The Capacitated Lot-Sizing Problem

Prerequisites

• Please look at Getting Started first for the most basic functions and the setup of OPTANO.Modeling

The mathematical Model

Sets: $$T= \text{the set of }t\text{ time steps}$$

Parameters: \begin{array}{l} d_t& = \text{ demand in time step }t\newline p_t& = \text{ unit production cost in time step }t\newline f_t& = \text{ machine setup cost in time step }t\newline h_t& = \text{ inventory cost in time step }t\newline c_t& = \text{ production capacity in time step }t \end{array}

Variables: $$ y_t = \begin{cases} 1, \text{ if we set up machines in time step }t\newline 0, \text{ else} \end{cases} $$

\begin{array}{l} x_t \in \mathbb{R}^+ = \text{ production amount in time step }t \newline s_t \in \mathbb{R}^+ = \text{ inventory amount in time step }t \end{array}

Objective: $$min \sum\limits_{t=1}^T( p_t x_t + f_t y_t + h_t s_t )$$

Restrictions:

\begin{array}{l} s_{t-1} + x_t = d_t + s_t & \forall t=1,...,T & \text{(Production, inventory & demand balance)}\newline s_0 = 0,s_T= 0 & & \text{(Inventory is empty at the start and the end)}\newline x_t \le c_t y_t & \forall t=1,...,T & \text{(Capacity can not be exceeded)} \end{array}

The Capacitated Lot-Sizing Problem

• Step 1: Create Business objects for your Model
• Step 2: Create your Model Class
• Step 3: Retrieve the Solution of your Model
• Step 4: Visualize your model (optional)