Home Work Set 3 MSIS 685, Linear Programming Fall 1998 Due : December 21st Questions with * are optional and are given so that students who wish to improve their grade have an opportunity to do so. Use the simplex method to solve the following transportation problem (red numbers Question 1 are costs, rows are demand nodes and columns are supply nodes.) Question 2 a b 3 9 6 11 15 10 4 7 6 22 8 6 17 3 10 25 15 2 19 a) Use the simplex method to solve the min-cost-flow problem for the network below. The numbers next to nodes are their demand (supply if negative). Those nodes without numbers are transshipment nodes b) Ignore the demand and supply and find the all shortest paths (i.e. all minimum cost paths) from node a to all other nodes, using Dijkstra's algorithm. 8 b 17 -20 f 7 Question 3 1 5 e 10 4 (10, 7) 10, 7) 5 g 5 6 a 12 15 10 c d 20 h 1 15 For the network shown below find the maximum flow from source nodes s to sink node t. 1 Question 3 b f 9 g 15 6 3 9 3 6 3 10 s t e 9 3 15 9 9 c 17 d 2 h Question 4 Do problem 7.1 on page 111 of Vanderbei's text. Question 5 Do problem 7.2 on page 111 of Vanderbei's text. Question 6* There is a general strong connection between the optimization theory and free competition. Here is an example: Suppose there are n economic activities (for example building factories, homes, shopping malls, etc.) that are to be individually located on n distinct parcels of land. If activity i is located on parcel j that activity yields sij units (e.g. dollars) value. If the assignment of activities to the land parcels are is made by a central authority, it might be made in such a way as to maximize the total value generated. In other words, the assignment would be made so as to maximize ijsijxij where 1 if activity i is assigned to parcel j xij otherwise. 0 More explicityly this activitiy leads to the optimization problem i j sijxij ixij =1 for all nodes jxij =1 for all nodes j 0xij xij=0, or 1 Max s.t. Actually, it can be shown that the final requirement xij=0 or 1 will be satisfied automatically. If one considers the problem from the view point of free competition, it is assumed that, rather than a central authority determining the assignment, the individual activities bid for the land and thereby establish prices. 2 Question 6* a) Show that there exists a set of activity proces pi for i=1,...,n and land prices qj, j=1,...,n such that for all i,j: pi+qj sij with equality holding if in the optimium activity i is assigned to parcel j. b) Show that part a) implies that if activity i is optimially assigned to parcel j, and if j' is any other parcel then sij-qjsij'-qj'. Give an economic interpreation of this result and explain the relation between free competition and optimiality in this context. c) Assuming that each sij is positive, show that the prices can be assumed to be nonnegative. Question 7* a) In a maximum flow problem suppose that we have multiple source and multiple sink nodes. How would you change the problem to a single source single sink maximum flow problem? Give an example. b) In a multiple (or single) source and sink maximum flow problem suppose that the capacity of source is not unbounded, but has a limit, say ks and/or the sink has a limited capacity of kt. How would you transform the problem to the ordinary unlimited source/sink capacity network flow problem? Give an example. Question 8 a) Consider the upper-bounded linear programming problem: Min s.t. T c x Ax=b 0x u Find the dual of this problem. If y is the dual variable corresponding to equations Ax=b then prove that at the optimum the complementary slackness theorem reduces to: If xi=ui then ci<aiTy if 0 < xi < ui then ci=aiTy if ci>aiTy the xi=0 b) Now take the special case of maximum flow problem in a network, in particular its minimum cost flow formulation: Min s.t. -xts ixik - ixki=0 for all nodes 0xijuij Show that the complementary slackness relations at the optimum in this case can be written as if yj-yi<cij then xij=0 if 0 < xij < uij then yj-yi=cij if yj-yi>cij then xij=uij ys-yt=1 3 Question 9* The minimum cost flow problem (without capacity constraints on the edges) can be converted to a transportation problem and thus solved by transportation algorithm. One way to do this conversion is to is to find the minimum cost path from every supply node to every demand node, allowing for possible shipping through intermediate transshipment nodes. The values of these minimum costs become the effective point-to-point costs in the equivalent transportation problem. Once the transportation problem is solved, yielding amounts to be shipped from origins to destinations, the result is translated back to flows in arcs and shipping along the previously determined minimal cost paths. Consider the transshipment problem with five shipping points defined by the symmetric cost matrix and the requirement vector s=(10,30,0,-20,-20) 0 3 C 4 6 4 3 3 6 4 0 5 4 8 5 0 2 5 4 2 0 5 8 5 5 0 ( (The requirement vector gives supply nodes if negative, demand nodes if positive and transshipment nodes if zero.) a) Set up the associated transportation problem and solve it. b) From the associated transportation problem find out the optimal flow in the original problem 4