Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 30, 1493-1501 Some Results in Duality Vanita Ben Dhagat and Savita Tiwari Jai Narain college of Technology Bairasia Road, Bhopal M.P., India vanita1_dhagat@yahoo.co.in Abstract We construct L.P.P. by considering Symmetric Matrix. These L.P.Ps. are such that that s dual have exactly same coefficients as it s primal for all conditions (The demand & Requirement are equal or not.) i.e. the constraints in the form of = & and the variables are positive or unrestricted. There are many such L.P.Ps. are constructed and solved with the help of Tora Package and the results are generalized and state in the form of theorem. Keywords : Linear Programming Problem (L.P.P.), Primal L.P.P., Dual L.P.P., Symmetric matrix, Tora package, [A]=[a ij ] m x m, [b]=[bi] i = 1,2 n. [C] = [ci] i=1,2.n. 1 Introduction: In every Linear Programming Problem (Maximization or Minimization) there always exist another Linear Programming Problem which is based upon the same data and having the same solution. The Original problem is called the Primal Problem while the associated one is called its Dual Problem. The relation between the primal and the dual problem constitutes very important basis for understanding the deeper structure of LP (compared to systems of linear equations). Objective value of the dual problem of a feasible dual solution provides lower bound on the optimal primal objective value and the dual problem can be derived for this purpose. However, the text derives it as a special case of the Lagrangian dual problem. The study of duality is very important in LP. Knowledge of duality allows one to develop increased insight into LP solution interpretation. Also, when solving the
1494 V. Ben Dhagat and S. Tiwari dual of any problem, one simultaneously solves the primal. Thus, duality is an alternative way of solving LP problems. The present study is concerned with an interesting class of L.P.Ps. where in [A], [b] and [C] are exactly same in primal and dual L.P.P. 2 Definition and Terminology: Definition 1: (Standard Primal Problem) Maximize z = c 1 x 1 + c 2 x 2 +.+ c n x n Subject to : a i1 x 1 + a i2 x 2 + +a in x n b i ; x j 0 ; i= 1,2,.,m. j= 1,2,,n. It has been shown that if the above problem has a finite solution then the following problem also has a finite solution Dual Problem: Minimize z* = b 1 x 1 + b 2 x 2 +.+ b n x n Subject to : a 1j x 1 + a 2j x 2 + +a nj x n c j ; x j 0 ; i= 1,2,.,m. j= 1,2,,n. Definition 2: (Standard Primal Problem) Minimize z = c 1 x 1 + c 2 x 2 +.+ c n x n Subject to : a i1 x 1 + a i2 x 2 + +a in x n b i ; x j 0 ; i= 1,2,.,m. j= 1,2,,n. It has been shown that if the above problem has a finite solution then the following problem also has a finite solution Dual Problem: Maximize z* = b 1 x 1 + b 2 x 2 +.+ b n x n Subject to : a 1j x 1 + a 2j x 2 + +a nj x n c j ; x j 0 ; i= 1,2,.,m. j= 1,2,,n.
Some results in duality 1495 3 Theorem: If, we construct Linear Programming Problems such that [A] is a symmetric matrix and [A] = [a ij ] m x m where a ij s are the terms of Progressive series (where all the terms are non zero and positive), zero or negative and [C] = [b] and the terms of [C] and [b] are a ij where i = 1, 2, 3, m. j= 1, 2, 3, m. also the variables are restricted or unrestricted, then the optimal value of the objective function (Z) is a 11. 4 PROPOSED METHOD : (i) Write the initial simplex table for the L.P.P. in this table x m is an incoming variable. (ii) We have consider LPP with 1, -2, 3, 4 and hence most negative (Z j - C j ) Corresponds to x 4. Hence by Simplex method x 4 is an incoming variable where m = 4 is the order of the [A]. If the order of [A] is m x m then it is obvious that x m is an incoming variable. (iii) It is clear that S 2 (the slack variable in the second constraint) leaves the basis. In this case x 4 leaves the basis because it corresponding to min (1/4, -2/5, ½, 4/7 ) = -2/5 = b 2 /c m = b2/(coefficient of xm in the objective function). (iv) i.e. x m = b 1 / c m. Hence, Max. Z = 1.1+(-2).0 + 3.0 + 4.0 i.e., Max. Z = 1 = a 11 = b 1 5 ILLUSTRATIONS: Consider the L.P.P. with: 1,-2,3,4 Max. z = x 1-2x 2 +3x 3 +4x 4 And x 1-2x 2 +3x 3 +4x 4 1-2x 1 +3x 2 +4x 3 +5x 4-2 3x 1 +4x 2 +5x 3 +6x 4 3 4x 1 +5x 2 +6x 3 +7x 4 4 x i 0, i = 1,2,3,4. (1) The dual of the above L.P.P. is: Min. z = x 1-2x 2 +3x 3 +4x 4
1496 V. Ben Dhagat and S. Tiwari And x 1-2x 2 +3x 3 +4x 4 1-2x 1 +3x 2 +4x 3 +5x 4-2 3x 1 +4x 2 +5x 3 +6x 4 3 4x 1 +5x 2 +6x 3 +7x 4 4 x i 0, i = 1,2,3,4....(2) The two problem (1) and (2) are primal and dual respectively. It is really very interesting to note that [A], [b] and [C] are exactly same in (1) and (2). We solve (1) using Tora Package. Optimum Solution Summary Final iteration No. = 2 Objective value (max) = 1 Alternative solution detected at x 1 Variable Value Obj. Coeff. Obj. Value Contrib. x1 1.00 1.000 1.00 x2 0.00-2.000 0.00 x3 0.00 3.000 0.00 x4 0.00 4.000 0.00 Constraint RHS Slack(-) / Surplus(-) x1 1 0.000 x2-2 0.000 x3 3 0.000 x4 4 0.000
Some results in duality 1497 Sensitivity Analysis Objective Coefficient Single Changes : Variable Current Coeff. Min. Coeff. Max. Contrib. Reduced Cost x1 1.000 -infinity 1.000 0.00 x2-2.000 -infinity -2.00 4.00 x3 3.000 -infinity 3.00 0.00 x4 4.000 -infinity 4.00 0.00 Right Hand Side Single Changes : Constraint Current RHS Min. RHS Max. RHS Dual price x1 1.000 1.000 1.000 1.000 x2-2.000-2.000-2.000 0.000 x3 3.000 3.000 infinity 0.000 x4 4.000 4.000 infinity 0.000 L.P.P. With Geometric Progression: Consider the L.P.P. with GP: 3, 9, 27, 81. Max. z = 3x 1 +9x 2 +27x 3 +81x 4 3x 1 + 9x 2 + 27x 3 + 81x 4 =3 9x 1 + 27x 2 + 81x 3 + 243x 4 = 9 27x 1 + 81x 2 + 243x 3 + 729x 4 = 27 81x 1 +243x 2 +729x 3 +2187x 4 = 81 And x i 0, i = 1,2,3,4. (3) The dual of the above L.P.P. is: Min. z = 3x 1 +9x 2 +27x 3 +81x 4 3x 1 + 9x 2 + 27x 3 + 81x 4 3 9x 1 + 27x 2 + 81x 3 + 243x 4 9 27x 1 + 81x 2 + 243x 3 +729x 4 27..(4)
1498 V. Ben Dhagat and S. Tiwari 81x 1 +243x 2 +729x 3 +2187x 4 81 And x i (i = 1,2,3,4) Unrestricted. The two problem (3) and (4) are primal and dual respectively. It is really very interesting to note that [A], [b] and [C] are exactly same in (3) and (4). We solve (3) using Tora Package Optimum Solution Summary Final iteration No. = 4 Objective value (max) = 3 Alternative solution detected at x 4 Variable Value Obj. Coeff. Obj. Value Contrib. x1 0.00 3.000 0.00 x2 0.00 9.000 0.00 x3 0.00 27.000 0.00 x4 0.40 81.000 3.00 Constraint RHS Slack(-) / Surplus(-) x1 3.000 0.00 x2 9.000 0.00 x3 27.000 0.00 x4 81.000 0.00 Sensitivity Analysis Objective Coefficient Single Changes : Variable Current Coeff. Min. Coeff. Max. Contrib. Reduced Cost x1 3.000 -infinity 3.000 0.00 x2 9.000 -infinity 9.000 0.00 x3 27.000 -infinity 27.000 0.00 x4 81.000 81.000 infinity 0.00
Some results in duality 1499 Right Hand Side Single Changes : Constraint Current RHS Min. RHS Max. RHS Dual price x1 3.000 0.000 3.000 3s.00 x2 9.000 9.000 infinity 0.00 x3 27.000 27.000 infinity 0.00 x4 81.000 81.000 infinity 0.00 L.P.P. With Arithmetic Progression : Consider the L.P.P. with AP: 2,4,6,8. Min. z = 2x 1 +4x 2 +6x 3 +8x 4 2x 1 + 4x 2 + 6x 3 + 8x 4 = 2 4x 1 + 6x 2 + 8x 3 + 10x 4 = 4 6x 1 + 8x 2 + 10x 3 + 12x 4 = 6 8x 1 + 10x 2 + 12x 3 + 14x 4 = 8 And x i 0, i = 1,2,3,4. (5) The dual of the above L.P.P. is: Max. z = 2x 1 +4x 2 +6x 3 +8x 4 2x 1 +4x 2 +6x 3 +8x 4 2 4x 1 +6x 2 +8x 3 +10x 4 4 6x 1 +8x 2 +10x 3 +12x 4 6...(6) 8x 1 +10x 2 +12x 3 +14x 4 8 And x i (i = 1,2,3,4) Unrestricted. The two problem (5) and (6) are primal and dual respectively. It is very interesting to note that [A],[b] and [C] are exactly same in (5) and (6). We solve (5) using Tora Package. Optimum Solution Summary Final iteration No. = 5 Objective value (max) = 2 Alternative solution detected at x 1
1500 V. Ben Dhagat and S. Tiwari Variable Value Obj. Coeff. Obj. Value Contrib. x1 1.000 2.000 2.000 x2 0.000 4.000 0.000 x3 0.000 6.000 0.000 x4 0.000 8.000 0.000 Constraint RHS Slack(-) / Surplus(-) x1 2.000 0.000 x2 4.000 0.000 x3 6.000 0.000 x4 8.000 0.000 Sensitivity Analysis Objective Coefficient Single Changes : Variable Current Coeff. Min. Coeff. Max. Contrib. Reduced Cost x1 2.000 -infinity 2.000 0.00 x2 4.000 4.000 infinity 0.00 x3 6.000 6.000 infinity 0.00 x4 8.000 8.000 infinity 0.00
Some results in duality 1501 Right Hand Side Single Changes : Constraint Current RHS Min. RHS Max. RHS Dual price 1(=) 2.000 0.000 2.000 1.00 2(=) 4.000 4.000 infinity 0.00 3(=) 6.000 6.000 infinity 0.00 4(=) 8.000 8.000 infinity 0.00 6. CONCLUSION: The present paper suggests an optimal heuristics approach for Duality in Linear Programming Problems. The present paper method deals with such type of Duality problems in which primal LPPs and dual LPPs are exactly same. REFERENCES 1. Bazaraa, M.S., J.J. Jarvis, and H.D. Sherali. Linear Programming and Network Flows. New York: JohnWiley & Sons, 1990. 2. Behere, Gayatri, Khot,P.G., Journal of Indian Acad. Math. Vol. 30, No. 1 (2008) pp. 23-31. 3. Dantzig, G.B., 1963. Linear Programming and Extensions, Princeton University Press, Princeton, NJ. 4. Mc.Carl, B.A. : Duality in quadratic programming, Quart. Appl. Math.,18 (1960). 5. Nering, E.D and Tucker, A.W., 1993, Linear Programming and Related Problems, Academic Press, Boston, MA. 6. Ortega, J.M. : Matrix theory a second course, New York, (1987). Received: January, 2009