General production optimization

Optimization

General production structure capacited multi-level lot-sizing model PochetWolsey, page 45

• $1 \le i \le m$: set of items to be produced
• $1 \le k \le K$: set of shared resources with limited capacity
• $1 \le t \le n$: time periods until final period n
• $p_t$: unit production cost
• $q_t$: fixed production cost
• $h_t$: unit inventory cost
• $d_t^i$: demand for item i to be satisified at period t
• $M_t$: large positive number, expressing an upper bound on the maximum lot size iin period t
• $L_t^k$: available capacity of resource k during period t
• $\alpha^{i k}, \beta^{i k}$: amount of capacity of resource k consumed
• $1 \le j \le m$: items
• $D(i)$ set of direct successors of part i in the BOM, i.e. the items consuming some amount of item i when they are produced
• $r^{i j}$: the amount of item i required to make one unit of item j.
• r is the dependent demand, and $d_t^i$ is the independent demand
• $\gamma^i$: lead-time to produce or deliver an lot of i
• $x_t^i$: the size of product or purchase order of item i launched in period t, and delivered in period $t + \gamma^i$.
• Decision variables:
  • $x_t$: production lot size in period t
  • $y_t$: binary variable indicating wether there is a positive production in period t
  • $s_t$:inventory at the end of period t

Objective Function

$$ \begin{aligned} \text{min} \sum_i \sum_t (p_t^i x_t^i + q_t^i y_t^i + h_t^i s_t^i) \cr \text{subject to} \cr s_{t-1}^i + x_{t-\gamma^i}^i & = [d_t^i + \sum_{j \in D(i)} r^{i j} x_t^j] + s_t^i \cr x_t^i & \le M_t^i y_t^i \cr \sum_i \alpha^{i k} x_t^i + \sum_t \beta^{i k} y_t^i & \le L_t^k \cr \end{aligned} $$

References