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of a considerable importance regarding the number of
items selected at it: it monopolizesthe biggest propor-
tion of loads. However, the LBLS is higher using U
2
:
about 25% of items are loaded before the last stage.
We notice that both curves have similar shapes but the
one of U
1
is lowest: this is due to the penalty of de-
lay. Indeed,U
1
penalizes delayed items more severely
than do U
2
.
To conclude, we can say that our algorithm has
proved to be efficient in solving the OKP. Compared
with the results provided in the static case, we reached
almost the same overall profits. Besides to respecting
the capacity constraint, we were able to fill at maxi-
mum the knapsack while making decision in oppor-
tune time. As to the utility functions, we noted that
the utility function U
2
proved to be more convenient
in terms of FLS and LBLS: it gives more interest in
newcomers if compared with U
1
, which prefers to de-
lay as much as possible and makes decision in latest
stages. However, the utility function does not con-
tribute in the overall reward: we reached all the time
the same values using either functions of utility.
The weak point in our algorithm is its high com-
plexity. As it can be seen in Table 3, the CPU time
is very high and increases exponentially as the size of
the problem increases. We think that this should be
given more attention in future works.
6 CONCLUSIONS
In this paper, we proposed a dynamic approach for
the OKP with delay that incorporates a stopping rule
at each stage of the loading process to enable the DM
to select his best items in an online manner. This ap-
proach was adopted to reduce the OKP to a series
of static knapsack subproblems. Using the optimal
stopping terminology, we stated our decision strategy
based on a dynamic formulation. Experimental re-
sults showed that we were able to reach optimal so-
lution using our online approach. Besides, the use of
two different utility functions allowed us to come up
to the desired solution while involving two different
attitudes to risk.
Future works may include improvements of the
present algorithm in order to reduce the CPU time. A
possible generalization of the present work is to study
the OKP with delay while considering the possibility
of losing a number of items during the selection pro-
cess. The other aspect that we would like to explore
in the future is the OKP with multiple DM.
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