We solve an optimization problem involving the assignment of transport routes to a delivery point. The general and particular cases of the optimization problem are described. An algorithm for the simplified case and ideas for the general case is presented

Ali, M.M.

Engelbrecht, M.

Ronkainen, T.

English

Mathematics in Industry Information Service Eprints Archive

OPTIMIZATION OF DELIVERIES FROM
DISTRIBUTION POINTS
We solve an optimization problem involving the assignment of trans-
port routes to a delivery point. The general and particular cases of the
optimization problem are described. An algorithm for the simplified
case and ideas for the general case is presented. The problem arose in
CaterPlus, a major food catering industry in South Africa.
Let us assume that we have a large geographical area filled by customers.
Each customer has a unique demand for a product we are delivering. We
want to organize the delivery process by splitting the customer base into
smaller regional units served by distribution points. Each distribution point
has some capacity of transportation to deliver products to the customers.
The problem is to find the location of these distribution points and organize
our transportation capacity to deliver products fulfilling the demand of every
customer. The total transportation capacity is determined by the fixed
number of trucks, where each truck can operate near one of the distribution
points and can carry products up to its capacity.
Our goal is to minimize the cost of the delivery. For simplicity, we model
the customers as points in ]R2 and use the total distance traveled by trucks
•School of Computational and Applied Mathematics, University of the Wit-
watersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. e-mail:
tom mi. ronkainen@gmail.com
tSchool of Computational and Applied Mathematics, University of the Witwatersrand,
Private Bag 3, Wits 2050, Johannesburg, South Africa. e-mail: mali@mail.cam.wits.ac.za
tCaterPlus, The BIDVest Group Ltd., Johannesburg, South Africa e-mail: mar-
tin. engelbrecht@caterplus.co.za
the number of delivery points,
the number of customers,
the number of trucks available,
location of delivery points P( i) E JR2,
location of customers C(i) E JR2,
demand of customers D(i) E JR,D(i) > 0,
capacity of trucks T(i) E JR,T(i) > O.
P(i),
C(i),
D(i),
T(i),
m
i = 1, ,q
i = 1, ,n
i = 1, ,n
i= 1, ,m
Let us first look at the simplified case with only one fixed distribution
point, i.e. assume that q = 1 and P(l) is given. All trucks are operating
from this distribution point. To model the route of a truck k we use an
ordered subset Rk = (rf, ... ,r~) C {l, ... ,n}, which defines the order of
the customers visited by the truck. The total cost of such a tour from the
delivery point p E JR2 is then
8-1
c(p, Rk) = d (p, C(r~)) +Ld (C(rf), C(rf+1)) + d (C(r;),p) ,
i=l
where d(x, y) is the cost of transportation between two points x and y.
Then the problem is to find an ordered m-partitioning R = {R1, ... , Rm}
of the integer set {I, ... ,n} such that RinRj = 0, when i -=I j, and U~lRi =
{I, ... ,n}, minimizing the function
m
f(1<-) =L (P(l), Rk)
k=l
IRkl
L D(rf) ::; T(k).
i=l
This partitioning corresponds to distributing customers to truck routes.
The general version of this problem involves the positioning of the dis-
tribution points and designating trucks and customers to each distribution
point. Assume that we can solve the reduced case above with an algorithm
:F mapping a distribution point p and subset S C {I, ... ,n} of customers
using the subset of trucks V = {VI, ... , Vt} C {I, ... ,m} to the optimal
routes, i.e.
F(p, 5,V) = arg minn V(p, V,R) : R is an ordered t-partitioning of 5} ,
where! is a generalization of the function f in (1), using only customers
given in the index set 5 and trucks given in the index set V. The minimiza-
tion over t-partitions of 5 is formally given by
t
A "kf(p, V,R) = L- (p, R )
k=l
IRkl:LD(Rf) :S T(Vk) \:jk = 1, ... , t.
i=l
Then the problem of positioning q delivery points is to locate the best
coordinates for delivery points and distribute trucks and customers to each
delivery point. Thus, we are looking for q points in ]R2 and two mappings
S : {I, ... , q} f-7 p({I, ... ,n}) and T : {I, ... ,q} f-7 p({I, ... ,m}) select-
ing the associated customers S(i) and trucks T(i) to each delivery point i
respectively. The general problem is then to minimize
q
g(P(l), ... ,P(q), 5,T) = :L j (P( i), T(i), F(P(i), S(i), T(i)) )
i=l
over possible locations P(l), ... , P(q) E R2 and over possible customer and
truck mappings 5 and T.
We will mainly concentrate on solving the simplified problem (1) and only
sketch a few ideas for solving the general problem in the last section. We
assume from now on a single fixed distribution point.
Solving this problem can be viewed as a combination of two separate
processes. We need to cluster customers into groups, whose demand can be
served by the capacity of one truck. We ignore now the situation where one
truck can perform two trips faster than another truck can perform one trip.
This is an unlikely possibility in the optimal case, if the capacities of trucks
are approximately the same. We also assume that we have enough trucks
to cover all demand at any time, otherwise condition (2) cannot be fulfilled.
The second part is to solve for the optimal cost for each single truck route.
Because we are using the distance metric as a cost, this reduces the problem
to the ordinary Euclidean TSP-problem.
Considering that the TSP-problem is NP-hard itself, it is clear that for
practical use we want to find an approximate algorithm. Instead of solving a
TSP-problem instance repeatedly in the clustering phase, we use simplified
metrics to determine the fitness of the clustering. Note also that distribut-
ing customers to trucks is a bin-packing problem, which is also known to be
NP-hard [2].
The approximate algorithm we implemented is as follows:
1. Choose the number of iterations N.
2. Denote trucks by Ti, i = 1, ... ,m, where each Ti is a set of customer indexes
served by the truck i. Denote the center of mass of the truck Ti by
1
rn(Ti) = ITiI L C(c).
cET;
q(Ti) =L (m(Ti),C(c)).
cET;
3. Sort customer indexes into decreasing order c'(l), ... , c'(n) by their demand.
4. For i := 1, ... ,m:
5. Set Ti -- {c'(i)}.
6. For i = m + 1, ... , n:
7. Find the j minimizing (m(1j), C(c'(i))) constrained by
D(c'(i)) +L D(c) :s; T(j).
cE7j
8. Set 1j --1j U {c'(i)}.
9. Repeat N times:
10. Select randomly i and j so that 1:S; i,j:S; m and i =/:-j.
11. Select randomly Cl E Ti.
12. If D(Cl) + I:cE7j D(c) S T(j):
13. If q(Ti \ {Cl}) + q(1j U {Cl}) < q(Ti) + q(1j):
14. Set Ti .- Ti \ {cd and 1j .-1j U {cd. Goto 10
15. Select randomly Cz E 1j.
16. If D(Cl) - D(cz) + I:cE7j D(c) :s; T(j) and
D(cz) - D(cI) + I:cET; D(c) :s; T(i):
17. If q((Ti \ {Cl}) U {cz}) + q((1j \ {cz}) U{cd) < q(Ti) + q(1j):
18. Set Ii ~ (Ii \ {Cl}) U {C2} and 7j ~ (7j \ {C2}) U {ct}.
19. For each Ii:
20. Construct a TSP-problem 9 using points P(l) and C(c), c E Ii.
21. Find the minimum spanning tree of 9 using Kruskal algorithm.
22. Duplicate all edges of g.
23. Find some Euler-path of 9 and sort the customers in Ii in the path order
by skipping duplicate nodes on the Euler path.
24. Find all intersections in the path and remove them by reordering customers.
Lines 3-18 form the clustering part of the algorithm. We start by sorting
the customers by their demand and fill m trucks with the m customers
having the biggest demand. Then we assign the rest of the customers to
the nearest truck with enough capacity. After this initialization step, we
use random sampling to improve clusters either by moving or swapping
customers. The clustering tries to minimize total distance of the customers
from the mass center of the cluster. This gives a good starting point for the
TSP-approximation [1J on lines 20-23 and some fine tuning by straightening
path intersections.
We also tested a modified K-means [3J algorithm, but it appeared that
this simple randomized algorithm gives better results.
The algorithm was implemented in C++ using GNU C++ compiler v4.0.2
and tested on an iBook G4 1333MHz Power PC using Ubuntu GNU /Linux.
We tested the algorithm using uniformly distributed data with randomly
placed circular holes. In Figure ?? there are 250 customers and 5 trucks.
The capacity of the trucks was randomly selected between 25 and 35 and the
demand for each customer was a random number between a and 1. We used
10000 x n = 2 500000 iterations. This is usually enough for clusters to sta-
bilize. In this example case, there were no changes after 2 000 x n = 500 000
iterations. When the algorithm finished, the capacities used in the trucks
were 99.6%, 99.5%, 88.9%, 68.4% and 50.1%. Figure?? shows how cus-
tomers were distributed to trucks and Figure 2 shows the routes discovered
by the TSP-approximation.
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Customers n Trucks m Average Time Min. - Max.
50 5 2.4s 1.9s - 2.4s
100 5 7.1s 7.0s - 7.1s
250 5 28.6s 22.1s - 34.4s
500 5 125.3s 123.3s - 126.6s
1000 5 482.6s 474.3s - 487.1s
50 10 LIs 0.9s - 1.3s
100 10 4.6s 3.6s - 5.4s
250 10 22.7s 22.3s - 23.0s
500 10 76.0s 74.6s - 78.1s
1000 10 95.6s 92.2s - 99.9s
We also made some performance testing. Table 1 summarizes the run-
ning times for uniformly distributed data using 10000 x n iterations. Note
that the algorithm is faster when there are more trucks. That is because
clusters are smaller and thus TSP-problem instances are simpler to solve.
We have demonstrated that it is possible to obtain some solutions to the
complex problem using relatively simple ideas. The solution presented here
is probably adequate for some real life problems. It also forms a basis for
further development. As we can see, the general approach of the algorithm
is independent of the quality and the cost functions, q and c. This leaves
opportunities to improve heuristics used in the clustering as well as having
more realistic modelling of the cost of transportation.
The generalized version of the problem was not in the scope of this
paper. However, this algorithm might be a useful tool for that as well. The
placement of the distribution points can be solved by the clustering part
of the algorithm. That can be done by combining mjq trucks to one truck
having the capacity of the sum of the capacities of the original trucks. Then
we run the clustering part of the algorithm, for this modified problem and
use the centers of mass m(1i) as delivery points P(i). The clusters also
determine the customer to delivery point mapping S.
[1] Kruskal, J.B. On the shortest spanning subtree of a graph and the trav-
eling salesman problem. Proceedings of the American Mathematical So-
ciety, (1956) pp 48-50.
[2] Coffman, E.G., Garey, M.R. and Johnson, D.S. Approximation algo-
rithms for bin packing: a survey. In Approximation Algorithms, editor
D. Hochbaum, PWS Pubishing (1996), pp 46-93.
[3] McQueen, J.B. Some Methods for Classification and Analysis of Multi-
variate Observations. Proceedings of 5-th Berkeley Symposium on Math-
ematical Statistics and Probability, Berkeley, University of California
Press, 1, (1967) pp 281-297.


Optimization of deliveries from distribution points

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