(IJACSA) International Journal of Advanced Computer Science and Applications,

Vol. 7, No. 12, 2016

Modeling and Solving the Open-End Bin Packing

Problem

Maiza Mohamed and Tebbal Mohamed and Rabia Billal

Laboratoire de Mathématiques Appliquées
Ecole Militaire Polytechnique
B.P. 17 Bordj-El-Bahri
Alger, 16111 Algérie

Abstract—In the Open-End Bin Packing Problem a set of items the automated machine when the worker leaves from work,
with varying weights must be packed into bins of uniform weight and the worker can unload the finished job next day.
limit such that the capacity of the bin can be exceeded only by the
last packed item, known as the overflow item. The objective is to
minimize the number of used bins. In this paper, we present our Whilst, a few work carried out on this problem, we present
Integer Linear Program model based on a modification of Cesili a modification of the 0-1 linear programming formulation
and Righini model [1]. Also, we propose two greedy heuristics to proposed by Ceselli and Righini [1] in which the authors
solve a problem. The first one is an adaptation of the Minimum studied a variant of the open-end bin packing called Ordered
Bin Slack heuristic where we have reduced to one unit capacity, Open-End Bin Packing Problem (OOEBBP). Then we present
the weight of the largest item in the current bin. While, the
our contributions based on the proposition of two greedy
second heuristic is based on the well-known First Fit Decreasing
heuristic. Computational results based on benchmark instances heuristics for solving the problem in an offline mode while
taken from the literature as well as generated instances show the ensuring high quality solutions in very short computational
effectiveness of the proposed heuristics in both solution quality time compared to solving the problem in optimal way.
and time requirement.
Keywords—Open-End Bin-packing; heuristics; discrete opti- The remainder of this paper is organized as follows. In
mization; combinatorial problem the next Section, we present a few existing lower and upper
bounds on which we based to develop our propositions.
I. I NTRODUCTION Suitable way to formulate the problem with an Integer Linear
Program formulation without considering the order between
The one dimensional open-end bin packing problem items is presented in Section III. In Section IV, we describe
(OEBPP) is a variant of the classical bin packing problem. and develop our proposed heuristics which are tested and
In the OEBPP, items with varying weights are packed into compared by means of computational tests on benchmark
identical bins such that in each bin, the total items weight instances in Section V. The last Section provides some final
content before packing the last item is strictly less than the conclusions and directions for future work.
bin capacity. The aim of the OEBPP is to minimize the
number of bins used to pack all items.
In the next, we assume the following notations:
C: Bin capacity;
This problem was introduced by Leung et al. [2], where I: Set of item;
the authors proved that the OEBPP is NP-hard. Various i;j: Index of items and bins respectively;
application of this problem can be found in a wide variety of wi : Weight or size of item i;
industries such as manufacturing, transportation, affectation rj : Residual capacity of bin j;
tasks, etc. For example, in the fare payment system in the S: Set of candidate items;
Hong Kong and Taipei subway stations, the passengers can O: Set of overflow items;
buy a ticket of a fixed value. A machine at the entrance R: Overall residual capacity of used bins;
gateway records the ticket’s value, while another machine
deducts the fare from the ticket’s value at the exit gateway. II. L ITERATURE R EVIEW
The machine at the exit gateway returns the ticket if the
remaining balances of the ticket is positive; otherwise, it keeps While there exists abundant research for the classical bin
the ticket. The objective from a passenger’s point of view is packing problem (interested reader can reffered to the chapter
to minimize the number of tickets they need to purchase as of Coffman et al. [3] for an excellent survey of this topic),
to minimize the number of bins generated by an algorithm. there is much less research on the OEBPP. Initially, Leung
Another application can be found in job scheduling in the et al. [2] proposed this variant of bin packing, in which the
manufacturing environment that operates one shift per day. A authors modelled the ticketing system at the subway station
worker loads jobs to a machine that can operate automatically. in Hong Kong. The authors was showed that any online
The objective is to schedule jobs to minimize the total time algorithm for the disscused problem have an asymptotic
(days) to complete all the jobs. Intuitively, the job with the worst-case ratio of at least 2. Thereafter, Yang and Leng [4]
longest processing time is scheduled to the last to fully utilize studied the Ordered Open Bin Packing Problem which extends
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 (IJACSA) International Journal of Advanced Computer Science and Applications,
Vol. 7, No. 12, 2016

the requirement corresponding to the order of passenger’s the current item in the first opened bin, else, a new bin is
itinerary in subway station problem. The same problem, opened.
which is the Ordered Open-End Bin Packing Problem, have
been also studied by Cesili and Righini [1] where authors
Other effective algorithm for solving the classical BPP
present branch and price algorithm for its exact optimization.
called Minimum Bin Slack (MBS). This algorithm was
proposed by Gupta and Ho [7]. At each step, an attempt is
made to find a set of items that fits the bin capacity as much
A. Lower bounds as possible. At each stage, a list of items not assigned yet
In this section we briefly describe two existing lower is sorted in the decreasing order of their weights. Each time
bounds from the OEBPP literature. a packing is determined, the items involved are placed in a
bin and removed from a set of unpacked items. Thereafter
Fleszar et al. [8] proposed a variant of MBS called MBS’, in
Ongkunaruk’s lower bound: Ongkunaruk [5] presented which the item of maximum size is not unstuck. MBS has
a simple idea used to compute a lower bound for the been used in several variants of bin packing problem, such as
OEBPP, it consists in determining the number of needed the bin packing problem with conflicts [9], and the variable
overflow items firstly in order to apply the continuous sized bin packing problem [10].
lower bound for the rest of items. The overflow items are
those selected to be packed as the last item which exceed the
capacity of bin. Generally, overflow items have a large weight. Ongkunaruk’s heuristic for the OEBPP: Ongkunaruk
[5] proposed a modification of the well-known first fit
decreasing algorithm for the OEBPP called Modified First Fit
The main target in the OEBPP is to maximize the overflow Decreasing (MFFD), this modification consists of determining
items in order to minimize the capacity of items inside bins, the overflow items firstly, then apply the first fit decreasing
so minimize the number of bins used. for the rest of items not yet packed.

For do, the author order items in the non-increasing size In MFFD, the author computes the lower bound K defined
firstly, then determine the smallest integer K which indicate above (equation (2)) and considered as overflow items, the
the number of overflow item, using the following formula: K−first large items , then apply the first fit decreasing
algorithm for the remaining items i not yet packed, where
n
X i = k + 1 to n. Finally, assign the overflow items into the
wj ≤ K Ċ (1) bins using FFD algorithm.
j=K+1

So the lower bound of Ongkunaruk L0OBP can be expressed III. P ROBLEM F ORMULATION
by:
In this section, we propose a modification of the integer
linear program initially proposed by Cesili and Righini
L0OBP = K (2) [1]. The authors studied a variant of the open bin packing
problem called the Ordered Open-End Bin Packing Problem
Ceselli’s lower bound: Ceselli and Righini [1] proposed a (OOEBPP) in which items to be packed are sorted in a
combinatorial lower bound algorithm called in what follows given order. For this formulation, the authors used the binary
CBA. This lower bound computes a set of overflow items variable yi to indicate whether item i is the overflow item in
iteratively in an optimal fractional packing by the following its bin, and the binary variable xij to indicate whether item i
way; the authors define R to be the overall residual capacity is assigned to the bin in which the overflow item is the item j.
of all the bins already initialized, whenever an item j is found, Because of the constraints on the order of the items, there is
whose size is greater than R, a new bin is initialized and the only xij variables with i < j. Instead, for the general OEBPP
overflow item of this bin is selected as the largest item among (without order), we have therefore xij decision variables with
those already packed but not yet used as overflow items, then i 6= j.
insert item j into the set O of overflow items in order to yield
the maximum residual capacity for the next iteration. So this
lower bound can be expressed by: X
M inimize yi (4)
LCLB = |O| (3) i∈N
s.t.
B. Upper bounds X
yi + xij = 1 ∀i ∈ N (5)
Well known algorithms for the classical BPP: First Fit i6=j
Decreasing(FFD) is one of the well-known algorithm often X
used to solve the classical BPP. Developed by Coffman et wi xij ≤ (C − 1)yj ∀j ∈ N (6)
i6=j
al. [3], FFD guarantees asymptotic worst case performance
bounds of 11/9 [6]. Let all items are sorted in the non- xij ∈ 0, 1 (7)
increasing weight order, the FFD algorithm consist on packing yi ∈ 0, 1 (8)
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Each binary variable yi indicates whether item i is the capacity. Open a new bin if it cannot be
overflow item in its bin. Each binary variable xij indicates assigned to any existing bin;
whether item i is assigned to the bin in which the overflow ◦ if a bin k is opened,
item is item j. Constraints (5) impose that each item must select the overflow item ok such that
be assigned to a bin, while constraints (6) impose that the wok = argmaxi∈Ib{wi } we pack firstly
overall weight of the items assigned to a bin, excluding the the overflow item;
overflow item, must fit into the bin and must leave at least Permute placement of the current item i
one capacity unit available for accommodating the overflow and selected overflow item ok
item. The number of bins used is indicated in the objective Set O = O ∪ ok and update I.b
function (4) by the number of binary variables yi set to 1. ◦ Set I = I \ i
◦ Update the residual capacity of opened bins
(We assume that the residual capacity of bin
IV. P ROPOSED H EURISTICS is the remaining of the capacity where we
consider that the over flow item require only
A. Adapted First Fit Decreasing AFFD one unit)
In this section, we will describe our heuristic algorithm, 3) Algorithm: Algorithm 1 details the AFFD procedure.
called in the following Adapted First Fit Decreasing(AFFD).
1) Main idea: In the open-end bin packing problem the Algorithm 1: Adapted First Fit Decreasing AFFD
capacity of any bin can be exceeded by only the last packed Data: I Set of items ordered in decreasing order of
item, called the overflow item. Then, it is wise to consider the weights, wi weight of item i, C Bins capacity.
weightest items as overflow items in order to minimize the Result: The number of used bins |O|
number of used bins. In this way, Ongkunaruk [5] proposed initialization
MFFD algorithm which is a modification of the FFD runway. O := ∅
The author determines overflow items firstly before packing, S := ∅
then apply first fit decreasing with bins capacity equal to C −1. for i = 1 to n do
S = S ∪ {i}
Our Proposition is an adaptation of the well-known first fit if ( i can be assigned in one of opened bins ) then
decreasing where each selected overflow item can occupy at assign item i with bin capacity C − 1 using
least one unit of bin capacity. Sorted items in decreasing order FFD procedure
of their weight, the AFFD algorithm consist on assignment of else
the current item i into the lowest indexed opened bin which j = argmaxj∈S {S}
accommodate it, else, a new bin is opened. An opened bin can S = S \ {j}
receive an item i if and only if the residual capacity -without O = O ∪ {j}
take into account the overflow item- is sufficient to contain it. Permute assignment between items i and j
In other words, an item i can be placed into an opened P bin
if the weight wi is less than or equal to C − 1 − j∈Ib wj
where Ib is the set of items assigned to the opened bin, except AFFD procedure can be implemented on O(n log n) time,
the overflow item. where n is the number of concerning items.

When a new bin is opened, a selection of an overflow

item j is to be made. The overflow item j is selected as the B. Adapted Minimum Bin Slack AMBS
weighted item among those already packed into all previous We describe in the follow the second proposed heuristic
opened bins, but not yet used as overflow items. Once the which consists in a simple adaptation of MBS heuristic
overflow item is selected, a permutation of the placement of Gupta and Ho [7]. We call this proposition, Adapted
between the selected item j and the current item i is performed. Minimum Bin Slack heuristic (AMBS). In this heuristic,
we execute the MBS procedure, in which we consider that
the weight of the largest item in the current bin is equal to
2) AFFD Pseudo code: Let Ibj a set of items assigned to
one capacity unit. Otherwise, this item is considered as the
bin j, except the overflow item oj . The AFFD heuristic can
overflow item of its bin, and this bin must leave at least one
be summarized in the following steps:
capacity unit available for the requirement of the overflow
• Step 0: Sort the items into a list I in decreasing order item. The AMBS heuristic is summarized in the following:
of their weight, set i = 1, S = ∅, O = ∅.
• Step 1: Until there are no items in a list I do Until there are no items in the unpacked items set I:
◦ S = S ∪ i;
◦ Pack the current item i into the lowest indexed • Select the largest item in I and assign this item as
opened bin j ∗ in which the total weight of an overflow item in the current bin j. The residual
items Ibj ∗ already assigned to it -except, the capacity of bin j becomes rj = rj − 1. Then, remove
over flow item- is strictly less than the bin the selected overflow item from the set I;
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• Apply the MBS-one search procedure on I with a Table headings are as follows:
capacity of bin equal to C − 1, in which we obtain Dev. : Deviation of solution obtained by the corre-
the best subset of unpacked yet items. sponding heuristic(H) from the best lower bound
0
provided by Ongkunaruk [5] LBOBP and the
• Remove the founded subset from I. lower bound provided by Cesili and Righini [1]
The complexity of MBS procedure is O(2n ), but the compu- LBCBA , computed as :
tational experience shows that it is quite efficient in solving 0
H − max(LBOBP , LBCBA )
practical problem instances with various parameters [8][9]. Dev(%) = 100 ∗ 0
max(LBOBP , LBCBA )
This is due to the fact that in practical problem instances, the
number of items involved in each packing is much smaller than sec. : Computational time in seconds.
n. If the maximum number of items that can be placed in one
bin is u, then the complexity of MBS-one-packing procedure TABLE I. C OMPUTATIONAL R ESULTS FOR THE U NIFORM I NSTANCES
is reduced to O(nu ) and thus packings for all bins are built in
O(nu+1 ). Problem MFFD AFFD AMBS
size Dev. sec. Dev. sec. Dev. sec.
u120 2.99 0 2.99 0 3.65 0.0031
V. C OMPUTATIONAL R ESULTS u250 1.41 0.0085 0.94 0.0063 2.50 0.0031
Heuristics and lower bounds have been coded in java and u500 1.33 0.0266 1.02 0.014 2.27 0.0078
run on an Intel i5 @ 2Go of RAM. The solver CPLEX 12.5 u1000 1.42 0.0312 0.95 0.0078 2.10 0.0328
was used to solve the linear programming to the optimality. Average 1.78 0.016 1.47 0.0070 2.63 0.0117

We tested our heuristics algorithm on three data-sets called TABLE II. C OMPUTATIONAL R ESULTS FOR THE T RIPLET I NSTANCES
uniform, triplet and random instances. The two first data-sets
contain eight instances classes were initially proposed by Problem MFFD AFFD AMBS
Falkenauer [11] and they were excessively used to test the size Dev. sec. Dev. sec. Dev. sec.
performances of several bin packing problem variants. Each t60 4.66 0 2.01 0 0 0.0032
class contains 10 different instances. The uniform data-set t120 2.66 0 2.33 0 0 0.0249
includes four instances classes of bins capacities C = 150 t249 1.26 0.0030 1.26 0 0 0.3557
and items with integer weights uniformly distributed in the t501 1.34 0.0157 1.19 0.0030 0 1.6488
interval of [20 − 100]. The number of items n for each class Average 2.48 0.0047 1.69 0.0007 0 0.5081
is respectively, 120,250,500 and 1000. The second data-set
was called the triplet bin-packing instances because in the
classical bin-packing problem, each bin can be filled with at Table I shows computational results for the Falkenauer
most three items. This data-set includes also four instances uniform instances, from these results we note that the average
classes of bin capacities C = 1000 and items with integer deviation decreases when the problem size increases, this is
weights uniformly distributed over the interval [25 − 50] with due to the convergence of the upper bounds and lower bounds
one decimal digit. The number of items n for each class is from the optimal solution. In average, our heuristic AFFD
respectively, 60, 120, 249 and 501. performs better than heuristic provided by Ongkunaruk,
If for the triplet instances in the classical bin-packing problem more particularly when the problem size increase, this is
(closed bin), each bin cannot contains more than three items, due to the way on which MFFD algorithm extract the set
so for the open bin cases we can add no more than one item of overflow items. In fact, the lower bound provided by
as an overflow item. Therefore, a valid lower bound (lower Ongkunaruk performs worse when the problem size increase.
number of used bins) can be expressed by LTOBP = d n4 e Ongkunaruk’s upper bound MFFD gives a further solution.
where n is the problem size or the number of items.
Concerning AMBS, this heuristic gives an important
The last data-set was generated randomly and contains deviation from the optimal solution, this is due to the fact
five classes of instances with different problem size, that from the MBS adaptation, some items with a large
n = 50, 100, 200, 500 and 1000. For each class, items weight weight will be packed inside the bin. While, the optimal
was generated using uniform distribution over four different solution requires that all largest weight items must be packed
intervals [20 − 140], [20 − 160], [40 − 140] and [40 − 160]. as overflow items in order to minimize the number of used bins.
The capacity of bins is fixed to 200. Ten different instances
was generated for a given problem size and weight interval
distribution. In Table II, we present the computational results of MFFD
and proposed heuristic for the Falkenauer triplet instances.
From these results, the overall attracting remarks that our
The performance and average run time of tested heuristics heuristic AMBS give the optimal solution for this instances,
on data-sets described above are shown in Table I, Table II because it is enough to find three items in each subset to
and Table III respectively. Each line contains average results be packed inside the bin, and one as overflow item. So the
over ten OEBPP instances and the best results are shown in solution given by AMBS coincides with the lower bound
bold faced characters. value LTOBP = d n4 e. Moreover, from this table we note that
both proposed heuristics AMBS and AFFD perform better
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 (IJACSA) International Journal of Advanced Computer Science and Applications,
Vol. 7, No. 12, 2016

than MFFD. paper, we have proposed and described two newly heuristics
for the OEBPP problem, these heuristics are an adaptation of
the well-known first fit decreasing algorithm and minimum
For a given problem size, the results given by both AFFD bin slack algorithm.
and MFFD are better on the uniform instances comparing
to the triplet instances. These remarks can be explaining by
the fact that the Falkenauer triplet instances are particularly Computational results based on a benchmark test bed
difficult because in the optimal solution all constructed bins show the good performance of proposed heuristics both on
should have maximally three items as no overflow item plus quality of solution, and on required execution time.
one overflow item, therefore it is difficult to find an optimal
subset of four items to each bin using first fit decreasing.
ACKNOWLEDGEMENT
In overall, all the results present reasonable average com- This work was supported by Algerian state and performed
putation time which confirms the good performances of our jointly at LMA/EMP (Laboratoire de mathématiques
proposed heuristics. The time required by MFFD and AMBS appliquées de l’Ecole Militaire Polytechnique - Alger,
are negligible. Furthermore, it is obvious to note that the time Algérie) and CRIL (Centre de Recherche en Informatique de
requirements increase with the problem size, but reasonable in Lens CNRS - UMR 8188, France). This support is gratefully
order of some millisecond until problem size of 500 items. acknowledged.
Table III shows computational results for the random
instances. In addition to both average deviation and execution
time columns, we show the column Dif f which represent the R EFERENCES
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size and 0.47% to 2.75% for the problems beyond to 200 [4] J. Yang and J. Y. T. Leung, “The ordered open-end bin-packing
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In overall proposed AFFD performs better with an overall State University, 2005.
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average computation time, in order to some milliseconds [10] M. Maiza, A. Labed, and M. S. Radjef, “Efficient algorithms for
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heuristics.

VI. C ONCLUSION
The open end bin-packing problem is an NP-hard
combinatorial problem often encountered in the practical
field. Few works are carried out to solve the problem in
polynomial or pseudo-polynomial time. Through the present
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Vol. 7, No. 12, 2016

TABLE III. C OMPUTATIONAL R ESULTS FOR THE R ANDOM I NSTANCES

MFFD AFFD AMBS

% Dev Diff Sec. % Dev Diff Sec. % Dev Diff Sec.
[20-140] 1.66 0.2 0.0000 0.83 0.1 0.0000 3.21 0.4 0.0000
[20-160] 0.00 0.0 0.0000 0.00 0.0 0.0000 5.07 0.7 0.0000
R-50
[40-140] 1.42 0.2 0.0000 1.42 0.2 0.0000 2.14 0.3 0.0000
[40-160] 1.91 0.3 0.0000 2.58 0.4 0.0000 3.83 0.6 0.0000
Avg. 1.24 0.2 0.0000 1.20 0.2 0.0000 3.56 0.4 0.0000
[20-140] 1.16 0.3 0.0016 1.16 0.3 0.0000 5.84 1.5 0.0000
[20-160] 0.35 0.1 0.0000 0.35 0.1 0.0000 5.09 1.4 0.0000
R-100
[40-140] 2.50 0.7 0.0000 2.15 0.6 0.0015 5.36 1.5 0.0000
[40-160] 2.07 0.6 0.0016 2.40 0.7 0.0000 5.79 1.7 0.0000
Avg. 1.52 0.4 0.0008 1.51 0.4 0.0003 5.52 1.5 0.0000
[20-140] 0.79 0.4 0.0000 1.18 0.6 0.0000 6.94 3.5 0.0000
[20-160] 0.75 0.4 0.0000 0.94 0.5 0.0000 7.72 4.1 0.0000
R-200
[40-140] 2.34 1.3 0.0016 1.44 0.8 0.0000 6.29 3.5 0.0015
[40-160] 1.71 1.0 0.0000 1.71 1.0 0.0016 7.00 4.1 0.0016
Avg. 1.39 0.8 0.0004 1.31 0.7 0.0004 6.98 3.8 0.0007
[20-140] 0.47 0.6 0.0046 0.72 0.9 0.0031 8.14 10.1 0.0031
[20-160] 0.82 1.1 0.0000 0.89 1.2 0.0000 7.85 10.5 0.0016
R-500
[40-140] 2.68 3.7 0.0061 1.74 2.4 0.0032 6.97 9.6 0.0093
[40-160] 1.56 2.3 0.0031 1.97 2.9 0.0047 7.55 11.1 0.0032
Avg. 1.38 1.9 0.0034 1.33 1.9 0.0027 7.62 10.3 0.0043
[20-140] 0.73 1.8 0.0265 0.77 1.9 0.0110 8.65 21.3 0.2511
[20-160] 0.53 1.4 0.0173 0.68 1.8 0.0094 8.61 22.6 0.0764
R-1000
[40-140] 2.75 7.6 0.0407 1.48 4.1 0.0111 7.35 20.3 0.2683
[40-160] 1.22 3.6 0.0264 1.63 4.8 0.0092 7.39 21.7 0.0014
Avg. 1.30 3.6 0.0277 1.14 3.1 0.0101 8.00 21.5 0.1493
Total Avg. 1.36 1.4 0.0064 1.29 1.3 0.0027 6.33 7.5 0.0308

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