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: 1, if item i is inventoried in location l during period t; 0, otherwise


: 1, if location l is reserved for item i for the planning horizon; 0, otherwise.
Note: Each location can only hold one unit at a time.

4.3. Formulation
The integrated problem is formulated as the following mixed integer programming model.
4.3.1. Objective Function




















= ∑ ∑+ ∑ ∑ ∑+ ∑ ∑ ∑+ ∑∑(

++ ℎ )

(1)



















=1 =1

=1 =1 =1

=1 =1 =1

=1 =1






The objective function is divided into four terms: the first term is the cost of reserving locations for items, in a dedicated storage policy. The second term is the cost associated with the travel of products from the production area to assigned storage locations. The third term is the cost associated with the transportation of products from the storage locations to the output point, which occurs once a product is demanded during a given time period. The fourth term is all the costs associated with production planning, production, setup, and holding inventory. The objective is to find the optimal production planning and the best locations of the products in order to minimize the total costs.


4.3.2 Constraints








∑ ≤ 1, ∀

(2)

=1



Constraints (2) limit the number of items that could be assigned to a reserved location to one. They ensure that a location can only be reserved by one item.













∑ ≤ 1,

∀ , ∀

(3)










=1







Constraints in (3) ensure that at most one item can be requested at each storage location in each period.













∑ ≤ 1,

∀ , ∀

(4)










=1
Constraints in (4) state at most one item can be moved from production area to a specific storage location l in period t.








∑ ≤ 1, ∀ , ∀

(5)

=1



Constraints in (5) state that at most one item can be inventoried at each storage location in each period.



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