RESEARCH SEMINAR, MOLDE, JUNE 6th

ARILD HOFFMOLDE UNIVERSITY COLLEGE

ARNE LØKKETANGENUROOJ PASHA

MILK COLLECTION IN

WESTERN NORWAY USING

TRUCKS AND TRAILERS

A DAIRY COMPANYTINE BA• The leading dairy company in Norway,

owned by 18000 milk producers• Core business is producing dairy

products from raw milk• We look at a subproblem: Collecting milk

from 990 milk producers in the northern part of Møre og Romsdal County in Western Norway

• 3 different dairy plants in the same district

DAIRY PLANTS

THE DAIRIES• Circle – Elnesvågen

– 77.2% of total delivery– Produces Jarlsberg, and other cheeses

• Star – Høgseth – 17.4% of total delivery– Produces milk for consumption

• Square – Tresfjord – 5.4% of total delivery– Produces cheeses Ridder and Port Salut

THE VEHICLES• Each dairy has associated a certain

number of heterogeneous cars, with or without trailers.

• The tanks have several compartments to avoid mixing:– Ecological milk from some producers.– Contaminated milk (antibiotics) from farms

with possible diseases.– Whey to be returned to the farms for animal

food (waste product when producing cheese).

MILK COLLECTION PROBLEM

• Collect milk from producers and deliver at the plants

• Each plant has a certain daily demand• Milk can be stored in cooler tanks at the producers

for at most three days• Small farms which are inaccessible for a truck

carrying a trailer• The plans generated are seasonal

– Fixed routes for the season (winter and summer)– Some slack (spare capacity) is incorporated due to varying

daily production

CURRENT COLLECTION STRUCTURE

• Milk can be stored for up to three days in cooler tanks at farms – Expensive to change

• Collection is according to prespecified frequencies:– 73 – Every third day, 7 days a week– 72 – Every second day, 7 days a week

• Needs a smaller cooler tank at the farm

– 62 – Every second day, not sundays• Needs same size tank as 73

SOLUTION STRUCTURE• A trailer can be used as a mobile

depot• The truck leaves the trailer at a

parking place and visits the farms to collect milk

• When the truck returns to the parked trailer, the milk can be transferred to the trailer tank and the truck are ready to collect milk from other farms

SOLUTION STRUCTURE

PARKING PLACES

SUPPLIERS

A REAL-WORLD PROBLEM• Vehicle Routing.

• Multi Depot (3 plants in this district, totally 49 in Norway).

• Pickup and Delivery (pickup milk, deliver whey).• Fleet Size and Mix.• Truck and Trailers / Satellite Depots• Two-Echelon VRP• Periodic VRP (frequency every 1, 2 or 3 days).• Time Windows (to a small degree at suppliers, but

also for meeting ferry times etc.).

INITIAL SOLUTION1. Compute the number of tours

necessary with reference to the available vehicle fleet, the visiting frequency and the needed supplies for each depot.

2. Select seed orders for each tour and cluster around these.

3. Optimize each tour using a simple local search.

4. Insert parking places for the tours that are served with a truck and trailer.

HEURISTIC

• Neighborhood structure– Move or swap orders between two

tours– Reduce the neighborhood by only

considering tours containing other close orders

– Partial neighborhood examination

HEURISTIC

• Reoptimization of subtours after change

• Try to improve each tour by moving orders to other subtours

• Recalculation to find optimal parking places

HEURISTIC• Tabu Search

– Variable Tabu Tenure– Diversification strategy to avoid that the

same moves are performed too often– Dynamic penalty for load-infeasible

solutions are added to the objective value

))()(()(

r r

ttir

Rir QLQL

r

17

• ξ(s) = ψ(s) + η τ(s)• ψ(s) is the number of h-neighbors order s has

inside its own tour, • τ(s) is the number of times order s is selected

from the current tour. • The order s with lowest value ξ(s) is selected

for an eventual move from the tour as long as it is not declared tabu.

• (h-neighborhood – the set s S which consist of the h suppliers closest to s. A tour with none of the h closest suppliers is not considered for a move.)

DIVERSIFICATION

18

• Our test results are not unambiguous about the value of η.

• We have chosen to use the value η = 0.75 for our subsequent tests, as this value gave a slightly better result than the other alternatives.

DIVERSIFICATION

19

• β(σ) is the penalty for solution σ which are added to the objective function

• (x)+ = max{0, x}. • r is all tours in the solution • Rr are all subtours in tour r. • Q is the capacity of the complete vehicle• Qt is the capacity of the truck, • Lr is the total load in tour r and • Lt

ir is the truckload on subtour i in tour r. • The penalty factor α is initially set to 1 and adjusted during the

search by multiplying or dividing it with a value κ when the solution is respectively feasible or infeasible. Preliminary tests show that the value κ = 1.1 gives best results in our search.

))()(()(

r r

ttir

Rir QLQL

r

PENALTY FACTOR

OBJECTIVE FUNCTION

γ(σ) : Driving distance

ε(σ) : Additional costs i.e. for using ferrys or toll roads

β(σ): Penalty for infeasible solutions

)()()()( f

COMPUTATIONAL RESULTS

• Want to find out the effect of – Truck/hanger size– Collection strategy– Effect of parking places

VEHICLE SIZESID Qv Qt Qx Freq. M Obj.value 1 10000 10000 0 7x3 95 6849.45 51 14500 14500 0 7x3 68 5312.21 5 18500 18500 0 7x3 53 4526.99 12 21000 10000 11000 7x3 47 4856.26 17 29000 10000 19000 7x3 36 4347.30 71 33500 14500 19000 7x3 32 3875.62 36 29500 18500 11000 7x3 35 3975.95 39 33500 18500 15000 7x3 33 3480.90

Qv – Total vehicle capacityQt – Capacity of the truckQx – Capacity of the hangerM – Number of tours

COLLECTION STRATEGIES

0.50

1.00

1.50

2.00

2.50

1 51 5 12 17 71 36 39

Vehicle type

Rel

ati

ve

ob

ject

ive

valu

e

7x1

7x2

7x3

- Clearly better to collect every third day

- Need bigger storage tanks at the farms

- In practice a mix of 72 and 73

ONE OR MORE PARKING PLACES

ID Qv Qt Qx Freq. M Obj.value 71 33500 14500 19000 7x3 32 3875.62 71p 33500 14500 19000 7x3 33 4292.43 39 33500 18500 15000 7x3 33 3480.90 39p 33500 18500 15000 7x3 32 3968.38

P – only one parking per tour

M - number of tours

CONCLUSIONS

• The visiting frequency should be as long as possible (3 days)

• A strategy where trailers are used as mobile depot are superior to only using single trucks

• Tours in the local neighborhood of the plant can be served by a single truck without a trailer

CONCLUSIONS

• When the total capacity is equal, a large truck with a smaller trailer is better than the opposite

• The possible use of more than one parking place on a tour can improve the solution quality significantly

• The advantage of extending the visiting frequency increases with the size of the vehicle

Objective Function

• Driving distance

• Vehicle acquisition cost

• Ferry / toll roads

• Extension– Cost of changing cooling tanks– Pay for driver (overtime, Sunday, etc.)

What to determine?

• Build an initial solution– How to determine optimal fleet mix?– How to decide number of clusters?– How to find seeding customers?– When stop clustering?– How to assign clusters to depots?

Cluster

• The process of grouping a set of customers into classes of similar/dissimilar customers.

• What criteria should be used for clustering?

• How it can be done?

• Is it good to use clustering approach?

How to cluster suppliers?

Proposed Methodology 1/3

• Cluster according to municipalities.

• Calculate demand for each cluster.

• Calculate number and types of vehicles needed for each cluster.

• Calculate distances between clusters and from/to depots.

• Merge two clusters if possible.

Proposed Methodology 2/3

• Sort clusters according to each depot.

• Find the most closest depot.

• Allocate further away cluster to this depot.– Minimize usage of ferry/toll.

Proposed Methodology 3/3

• Main Methodology – Tabu Search, Iterated Local Search,

Guided Local Search• Try to find appropriate fleet mix using Shrinking

and Expanding Heuristic.

EXAMPLE OF A TOUR

AB