JS and SS Mathematics JS and SS TSM Mathematics TRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN School of Mathematics MA3484 Methods of Mathematical Economics Trinity Term 2015 Saturday GOLDHALL 09.30 11.30 Dr. D. R. Wilkins Instructions to Candidates: Credit will be given for the best 3 questions answered. Materials Permitted for this Examination: Formulae and Tables tables are available from the invigilators, if required. Non-programmable calculators are permitted for this examination, please indicate the make and model of your calculator on each answer book used. You may not start this examination until you are instructed to do so by the Invigilator. Page 1 of 6
1. Let c i,j be the coefficient in the ith row and jth column of the cost matrix C, where 6 8 4 5 10 7 C =. 3 4 9 5 6 12 and let s 1 = 6, s 2 = 12, s 3 = 5, s 4 = 7, d 1 = 8, d 2 = 13, d 3 = 9. Determine non-negative real numbers x i,j for i = 1, 2, 3, 4 and j = 1, 2, 3 that minimize 4 3 c i,j x i,j subject to the following constraints: i=1 j=1 3 x i,j = s i for i = 1, 2, 3, 4, j=1 and x i,j 0 for all i and j. 4 x i,j = d j for j = 1, 2, 3, i=1 Also verify that the solution to this problem is indeed optimal. (20 marks) Page 2 of 6
2. Let A be an m n matrix of rank m with real coefficients, where m n, let b R m be an m-dimensional column vector, let c R n be an n-dimensional column vector, and let c T denote the row vector that is the transpose of the vector c. Let a (j) denote the m- dimensional vector specified by the jth column of the matrix A for j = 1, 2,..., n. We consider throughout this question the following linear programming problem in Dantzig standard form: Linear Programming Problem Determine an n-dimensional vector x so as to minimize c T x subject to the constraints Ax = b and x 0. Given any n-dimensional vector x, we denote by (x) j the jth component of x for j = 1, 2,..., n. Then x 0 if and only if (x) j 0 for j = 1, 2,..., n. We recall some terminology relevant to the above linear programming problem. A feasible solution to this problem is a vector x satisfying the constraints Ax = b and x 0. An optimal solution is a vector x that is a feasible solution that minimizes the cost c T x amongst all feasible solutions of the problem. A subset B of {1, 2,..., n} is said to be a basis for the linear programming problem if (a (j) : j B) is a basis of the vector space R m. Any basis B has exactly m elements. A feasible solution x is said to be basic if there exists a basis B for the linear programming problem such that (x) j = 0 when j B. Prove that if there exists a feasible solution to the programming problem then there exists a basic feasible solution to the problem. Prove also that if there exists an optimal solution to the programming problem then there exists a basic optimal solution to the problem. (20 marks) Page 3 of 6
3. Consider the following linear programming problem: find real numbers x 1, x 2, x 3, x 4, x 5, x 6 so as to minimize the objective function 3x 1 + 5x 2 + 3x 3 + 4x 4 + 7x 5 + 8x 6 subject to the following constraints: 4x 1 + 3x 2 + 5x 3 + 2x 4 + x 5 + 3x 6 = 20; 2x 1 + 4x 2 + 3x 3 + 6x 4 + 2x 5 + 5x 6 = 23; x j 0 for j = 1, 2, 3, 4, 5, 6. A feasible solution to this problem is the following: x 1 = 0, x 2 = 5, x 3 = 1, x 4 = 0, x 5 = 0, x 6 = 0. (This feasible solution satisfies the constraints but does not necessarily minimize the objective function.) Find an optimal solution to the linear programming problem and verify that it is indeed optimal. (20 marks) Page 4 of 6
4. Let a i,j, b i and c j be real numbers for i = 1, 2 and j = 1, 2, 3, and let the Primal Linear Programming Problem and the Dual Linear Programming Problem be specified as set out below: Primal Linear Programming Problem. Determine real numbers x 1, x 2, x 3 so as to minimize c 1 x 1 + c 2 x 2 + c 3 x 3 subject to the following constraints: a 1,1 x 1 + a 1,2 x 2 + a 1,3 x 3 = b 1 ; a 2,1 x 1 + a 2,2 x 2 + a 2,3 x 3 b 2 ; x 1 0 and x 2 0. Dual Linear Programming Problem. Determine real numbers p 1, p 2 so as to maximize p 1 b 1 + p 2 b 2 subject to the following constraints: p 1 a 1,1 + p 2 a 2,1 c 1 ; p 1 a 1,2 + p 2 a 2,2 c 2 ; p 1 a 1,3 + p 2 a 2,3 = c 3 ; p 2 0. (a) Prove that if (x 1, x 2, x 3 ) is a feasible solution of the Primal Linear Programming Problem and (p 1, p 2 ) is a feasible solution of the Dual Linear Programming Problem then p 1 b 1 + p 2 b 2 c 1 x 1 + c 2 x 2 + c 3 x 3. Prove also that p 1 b 1 + p 2 b 2 = c 1 x 1 + c 2 x 2 + c 3 x 3 if and only if the following Complementary Slackness Conditions are satisfied: a 2,1 x 1 + a 2,2 x 2 + a 2,3 x 3 = b 2 if p 2 > 0; p 1 a 1,1 + p 2 a 2,1 = c 1 if x 1 > 0; p 1 a 1,2 + p 2 a 2,2 = c 2 if x 2 > 0. (8 marks) Question continues on next page Page 5 of 6
(b) Using the proposition quoted below, or otherwise, prove that if (x 1, x 2, x 3 ) is an optimal solution of the Primal Linear Programming Problem, then there exists an optimal solution (p 1, p 2 ) of the Dual Linear Programming Problem which satisfies p 1 b 1 + p 2 b 2 = c 1 x 1 + c 2 x 2 + c 3 x 3. (12 marks) Minimum of a Linear Functional on a Convex Polytope The following proposition may be used, without proof, in answering question 4(b). Let n be a positive integer, let I be a non-empty finite set, and, for each i I, let η i : R n R be a non-zero linear functional and let s i be a real number. Let X be the convex polytope defined such that X = R : η i (x) s i }. i I{x (Thus a point x of R n belongs to the convex polytope X if and only if η i (x) s i for all i I.) Let ϕ: R n R be a non-zero linear functional on R n, and let x 0 X. Then ϕ(x 0 ) ϕ(x) for all x X if and only if there exist real numbers g i satisfying g i 0 for all i I such that ϕ = g i η i and g i = 0 whenever η i (x 0 ) > s i. i I c THE UNIVERSITY OF DUBLIN 2015 Page 6 of 6