Department of Mechanical and Industrial Engineering
MIE1620: Linear Programming and Network Flows
(Fall 2012)
Quiz 2
December 5, 2013
1. (17 points) Consider the uncapacitated network flow problem shown in Figure 1. The label next to each arc (in boxes) is its cost. The bold numbers beside the arrows are the supplies/demands. Consider the spanning tree indicated by the dashed arcs in the figure and the associated basic solution.
(a) (4 points) Determine the values of the arc flows corresponding to this basic solu- tion. Is the solution feasible? Nondegenerate?
(b) (3 points) For the basic solution from part (a), find the reduced cost of each arc in the network.
(c) (4 points) Determine an optimal solution to the network flow problem.
(d) (3 points) By how much can we decrease c45 and still have the same optimal basic feasible solution?
(e) (3 points) If we increase the supply at node 1 and the demand at node 5 by a small positive amount δ, what is the change in the optimal cost?
Figure 1: Network for Problem 1
2. (10 points) Consider two nonempty polyhedra P = {x ∈ ℜn | Ax ≤ b} and Q = {x ∈ ℜn | Dx ≤ d}. We are interested in finding out whether the two polyhedra have a point in common.
(a) (2 points) Formulate an LP problem with the following properties: if P ∩ Q is nonempty, the LP returns a point in P ∩ Q; if P ∩ Q is empty, the LP is infeasible.
(b) (3 points) Form the dual of the LP from part (a).
(c) (5 points) Suppose that P ∩ Q empty. Use the dual from part (b) to show that there exists a vector c such that c′x < c′ y for all x ∈ P and y ∈ Q.
3. (6 points; 3 points each) Consider a linear programming problem of the form.
(a) Form the dual of the problem.
(b) Explain how Dantzig-Wolfe decomposition can be applied to the dual. It is suffi- cient to provide a few sentences to identify the coupling constraint and to describe the constraints of the subproblems solved during a typical iteration is sufficient.
4. (9 points) Consider a transportation problem with two source nodes s1 , s2 , and n demand nodes 1, . . . , n. All arcs of the form (si , j), i = 1, 2; j = 1, . . . , n are assumed to be present and to have infinite capacity. Let D =Σ di be the total demand. Let the supply at each source node be equal to D/2. Assume that di > 0 for all i.
(a) (3 points) Draw a figure representing the network flow problem. Include all nodes, arcs, supply and demand arrows, etc.
(b) (2 points) How many basic variables are there in a basic feasible solution for this network?
(c) (4 points) Show that there exists a degenerate basic feasible solution if and only if there exists some set S ⊂ {1, . . . , n} such that Σi∈S di = D/2.
5. (8 points) Consider the following LP
Supposex(ˆ) is a feasible solution to formulation (1). Now consider an inverse formulation to formulation (1), which we assume to be feasible:
We wish to understand the structure of optimal solutions to formulation (2).
(a) (2 points) Prove that formulation (2) has an optimal solution.
(b) (3 points) Determine if p = 0 can be a part of any feasible solution (c, p, ϵ) to formulation (2).
(c) (3 points) If at an optimal solution (c* , p* , ϵ* ), the constraint A(x(ˆ) − ϵ* e) ≥ b is
satisfied, show that there must exist some i such that ai(′)(x(ˆ) − ϵ* e) − bi = 0.