polimd/OR_Lab/lab_2.md

Lot sizing

Sets

• $I$: Production ${A_1,A_2,A_3}$
• $J$: Month {1,2,3,4}

Parameters

• $b_J$ Work days $\forall j \in J$
• $d_{i,j}$ Max Dem $\forall j \in J, \forall i \in I$
• $r_i$ Revenues $\forall i \in I$
• $c_i$ Cost $\forall i \in I$
• $q_i$ Max Production Lev. $\forall j \in J$
• $m_1$ Inv. $\forall i \in I$

Variables

$X{ij}$ Prod $i \in I, j \in J$ $V{ij}$ Sold $i \in I, j \in J$ $Z_{ij}$ Store $i \in I, j \in J$

Objective

$$max \sum_{I,J}{Vi*r_i-(ciX{ij}+miZ{ij})}$$

Constraints

Max production $$\sum{I}{Z{ij}\le K \forall j \in J}$$ Sold cannot exceed produced $$V{ij}\le d{ij} \forall i \in I, \forall j \in J $$ Inventory balance $$Z{ij}-Z{ij-1}+X{ij}-V{ij} \forall i \in I, \forall j \in J$$ Initial production $$X_{i0}=0 \forall i \in I $$

$$\sumI{\frac{x{ij}}{q_i}}\le b_j \forall j \in J$$

$$x_{ij} \le 0, Integer$$

$$v_ij,z_ij\ge 0$$

• To define the variable $Z$ we can use

  VAR Z{I,J,UNION{0}}