1. Log Truck Scheduling Problem

2. Finding Efficient Routes
It consists in finding one feasible route for each vehicle in a fleet in order to satisfy the demands of the customers and in such a way that the total transport cost is minimized. We use a mathematical formulation of the log truck scheduling problem where each column represents a feasible route. We generate a large pool of columns based on solving a transportation problem. Then we apply a composite pricing algorithm, which mainly consists of pricing the pool of columns and maintain an active set of these, for solving the LP relaxed model. A branch and price approach is used to obtain integer solutions in which we apply composite pricing and constant branching.

3. Mathematical Model

4. The truck in the example starts with a pick-up of 40 tonnes at supply point 1, and drives to the customer where it is totally emptied.
The truck drives back to the same supply point for another trip.After delivering 40 more tonnes to the demand point, it visits supply point 3 and picks up 30 tonnes there. Since the truck is not fully loaded after that visit, it continues to supply point 2 where it picks up an additional 10 tonnes.
Finally, the truck delivers the last load and travels back home. It is clear that the information in the column involves the sum of the amounts picked up or delivered along the whole route. Therefore, we need to save all data concerning the exact schedules to be able to reproduce the real route for each truck.
Collecting Data

5. Variables for Data Three sets of constraints must be met.
(1) forces the model to choose one route for each truck. Each truck is permitted to stay at the home base during the whole day, therefore a column with cost zero representing this ‘route’ is added to the set of feasible routes.
(2) ensures that the total quantity picked up does not exceed the supply
(3) guarantees that the demand is met.

6. Why It works Our model is related to the transportation problem.
If we consider the special case where all supplies and demands are multiples of the capacity of the truck, then each supply/demand can be expressed as the number of fully loaded trucks. so we can split the number of supplies/demands in order to get a supply/demand equal to one load.
Then, the problem turns into a set partitioning problem. At the same time, constraints (2) and (3) are connected to the constraints in the transportation problem. The difference is that there are no continuous flow variables, the flow is decided through the choice of the routes
Column generation is a suitable and natural solution approach for our problem. This is due to two reasons:
1: The columns have a practical interpretation. They represent a whole day’s route for one truck.
2. The number of possible columns is huge even for small problems and in practice it is impossible to enumerate all feasible routes.

7. Transportation problem
Mines and factories
Suppose that we have a collection of mines mining iron, and a collection of factories which consume the iron that the mines produce.
the cost of transporting one shipment of iron from x to y.
For simplicity, we ignore the time taken to do the transporting.
We also assume that each mine can supply only one factory (no splitting of and that each factory requires precisely one shipment to be in operation
Having made the above assumptions, a transport plan is a bijection — i.e. an arrangement whereby each mine supplies precisely one factory .
We wish to find the optimal transport plan, the plan whose total cost
is the least of all possible transport plans from to .
This motivating special case of the transportation problem is in fact an instance of the assignment problem.

8. Log routes
Routes