A Class of Continuous Network Flow Problems

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Erratum: A Class of Continuous Network Flow Problems

E. J. Anderson, P. Nash and A. B. Philpott

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Mathematics of Operations Research

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Erratum: A Class of Continuous Network Flow Problems Author(s): E. J. Anderson, P. Nash and A. B. Philpott Source: Mathematics of Operations Research, Vol. 8, No. 3 (Aug., 1983), p. 478 Published by: INFORMS Stable URL: http://www.jstor.org/stable/3689316 Accessed: 18-09-2016 01:12 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.  Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms  INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Mathematics of Operations Research  This content downloaded from 178.250.250.21 on Sun, 18 Sep 2016 01:12:55 UTC All use subject to http://about.jstor.org/terms  MATHEMATICS OF OPERATIONS RESEARCH  Vol. 8, No. 3, August 1983 Printed in U.S.A.  ERRATUM* E. J. ANDERSON,t P. NASHt AND A. B. PHILPOTTt  In [1, ?5], we state two corollaries to a continuous maximum flow-mi theorem. The second corollary is incorrect, owing to an implicit assumption that the  optimal cut is unique. A counterexample is given by the following network 2, comprising three nodes, and  arcs having zero capacity except for (1,2) and (2,3) which have capacities b12(t) =2, 0 < t < 1, =1, < t<2,  b23(t)= 1 0 < t < 2. Furthermore, storage is allowed only in node 2 which has capacity equal to 1.  The zero-storage solution has unit flow in both arcs and is also optimal for CNP in s2; an optimal cut C for the zero-storage solution can be defined as follows.  c(t) =1,2, 0 < t < 1,  C(t {1},2} < t < 2, O<=0. Since a2(1) > 0, it is clear that C violates Corollary 2. A correct restatement of the corollary can be made as follows.  Let 0O be a network exactly similar to Q except that aj is;  identically zero for every nodej. Suppose {xjk} is an optimal solution to CNP posed for S20 and this solution has value v0. Then there is a nonempty collection Co of cuts in Q20 having value v0.  COROLLARY 2. The zero-storage solution { xjk) is optimal for CNP posed in Q if and only if some cut C in CO is such that no node j leaves C at a time t when aj(t) > 0. PROOF. Clearly { xjk } will be feasible for CNP posed in Q2, and if C is such that no  storage nodes leave the cut in the above fashion then the value of C in Q is also v0, and  by the theorem we have an optimal solution to CNP. Conversely, if the zero-storage solution is optimal (with value v0), there exists a cut C in Q2 with value vo. Thus C is in C0, since the value of C in 20 must be at most vo; furthermore, if some nodej leaves C  at t where aj(t) > 0, then the value of C in 2 is greater than v0, which is a  contradiction.  Reference [1] Anderson, E. J., Nash, P. and Philpott, A. B. (1982). A Class of Continuous Network Flow Problems. Math. Oper. Res. 7 501-514.  * Received February 3, 1983. tCambridge University.  *Massachusetts Institute of Technology. 478  0364-765X/83/0803/0478$01.25  Copyright ? 1983, The Institute of Management Sciences  This content downloaded from 178.250.250.21 on Sun, 18 Sep 2016 01:12:55 UTC All use subject to http://about.jstor.org/terms              

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