Econ 172A W2002 Midterm Examination I Instructions 1 Please check to see that your name is on this page If it is not then you are in the wrong seat 2 The examination has 6 questions Answer them all 3 If you do not know how to interpret a question then ask me 4 You must justify your answers to the rst three questions No justi cation is needed on the last three questions 5 The table below indicates how points will be allocated on the exam I II III IV V VI Exam Total Score Possible 30 30 20 20 20 20 140 1 1 Consider the linear programming problem max x0 subject to 5x1 5x1 3x2 x2 x2 30 10 7 a Graphically represent the feasible set of this problem b Graphically solve the problem for the following values of x0 i x0 x1 x2 ii x0 x1 x2 iii x0 3x1 4x2 In each case graphically identify the solution write down the values for x1 and x2 that solve the problem write down the value of the problem If the solution to the problem is not unique then give two solutions 2 2 This problem concerns the linear programming problem from question 1 with x0 x1 x2 a Write the problem in the form max c x subject to Ax b x 0 b Write the problem in the form max c x subject to Ax b x 0 c Write the dual of the problem 3 3 Consider the linear programming problem max subject to 2x1 x1 2x1 x2 3x2 3x2 x 30 20 0 a Write the initial simplex array for the problem That is write the problem in a form suitable for a simplex algorithm pivot b Make one simplex algorithm pivot using the array you provided in part a If it is not possible to make a pivot explain why not If it is possible to make a pivot state the guess for the problem provided by your pivot and state whether this guess solves the optimization problem Use the tables below which may contain extra rows and or columns for your answers Row 0 1 2 3 Basis x1 x0 x2 x3 x4 x5 x6 x7 V alue Row 0 1 2 3 Basis x1 x0 x2 x3 x4 x5 x6 x7 V alue 4 4 Which of the tables below correspond to arrays that could arise in a correct simplex algorithm computation The objective is to maximize x0 and all of the variables in the problem are constrained to be nonnegative I did not include a column for the variable x0 Each part of the question is independent from the other parts To answer the question simply circle the letter or letters corresponding to correct simplex arrays a Row 0 1 2 3 Basis x1 x2 x3 x0 1 1 2 x7 1 1 2 x5 0 2 1 x4 0 1 0 x4 0 0 0 1 x5 0 0 1 0 x6 x7 2 0 1 1 1 0 1 0 V alue 6 6 4 10 Row 0 1 2 3 Basis x1 x0 0 x1 1 x5 0 x4 0 x2 x3 1 2 1 2 2 1 1 0 x4 0 0 0 1 x5 0 0 1 0 x6 x7 2 0 1 1 1 0 1 0 V alue 6 6 4 10 Row 0 1 2 3 Basis x1 x0 0 x7 1 x5 0 x4 0 x2 x3 1 2 1 2 0 1 1 0 x4 0 0 0 1 x5 0 0 1 0 x6 x7 2 0 1 1 1 0 1 0 V alue 6 6 4 10 Row 0 1 2 3 Basis x1 x2 x3 x0 0 1 2 x1 1 1 2 x5 0 2 1 x4 0 1 0 x4 0 0 0 1 x5 2 0 1 0 x6 x7 2 1 1 1 1 0 1 0 V alue 6 6 4 10 Row 0 1 2 3 Basis x1 x0 0 x1 1 x5 0 x4 0 x4 0 0 0 1 x5 0 0 1 0 x6 x7 2 1 1 1 1 0 1 0 V alue 6 6 4 10 b c d e x2 x3 1 0 1 2 2 1 1 0 5 5 Which of the following can be a feasible set for a linear programming problem a b c d e 6 6 For each of the statements below circle TRUE if the statement is always true circle FALSE otherwise No justi cation is required These problems refer to the linear programming problem P written in the form max c x subject to Ax b x 0 and its dual min b y subject to yA c y 0 a TRUE FALSE If D is not feasible then P is unbounded b TRUE FALSE Let u be a vector of ones with the same number of components as x If P has a solution then max c x u subject to Ax b x 0 has a solution c TRUE FALSE If P has a solution and c c then max c x subject to Ax b x 0 has a solution c may be di erent from c d TRUE FALSE If a linear programming problem is infeasible then it will continue to be infeasible if the objective function changes 7