Answer Happy

Problem Set 7.1 1. For each of the following distribution problems, determine either a feasible ship- ping schedule or the set of rows R' and columns C of Theorem 7.1.2, and verify that these rows and columns satisfy the inequality of that theorem. (A capacity of means that there is no limit on the number of units that can be shipped through the corresponding link.)
(h) OO OO oo 10 0 0 oo oo 00 OO 0 5 OO 8 0 88 oo 0 14 0 0 oo oo 0 00 oo 5 0 4 3 6 7 12 2
2-58 PM Sat Apr 23 Το Theorem 7.1.2. Let Rand C be as defined in Theorem 7.1.1, and let R' and C' denote their complements. Then Σ Σ», > Σας +Σ(ΣΚ) jEC IER jEC VIER That is, the total demand in the C' columns is strictly greater than the total supply in the R' rows plus the sum of the capacities of the links from the remaining rows to the C' columns. Proof. For any j, Xij <bj, and for at least one j E C', L_1 Xij <bj. Also, from part (b) of Theorem 7.1.1, i E R' and j e C implies that Xij = 0. Thus Σ» »ΣΣ) - ΣΣ ΣΣ +) jec 1 ' 276 -ΣΣΑ; +ΣΣki - ΣΣ Χ. + ΣΣΕ , iER' jeC' jEC' IER 476 11