Model Details

Overview

The Delivery Schedule Optimization application uses a time-expanded min-cost flow model to optimize warehouse-to-customer deliveries. This document explains the key concepts and constraints of the model.

Model Structure

Time Expansion

The model creates a time-expanded network where:

• Each warehouse has a node for each day in the planning horizon
• Each customer has nodes for their valid shipping windows
• Arcs represent possible shipments between warehouses and customers

Network Components

1. Warehouse Nodes

   • One node per warehouse per day
   • Capacity constraints on daily shipments
   • Connected to source node with total capacity

2. Customer Nodes

   • One node per customer per shipping window
   • Demand constraints based on order volumes
   • Connected to sink node with total demand

3. Shipping Arcs

   • Connect warehouse nodes to customer nodes
   • Costs based on transportation policies
   • Minimum flow constraints for consolidation
   • Capacity constraints based on available shipping capacity

Constraints

1. Capacity Constraints

• Warehouse daily capacity limits
• Minimum shipment sizes for consolidation
• Maximum flow on shipping arcs

2. Time Constraints

• Urgent orders must be shipped within 3 days
• Batch orders must be shipped in next calendar week
• Custom shipping windows based on order type

3. Flow Constraints

• Flow conservation at each node
• Minimum flow requirements for consolidation
• Maximum flow limits based on capacity

4. Cost Structure

• Tiered shipping rates based on volume
• Different costs for different transportation modes
• Consolidation discounts

Optimization Objective

The model minimizes total transportation cost while satisfying all constraints:

• Minimize: Σ (flow on arc × cost per unit)
• Subject to:
  • Capacity constraints
  • Time window constraints
  • Flow conservation
  • Minimum shipment sizes

Solution Process

1. Graph Construction

   • Create time-expanded network
   • Add source and sink nodes
   • Set up arcs with costs and capacities

2. Flow Optimization

   • Solve min-cost flow problem
   • Determine optimal flows on each arc
   • Calculate total cost

3. Solution Extraction

   • Convert flows to shipment assignments
   • Calculate per-order costs
   • Generate output reports

Performance Considerations

Model Size

• Number of nodes: O(W×D + C×W)
  • W: number of warehouses
  • D: number of days
  • C: number of customers
• Number of arcs: O(W×C×D)

Solution Time

• Typically solves in seconds to minutes
• Depends on:
  • Number of orders
  • Planning horizon length
  • Number of warehouses and customers
  • Complexity of shipping rules

Limitations

1. Deterministic model (no uncertainty)
2. Linear cost structure
3. No consideration of:
   • Loading/unloading times
   • Vehicle capacity
   • Driver schedules
   • Weather conditions