology, University Kebangsaan Malaysia, Bangi, Selangor, Malaysia 3 Department of Mathematics, Faculty of Mathematical Sciences & Computer, Shahid Chamran University, Ahvaz, Iran 1 mojtaba_borza@yahoo.com, 2 asr@ukm.my, 3 msaraj@scu.ac.ir Copyright 2013 Mojtaba Borza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper considers a fractional functional programming problem with interval coefficients of the type:

2 3642 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj,,,,, 0. According to the positivity or negativity of the function, two linear programming problems are resulted to be solved to achieve the optimal solution. Combination of the two linear programming problems finally yields a mixed 0-1 linear programming problem which can be used to obtain the optimal solution of an absolute value linear fractional programming problem with interval coefficients in the objective function. A numerical example is given to illustrate the efficiency and the feasibility of the method. Keywords: Convex combination, Interval coefficients, Absolute-value linear programming, Linear fractional programming, mixed 0-1 linear programming. 1. Introduction The fractional programming (FP) is a special case of a nonlinear programming, which is generally used for modeling real life problems with one or more objective(s) such as profit/cost, actual cost/standard, output/employee, etc, and it is applied to different disciplines such as engineering, business, finance, economics, etc. One of the earliest fractional programming problems is an equilibrium model for an expanding economy introduced by Von Neumann [8] in In 1962, Charnes and Cooper [4] showed that a linear fractional programming can be equivalently transformed into a linear programming using nonlinear variable transformation. In 1967, the non linear model of fractional programming problem was studied by Dinkelbach [5]. He showed a fractional programming can be transformed into a non fractional parametric programming, and an iterative procedure must be followed to achieve the approximation of optimal solution. Problem of the following type was considered by Chadha [2] and Chang [3]. Maximize, is unrestricted.

3 Mixed 0-1 linear programming 3643 Chadha [2] showed that above problem under some additional assumptions can be solved by the adjacent extreme point (simplex-type) methods introduced by Martos [6]. Chang [3] used fuzzy goal programming to achieve the global solution of the above problem. Stancu-Minasian [7] gives a survey on fractional programming which covers applications as well as major theoretical and algorithmic developments. As we know, there are many phenomena in the real physical world in which the coefficients are not certain when they are modeled mathematically. Therefore, it is much better to select coefficients as the intervals instead of fixed numbers in such cases. For example, one of these situations occurs when the coefficients are fuzzy numbers. In these cases if decision makers specify an α-level of satisfactory, then the fuzzy numbers are transformed into intervals. Recently, the linear fractional programming problem with interval coefficients in the objective function was considered by Borza et al. [1]. They showed that the problem can be equivalently transformed into the linear programming problem. In this paper, the absolute value linear fractional programming problem with interval coefficients in the objective function is considered. For solving the problem, a method based on variable transformation by Charnes and Cooper [4] and convex combination of intervals is used. 2. Formulation of the problem The general extended form of an absolute value linear fractional programming problem with interval coefficients in the objective function which is considered in this paper is as follows: Problem 1 1, 1 1, 1, 1 1, 1 1, 1, 1 s.t, 0,, 0, where, for 1,, and are -dimensional constant column vectors. It is assumed that

4 3644 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj,,, 0 or,,, 0 for all,,, where is the compact feasible region of problem 1. Two different cases must be considered to deal with problem 1. Case1:,,, 0 for all,,. In this case, problem 1 is reduced into the following problem. Problem 2 1, 1 1, 1, 1 1, 1 1, 1, 1, 0,, 0. Case2:,,, 0 for all,,. In this case, problem 1 is changed into the following problem. Problem 3 1, 1 1, 1, 1 1, 1 1, 1, 1, 0,, 0. Since the numerators in the objective functions of problems 2 and 3 are both positive, then the last point of intervals in the denominators must be selected to obtain the optimal solutions. Problems 2 and 3 are therefore reduced to problems 4 and 5, respectively. Problem 4 Problem 5 1, 1 1, 1, , 0,, 0. 1, 1 1, 1, , 0,, 0.

5 Mixed 0-1 linear programming 3645 On using variables and, and setting for 1,,, problems 4 and 5 are transformed into problems 6 and 7, respectively. Problem 6,,, Problem 7 0, 1, 0,, 0,0.,,, 0, 1, 0,, 0,0. On using auxiliary variable, non linear problems 6 and 7 are transformed into the following linear programming problems 8 and 9, respectively. Problem 8,,,,,,,, 0, 1, 0,, 0,0, 0.

6 3646 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj Problem 9,,,,,,,, 0, 1, 0,, 0,0, 0. To solve problems 8 and 9, the interval coefficients must be replaced by fixed numbers. For this purpose, the following proposition is useful. Proposition 1: Let (,,,, be a feasible point of problem 8 or 9 and, for 1,,1. The following relations hold true: a) If then. b) If then. Proof: The proof is straightforward and is omitted. Following the above proposition yields the biggest feasible region and the best possible solution is consequently obtained. Problems 8 and 9 are therefore transformed into problems 10 and 11, respectively. Problem 10,, 0, 1, 0,, 0,0, 0.

7 Mixed 0-1 linear programming 3647 Problem 11,, 0, 1, 0,, 0,0, 0. In brevity, two different linear programming problems 10 and 11 must be considered simultaneously, and then by comparing their optimal solutions the best one is selected as an optimal solution of problem Mixed 0-1 linear programming to problem 1 In the previous section, we saw that two different linear programming problems are related to an absolute value linear fractional programming problem with interval coefficients in the objective function. In this section, on using combination of problems 10 and 11, only one linear programming problem is found to be utilized in replace of problem 1. The only difference between problems 10 and 11 is related to their equality constraints, one of these equality constraints must therefore be active at any feasible point especially at the optimal solution and the other is not active. According to this description, the following mixed 0-1 linear programming problem is introduced as combination of problems 10 and 11.

8 3648 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj Problem 12,, 0, 1, 1, 0,, 0,0, 0, 0 or 1, where 0 1 is a fixed arbitrary enough small real number. Because of the equality constraints in the above problem, there is a possibility that problem 12 loses feasibility in practice. To remove this fault, the equality constraints must be replaced by suitable inequality constraints. For this purpose, the following two lemmas are needed. Lemma1: Constraint 1 replace of 1. can be equivalently used in Proof: By introducing variables,,, and for 1,,, problem 2 is transformed into the following problem. Problem 13,,, 0,,,, 1, 0,, 0,0. The linear combination of each interval in the constraint yields the following problem.

9 Mixed 0-1 linear programming 3649 Problem 14,,, 0, , 0,, 0, 0,0 1 for 1,,1. The equality constraint in problem 14 can be further reduced to 1, (1) since 0 for 1,,,0, 0 1, 0 for 1,,1. Therefore (1) can be written as: (2) Combining (1) and (2) results: 1 1 (3) which further reduced to: 1 (4) and 1. (5) According to Lemma 1, inequality constraint 1 can be used in replace of constraint 1. Lemma 2: Inequality constraint 1 can be used in replace of equality constraint 1. Proof: This lemma is proved in similar way to Lemma 1. The proof is therefore omitted.

10 3650 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj According to Lemmas 1 and 2, problem 12 is transformed into the following problem. Problem 15,, 0, 1 12, , 0,, 0,0, 0, 0 or 1, where 01 is a small enough arbitrary fixed real number. If we set to be less than 110, thus the expected result can be obtained. Let (,,,,, be the optimal solution of problem 15, then the optimal solution of problem 1 is,, with the optimal objective function value. 4. Numerical example The following example is considered to illustrate the efficiency of the method. Problem 16,,,,,, 2 13, , 2 17, ,

11 Mixed 0-1 linear programming , , 0, 0. Problem 15 is formulated for the above problem as follows: Problem 17 where , , 3 21, 3 1, 3 240, , 2 130, , 2 170, , 4 110, , 0, 0, 0,0, 0 or 1, The optimal solution of problem 17 is: , , , 0, Using relation, 1,2, results 5, 9 as the optimal solution of problem 16.

12 3652 Mojtaba Borza, Azmin Sham Rambely and Mansour Saraj Conclusion In this paper, the absolute value linear fractional programming problem with interval coefficients in the objective function is considered. We showed that to solve the problem two different linear programming problems must be considered. The combination of the two linear programming problems results a mixed 0-1 linear programming problem. It is not difficult to see that the two linear programming problems are derived from the mixed 0-1 linear programming problem by setting variable be equal to 0 and 1. Acknowledgment The authors would like to acknowledge the research grant GUP from the University Kebangsaan Malaysia. References [1] M. Borza, A. S. Rambely and M. Saraj, Solving linear fractional programming with interval coefficients in the objective function. A new approach, Applied Mathematical Sciences 69( 2012), [2] S.S. Chadha, Fractional programming with absolute-value functions, European Journal of Operational research, 141(2002), [3] Ching-Ter Chang, Fractional programming with absolute-value functions: a fuzzy goal programming approach, Applied Mathematics and Computation, 167 (2005), [4] A. Charnes and W. W.Cooper, Programming with linear fractional functions Naval Research Logistics Quaterly, 9 (1962),

13 Mixed 0-1 linear programming 3653 [5] W. Dinkelbach, On nonlinear fractional programming, Management Science, 13(1967), [6] B. Martos, Hyperbolic programming, Naval Research Logistics Quarterly II (1964), [7] I. M. Stancu-Minasian, Fractional programming: Theory, methods and applications, Kluwer Dordrecht, [8] J. Von Neumann, Ube rein es gleichungs system und eine verallgemeierung des brouwerschen fixpuntsatzes, K. Menger, eds., Ergebnisse eines mathematicschen kolloquiums 8, Leipzig und Wien, (1973) Received: March 31, 2013