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ORIE PRELIM 3310/5310
February 19, 2009
Closed book exam. Justify all work. 1. Consider again the the following problem from the homework assignment. The standard decomposition procedure is applied using two subproblems (SUB1, SUB2): ORIGINAL PROBLEM max 6x1 s.t. x1 2x1 x1 + 4x2 + x2 - x2 + x2 + x2 + 3x3 + + x3 x3 x4 x4 x3 , + x5 + 2x5 x4 , x5 + + 3x4 x4 + - x5 x5 3 5 4 6 3 4 0
x1 ,
x2 ,
We begin the solution procedure with a restricted master program consisting of slack variables s 1 and s2 associated with the two linking constraints and variables 1 and 1 corresponding, respectively, to the two subproblem solutions (x1 , x2 , x3 ) = (0, 0, 0) and (x4 , x5 ) = (0, 0) . At a later stage of the computation the following tableau for the restricted master program is encountered. basis 1 2 (-z) -12 0 s1 -2 0 2 -4/3 0 2 1 1 1 4/3 0 1 0 0 0 0 1 2 0 0 1 0 0 s1 0 1 0 0 0 s2 r.h.s. -3 -27 0 1 1/3 1/3 0 1 -1/3 2/3
(a) (5 points) Write the initial restricted master as specified above. (b) (5 points) Determine by inspection the inverse of the basis matrix (i.e., B -1 ) for this tableau. (c) (5 points) What are the shadow prices 1 , 2 , 1 , 2 associated with the four constraints of the restricted master program? ( 1 , 2 correspond to the two linking constraints, while 1 , 2 correspond, respectively, to the and constraints.) (d) (10 points) Using only information from part (b) and the above tableau, determine the column in the master program corresponding to 2 . (e) (5 points) Using the shadow prices of part (c), determine the LP problem which should now be solved by SUB1. (f) (5 points) Determine the column of the (original) master program corresponding to the extreme point solution (x1 , x2 , x3 ) = (0, 4, 2) for SUB1. (g) (5 points) Enter the column generated in (f) part into the updated tableau given above. (h) (5 points) Compute the reduced cost for the updated column in part (g). Can this new proposal of SUB1 improve the solution to the restricted master program given above? Explain. (i) (5 points) Show that the 2 column of the above restricted master tableau corresponds to the extreme point (x4 , x5 ) = (3, 0) . (j) (15 points) Determine the solution (x1 , x2 , x3 , x4 , x5 ) for the original problem which is specified by the above restricted master tableau. (k) (10 points) Determine an initial Phase I restricted master program which results if (only) the extreme points (x1 , x2 , x3 ) = (0, 4, 2) for SUB1 and (x4 , x5 ) = (0, 2) for SUB2 are used.
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2. The following model is an example of a so-called generalized linear programming problem. max 2x1 s.t. x1 3x1 x1 , 5x2 2x2 6x2 21 x2 , x3 , - + - + 4x3 + 1 x3 + 2 x3 + 32 1 , 2
=
6 5 4 0
Apparently, the complicating factor here is the form of the x3 -column stipulated by 1 and 2 . Initially we avoid this difficulty by ignoring the x3 -column in an attempt to produce a good solution; i.e., we restrict attention to x1 , x2 and solve: max 2x1 s.t. x1 3x1 - 5x2 + 2x2 - 6x2 x1 , x 2 6 5 0
Suppose that for this restricted problem the optimal solution is x , x , with shadow prices 1 , 2 for 1 2 the two constraints. (a) (10 points) State a condition which must be satisfied by any column of the form required for x 3 that could improve the solution given by x , x . 1 2 (b) (15 points) Determine a linear programming problem whose solution will specify, if one exists, an improving x3 -column of the required form. (c) (25 points) Give a brief, but precise and complete, description of how the decomposition/column generation technique could be used to solve this problem.
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