Econ 172A, W2002: Midterm Examination IInstructions.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 first three questions. No justification is needed on the last threequestions.5. The table below indicates how points will be allocated on the exam.Score PossibleI 30II 30III 20IV 20V 20VI 20Exam Total 14011. Consider the linear programming problem:max x0subject to 5x1+3x2≤ 305x1− x2≥ 10x2≥−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 x1and x2that solve theproblem; write down the value of the problem. If the solution to the problem is not unique, thengive two solutions.22. 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.33. Consider the linear programming problem:max 2x1− x2subject to x1+3x2≤ 302x1− 3x2≤ 20x ≥ 0(a) Write the initial simplex array for the problem. (That is, write the problem in a form suitable fora simplex algorithm pivot.)(b) Make one simplex algorithm pivot using the array you provided in part (a). (If it is not possibleto make a pivot explain why not. If it is possible to make a pivot, state the guess for the problemprovided 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 Basis x1x2x3x4x5x6x7Value(0) x0(1)(2)(3)Row Basis x1x2x3x4x5x6x7Value(0) x0(1)(2)(3)44. Which of the tables below correspond to arrays that could arise in a correct simplex algorithm compu-tation? The objective is to maximize x0and all of the variables in the problem are constrained to benonnegative. I did not include a column for the variable x0. (Each part of the question is independentfrom the other parts.) [To answer the question, simply circle the letter (or letters) corresponding tocorrect simplex arrays.](a)Row Basis x1x2x3x4x5x6x7Value(0) x01 −1 2 0 0 −2 0 6(1) x7−1 1 2 0 0 1 1 6(2) x50 2 1 0 1 −1 0 −4(3) x40 −1 0 1 0 1 0 10(b)Row Basis x1x2x3x4x5x6x7Value(0) x00 −1 2 0 0 −2 0 6(1) x11 1 2 0 0 1 1 6(2) x50 2 1 0 1 −1 0 4(3) x40 −1 0 1 0 1 0 10(c)Row Basis x1x2x3x4x5x6x7Value(0) x00 −1 2 0 0 −2 0 6(1) x71 −1 2 0 0 1 1 6(2) x50 0 1 0 1 −1 0 4(3) x40 −1 0 1 0 1 0 10(d)Row Basis x1x2x3x4x5x6x7Value(0) x00 −1 2 0 2 −2 1 6(1) x1−1 1 2 0 0 1 1 6(2) x50 2 1 0 1 −1 0 4(3) x40 −1 0 1 0 1 0 10(e)Row Basis x1x2x3x4x5x6x7Value(0) x00 1 0 0 0 2 1 6(1) x11 1 2 0 0 1 1 6(2) x50 2 1 0 1 −1 0 4(3) x40 −1 0 1 0 1 0 1055. Which of the following can be a feasible set for a linear programming problem?(a)(b)(c)(d)(e)66. For each of the statements below, circle TRUE if the statement is always true, circle FA L S E otherwise.No justification is required.These problems refer to the linear programming problem (P) written in the form:max c · x subject to Ax ≤ b, x ≥ 0and its dualmin b · y subject to yA ≥ c, y ≥ 0.(a) TRUE FALSEIf (D) is not feasible, then (P) is unbounded.(b) TRUE FALSELet u be a vector of ones (with the same number of components as x. If (P) has a solution, thenmax c · (x − u) subject to Ax ≤ b, x ≥ 0has a solution.(c) TRUE FALSEIf (P) has a solution andc ≥ c,thenmaxc · x subject to Ax ≤ b, x ≥ 0has a solution (c may be different from c).(d) TRUE FALSEIf a linear programming problem is infeasible, then it will continue to be infeasible if the objectivefunction