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5.2. Subgradient optimization
To improve the lower bound, we update the Lagrangian multipliers by solving the following Lagrangian dual
problem.
(LDP)

max










min ( , , , , )




























,

, ,


































subject to the constraints in problem LRP, where min ( ,, , , ) is concave and piecewise linear

















































continuous to, ,

. We use the subgradient method to optimize the dual problem by updating the




















































multipliers at the th iteration as follows. For example, the Lagrange multiplier ,( ) is updated by




























(

( ))(∑




− 1)
















,( +1) = ,( )

+













=1







(22)






























(∑



1)2































=1

=1


















where ,( ) is the Lagrange multipliers at function value at th iteration respectively. results.


th iteration. , ( ) are the value of upper bound and Lagrange is the step size which is determined by preliminary experimental



5.3. Construction of a feasible solution
The solution to the relaxed problem is usually infeasible to the original problem due to the relaxation of the coupling constraints. A feasible solution, i.e., upper bound, is constructed by the following heuristic algorithm. The heuristic algorithm successively checks the violation of constraints (2), (4), (5) and (15) and modifies the solution of the LRP problem. We also propose a new algorithm to obtain better upper bound in Section 6.

Heuristic algorithm
Step 1: Initialization.

Set the parameter ← , ← ∅.


Step 2: Remove those items with reserved space larger than needed: i.e., if there exists item that satisfies the condition ∑ > max , item is removed from the planning. ← ∖ { }.
Step 3: Check if there are any items that do not satisfy the constraints (2). If item does not satisfy (2),

then ∪ { }.
Step 4: For each item ∈ , if there exists an feasible storage location 2 , i.e., ∑ 2 = 0, then the item is reassigned to the location 2; otherwise, the planning of item is removed.
14

Update ← ∖ { }. ← ∖ { }. If error = , go to Step5.
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