The multi-product, multi-period model in network design

From my doctoral course

This note comes from the doctoral course “Supply Chain Network Design”. I rewrote the explanations in my own words and recomputed the numerical example. The course’s main sources include Watson et al. (2013), Chopra and Meindl, and Dekker et al. (2004).

Why is the basic model not enough?

The basic location model sees one product and one period. In practice:

• Several products use the same lines, warehouses and fleet (shared capacity).
• Demand changes over time (seasonal, growth) and decisions such as adding capacity are multi-period.
• Inventory carries over between periods.

Data

Fixed costs of facilities (plant, warehouse, vendor), capacities (plants, warehouses, roads and vehicles), the transport cost of products from plant to distribution centres and from these to customers, production and holding costs, and in an international network duties on transport, taxes on product or profit, exchange rates and prices.

Model structure

Indices: product $p$, period $t$, facility $j$, customer $i$.

• Decision variables: $y_{jt}$ (facility $j$ is open in period $t$), $x_{ijpt}$ (quantity of product $p$ from $j$ to $i$ in period $t$) and $I_{jpt}$ (inventory).
• Objective: the sum (discounted) of fixed, production, transport, inventory and shortage-penalty costs; or maximising profit after tax.
• Constraints:
  • Inventory balance: $I_{jpt}=I_{jp,t-1}+\text{inflow}-\text{outflow}$
  • Shared capacity: ${\sum }_p{a}_p({\text{flow}}_{jpt})\le s_j{y}_{jt}$ (each product takes a share of capacity)
  • Meeting demand in each period for each product
  • Open-and-close logic: once a facility is opened, it stays until the end of the horizon.

The result is a large MIP, solved with commercial solvers and sometimes heuristics, or with decomposition (such as Lagrangian or Benders) for large sizes.

Decisions seen only in the multi-period case

• Timing of capacity expansion (in which year do we build a new warehouse?)
• Shifting production between plants when costs or taxes change.
• Safety stock for peak seasons.

My note

With the addition of time and product the model becomes very large; so the modeller’s decision matters: what detail do I drop? Aggregation of products and customers and the size of periods affect tractability and credibility more than any algorithm.

Related: Multi-objective network design: cost, sustainability and resilience.

multi_productmulti_periodnetwork_design