# 3 Suppose instead of having one depot you had n depots, each with d

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each with di vehicles stationed at it for i = 1, .... , n.
(a) How does the formulation in (1) change for multiple
depots?
(b) Could Dantzig-Wolfe or Bender\'s decomposition be
used with multiple depots? How could you apply these
techniques, or why are they not very practical on these
problems?
Solution
Dantzig-Wolfe (DW) DW approach deals with the problem:
min c_1^T x_1 + c_2^T x_2
s.t. A_1 x_1 = b_1 (C1)
A_2 x_2 = b_2 (C2)
B_1 x_1 + B_2 x_2 = b_0 (C3)
x_1 >= 0, x_2 >= 0.
Benders (B) decomposition deals with the problem
min f_1^T x_0 + g_1^T y_1 + g_2^T y_2
s.t. T_1 x_0 + W_1 y_1 = h_1
T_2 x_0 + W_2 y_2 = h_1
x_0 >= 0, y_1 >= 0, y_2 >= 0.

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3. Suppose instead of having one depot you had n depots, each with di vehicles stationed at it for i = 1, .... , n. (a) How does the formulation in (1) change for multiple depots? (b) Could Dantzig-Wolfe or Bender\'s decomposition be used with multiple depots? How could you apply these techniques, or why are they not very practical on these problems? Solution Dantzig-Wolfe (DW) DW approach deals with the problem: min c_1^T x_1 + c_2^T x_2 s.t. A_1 x_1 = b_1 (C1) A_2 x_2 = b_2 (C2) B_1 x_1 + B_2 x_2 = b_0 (C3) x_1 >= 0, x_2 >= 0. Benders (B) decomposition deals with the problem min f_1^T x_0 ...
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