Unformatted text preview: Engineering Optimization: ISYE3133C Practice Midterm #2
1. (4 pts each for a total of 20 pts) For each of these statements, mark True or False (no justification needed): a) T - F - Every dual feasible basis for a standard form LP is optimal.. b) T - F - There is a one to one correspondence between the extreme points of a standard form linear program and its basic feasible solutions. c) T - F - Every transportation problem is balanced. d) T - F - In a basic feasible solution only the nonbasic variables can have value zero. e) T - F - The dual of a standard form LP always has non-negative variables. 2. (50 pts Total) Consider the following linear program and its graph: Maximize Subject to z = 3x1 + x2 -x1 + x2 2 (1) x1 + x2 4 (2) x1 3 (3) x1, x2 0 a) Convert the problem into standard form. b) Write a basis for the standard form problem. Indicate what are the associated primal and dual solutions. Indicate if the basis is primal/dual feasible/infeasible. Write the reduced costs of the non-basic variables. Identify the associated extreme point in the graph. c) Find the optimal basis for the standard form problem. Indicate what are the associated primal and dual solutions. Why is it optimal? d) For the standard form problem, suppose we include a new variable into the formulation with objective coefficient 2 and whose coefficients for constraints (1), (2) and (3) are -1, 1 and 2 respectively. Is the basis from c) still optimal? Justify your answer. 3. (30 pts) Touche Young has three auditors. Each can work as many as 160 hours during the next month. During this time three projects must be completed. Project 1 will take 130 hours; project 2, 140 hours; and project 3, 160 hours. The amount per hour that can be billed for assigning each auditor to each project is given bellow. a. Formulate a transportation problem to maximize total billings during next month. Clearly indicate the supply and demand constraints. b. Draw the Graph or diagram of your transportation problem. Clearly label the nodes indicating if they are supply or demand nodes and their respective capacities or demands. Write the profit/cost per unit of flow besides each one of the arcs. c. Is the transportation problem from a)-b) balanced? Auditor 1 2 3 Project 1 120 140 160 Project 2 150 130 140 Project 3 190 120 150 Remember to describe (in words) the variables, constraints and objective function. redit Question
You can use the following in any of the problems. normal max problem of Standard Form problem Credit Question The Dual
normal max problem max
j=1 n cj xj
j=1 n max s.t. z = s.t.
n j=1 j=1 aij xj = bi
aijx j 0 i x =b
j i = 1, . . . , m
i = 1, . .. .. ., , n m j = 1, j = 1, . . . , n (P)
(P) is m xj 0 min w =
min w = s.t. s.t. m i=1 bi y i
bi y i i=1 m m i=1 aij yi cj
aij yi cj j = 1, . . . , n
j = 1, . . . , n (D)
(D) (1) (1) i=1 hat that asible solution to (D) Also, the dual of (D) is (P). easible solution to (D) of (P) which can take the following values: optimal objective value he optimal objective value of = + if (P) is unbounded. (P) which can take the following values: = = + if (P) is unbounded. - if (P) is infeasible. s = - if (P) (P) has an optimal solution (i.e. (P) is feasible and is not unbounded). a number if is infeasible. is a number if (P) has an optimal solution (i.e. (P) is feasible and is not unbounded). + for any number d. d + for any number d.
m i=1 d consider the following three possibilities for (P): uld consider the following three possibilities for (P): nbounded. By lemma 3 if the primal is unbounded then the dual is infeasible. Then this unbounded. By lemma if the primal y unbounded solution to (D). not happen because we3assumed that is is a feasible then the dual is infeasible. Then this nnot happen because we assumed that y is a feasible solution to (D). feasible. Then z = -. (2) holds because - is less than or equal to any number and nfeasible. Then z = -. (2) holds because - is less than or equal to any number and i is a number. i y i is a number. an optimal solution. Lemma 1 states that s an optimal solution. Lemma 1 states that z m bi y i . z i=1 bi y i . (2) (2) m m ...
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- Spring '08
- Optimization, standard form problem