Automated Air Load Planning

Automated Air Load Planning
Kurt R. Heidelberg,1,
* Gregory S. Parnell,2
James E. Ames IV1
1
Virginia Commonwealth University, Department on Mathematical Sciences, P.O. Box
842014, Richmond, Virginia 23284-2014, USA
2
Toffler Associates, 302 Harbor’s Point, 40 Beach Street,
Manchester, Massachusetts 01944, USA
Received September 1996; revised April 1998; accepted 31 May 1998
Abstract: We describe the development of a heuristic algorithm for determining efficient
2-dimensional packings in cargo aircraft where cargo placement constraints are critically
important in determining the feasibility of packing locations. We review the performance of a
new algorithm versus some traditional ones for aircraft loading. The algorithm is also tested in
a more generalized setting where there exist no additional constraints on items, to suggest
applicability in other environments. The new algorithm has been used worldwide in the
Automated Air Load Planning System (AALPS) for cargo aircraft loading, with much success.
© 1998 John Wiley & Sons, Inc.* Naval Research Logistics 45: 751–768, 1998
1. INTRODUCTION
Creating tentative air load plans for military division sized deployments is a time-consuming
process. For example, preparing load plans for the 82nd Airborne division could take more than
300 man-hours. In 1978, computer assisted load planning began with the prototype version 1 of
the Army’s Automated Air Load Planning System (AALPS). AALPS was developed as a gross
load planning tool, with an item database containing specifications on cargo and an aircraft
database for the specifications on the cargo aircraft. An Automatic Load Planner (ALP) module
provides the user with an expert system which will generate acceptable load plans and aids the
user to determine the placement of cargo among aircraft, and provides an estimate of aircraft
quantities in a given airlift scenario. The user can use the Type Load Editor (TLE), a graphical
editor that lets the user manipulate loads generated in the ALP or to develop new loads from
scratch.
AALPS is data-driven in nature. Aircraft in the database are: C-130, C-141, C-5, C-17,
KC-10, KC-135, and most other cargo aircraft used by the armed forces to include Civil Reserve
Air Fleet (CRAF) aircraft such as the Boeing 7X7 series and DC-10, MD-11, etc. For its cargo
database, AALPS uses the Military Transportation Management Command Transportation
Engineering Agency (MTMC-TEA) TB 55-46-1 and TB 55-46-2 Equipment Characteristics
* Present address: The University of California, Department of Anthropology, 1334 Watkins Hall,
Riverside, California 92521.
Correspondence to: G.S. Parnell
© 1998 John Wiley & Sons, Inc. *This article is a US government work and, as such, is in the public
domain in the United States of America. CCC 0894-069X/98/080751-18

File (ECF), which contains approximately 15,000 end-item configurations. Attributes of these
cargo items can be manipulated by AALPS users to accommodate specific configuration
requirements. AALPS is currently used at over 300 sites throughout the Continental United
States (CONUS) Army community, and has been used effectively in recent operations such as
JUST CAUSE, DESERT SHIELD, DESERT STORM, and RESTORE DEMOCRACY.
AALPS has been used by Lockheed Corporation to aid in the design of cargo aircraft, and is
being used by the U.S. Air Force to test and configure the C-17.
The greatest difficulty in the development of AALPS is the bin packing problem associated
with the ALP module. Bin packing is the process of placing a set of items into a set of bins such
that wasted spaced is minimized. Like the Traveling Salesman Problem and the Minimal Graph
Coloring Problem, the Bin Packing Problem is NP-complete, which implies that, in a worst-case
situation, no means of finding an optimal solution is significantly faster than exhaustive search
of all possible permutations. Often, it is intractable to obtain an optimal packing, since any large
quantity of cargo would quickly become unmanageable and impractical for most time and
computational bounds. A heuristic approach must be taken to achieve an acceptable result,
therefore trade-offs of time, optimization, resources, complexity, and additional packing con-
straints must come into play. The problem becomes more difficult when applied to cargo
conveyance systems, where additional packing constraints often prohibit any piece from being
packed in its ideal location. These include unital constraints, those that violate the integrity of
the load by themselves. Examples of unital constraints are contact and axle weights, station
loading requirements, shoring requirements, pounds per square inch, pounds per linear foot, and
height (since cargo is not stacked inside of aircraft, height is not dealt with as a third dimension).
To further complicate the matter, collective constraints, those that involve all cargo as a group,
need to be satisfied. These include aircraft compartment limitations, longitudinal and latitudinal
balancing, axle configuration, Allowable Cabin Load (ACL), and Hazardous Material (HAZMAT)
incompatibility.
This paper considers heuristic approaches to the 2-dimensional problem of packing cargo on
standard cargo conveyance systems, particularly aircraft. There is a wealth of literature which
discusses bin-packing, including comparative studies [3, 7], which can give an understanding of
the relative behavior of varying heuristics in controlled environments. The 2-dimensional bin
packing problem has been attacked with many approaches, with some algorithms geared
specifically toward cargo conveyance [2]. This paper focuses on the possibility of items being
rejected due to additional cargo-conveyance constraints, and how existing heuristics, as well as
a new heuristic, will perform as a result of this rejection.
The particular problem involves an off-line packing environment. Since all items are known
to the heuristic from the beginning, a packing plan can be developed off-line, before any packing
actually begins. This is generally beneficial over on-line environments, where an item is packed
into a bin immediately after it becomes visible.
2. OFF-LINE ALGORITHMS
Off-line packing algorithms generally produce results more efficient than on-line algorithms,
since they consider attributes of all the items before the packing is started [3]. There are a great
number of algorithms that fall into the Level variety: those that partition the bin into sections of
some height where each section is packed with items side-by-side. However, there are also
algorithms that deviate from this approach. Some of these algorithms are more computationally
expensive than Level algorithms, as they require more partitioning of items before packing can
begin.
752 Naval Research Logistics, Vol. 45 (1998)

2.1. Level Algorithms
Consider the on-line algorithms in Figure 1. The results of many simple on-line algorithms
can be considerably improved upon, if the packing environment allows items to be sorted with
respect to their dimensions. Typically, these sorts are based on descending height. For example,
consider the BFL algorithm [5]. When items are being considered for packing in this algorithm,
a level created to accommodate a very long item i may ultimately be occupied by other very
short items, resulting in considerable wasted space. If the items were sorted by height however,
items subsequent to i will have a first chance to occupy its same level, minimizing the amount
of waste. Figure 2 shows the benefit of a sort by height on the BFL algorithm by comparing it
with Best Fit Level Decreasing (BFLD). The items to the right of each bin show the order in
which they were packed. Notice that BFLD wastes much less space than BFL for this case. A
variant of Level algorithms called Shelf algorithms are similar to their Level counterparts in
every way except the Level heights are predetermined and limited to a finite set. These are
practical in environments where items are partitioned within the packing area or containerized
prior to packing. Sorting items usually makes off-line Shelf algorithms more effective than their
Figure 1. Level algorithms.
Figure 2. Comparison of BFL and BFLD algorithms.
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Heidelberg, Parnell, and Ames: Automated Air Load Planning

on-line counterparts, since there now exists some data on which to base the selection of shelf
sizes [9].
2.2. Nontraditional Packing Algorithms
There exists a collection of other bin packing algorithms that may be practical in some
environments with uniform item size distribution. Some notable examples are the CLS and
KLM algorithms which are discussed in detail, particularly in the context of cargo conveyance
systems, in other sources [8]. Comparison of these algorithms versus Level algorithms is
described next.
2.2.1. Karp, Luby, and Marchetti–Spaccamela Algorithm (KLM)
Figure 3 summarizes the algorithm. Figure 4 is an example of a KLM packing. KLM can yield
good results when item dimensions are distributed evenly [2]. If the distribution is not even,
however, the efficiency can degrade considerably. This is obvious in a case where all items to
be packed are such that they are categorized in the same group. For example, consider an
environment where all items are in Group 1. There would be no pairwise matchings, resulting
in a large amount of wasted space. KLM is also a more complex algorithm, and subsequently
requires more processing time than traditional Level algorithms.
Figure 3. Karp, Luby, and Marchetti–Spaccamela algorithm.

2.2.2. CLS Algorithm
The CLS algorithm disposes of the requirement of having levels or shelves to partition the bin
into smaller pieces (see Figure 5). This major difference from the previously discussed
algorithms introduces some important considerations. Figure 6 is an example of a CLS packing.
Refer to Figure 4 for item dimensions.
Like the KLM algorithm, CLS does well when items are randomly distributed [3]. The CLS
algorithm is straightforward and tends to require less processing time than KLM when item sizes
are randomly distributed.
3. APPLYING BIN PACKING ALGORITHMS TO CARGO CONVEYANCE
SYSTEMS
Much effort has been spent in applying bin packing algorithms to cargo conveyance systems [4],
particularly aircraft [1], ships [5], packing crates [2], railcars, and trucks [6]. Loading cargo
conveyance systems can initially be taken as a classical application of bin packing algorithms.
Indeed, some commercial carriers who containerize all of their cargo prior to loading can apply any
unidimensional packing routine conducive to their loading environment. The containers, all equal in
width and categorized into a few groups based on length, can generally be packed in an off-line
fashion to yield acceptable results. However, in some environments, containerizing cargo is not
Figure 4. KLM packing.
755

practical. For example, when construction equipment, trucks, and tracked vehicles are moved by
aircraft, tractor-trailer, or railcar, they often are simply driven onto the carrying vehicle.
Special concerns and constraints apply to these environments, particularly with aircraft,
which make the problem more difficult than in the classical bin packing environment. For
example, no assumptions of any even distribution of sizes can be made, since cargo lists can
vary from many of a single type of cargo, to a large deployment with thousands of different
sized items.
In a standard air load planning environment, the task of bin packing is approached as a
2-dimensional problem using length and width of the cargo items and of the aircraft’s cargo
hold. Height is not as significant, since cargo is generally not stacked inside of the aircraft
(smaller items are typically loaded into containers or palletized, and loaded onto the aircraft as
a single, aggregate item). Height is, however, a constraint that may prohibit the loading of a
cargo item in particular areas of an aircraft or onto a particular aircraft entirely.
3.1. Constraints on Packing Cargo Conveyance Systems
Cargo plane loading introduces significant constraints that require changes in more common
bin packing approaches. Constraints include longitudinal and lateral balancing requirements,
maximum weights for axles and contact points resting on the cargo floor, maximum weights
within regions on the cargo floor, hazardous material compatibility, and others. Following is a
Figure 5. CLS algorithm.

discussion of some of the more difficult constraints that pose particular problems to the previous
packing algorithms.
3.1.1. Longitudinal Balancing Constraints
Under a realistic sorting scheme for a Level algorithm, such as BFLD, or BFLDWt (Best Fit
Level Decreasing Weight), heavier cargo items would tend to be loaded toward the aft and left
Figure 7. C-5 Airframe loaded using BFLD.
Figure 6. CLS packing.
757

ends of the aircraft. This frequently would shift the Center of Balance (CB) to an intolerable
extent, causing a longitudinal balancing problem (see Figure 7).
To overcome the longitudinal balancing requirement, load planners employ a system called
“Pyramid Loading [10].” Here, the cargo is sorted based on factors including weight, priority,
and bulkiness, and is loaded on the aircraft from the desired Center of Balance (CB) toward the
fore and aft of the cargo hold. Basically, cargo is loaded alternating fore and aft until the aircraft
floor space is full or the cargo exceeds its maximum cargo weight. Generally, the load will be
close to balanced throughout this process, and any error can usually be corrected by sliding the
cargo (see Figure 8).
3.1.2. Lateral Balancing Constraints
Often, the most significant constraint is the requirement for lateral load balancing. Through-
out different compartments in the aircraft there are maximum allowable deviations for the center
of balance from the center (X 5 0), based on the weight of the load. Because of this constraint,
a simple level packing algorithm may fail, since packing from left to right can yield an out of
balance load. An example would be a loaded tractor-trailer combination on a C-5 Galaxy
airframe. A C-5 aircraft has lateral balancing compartments of 40 in. in length and a width the
same as the packing area. A tractor-trailer with an axle weighing 34,000 pounds and two
motorcycles each with an axle weighing 1000 pounds are packed, left to right, so that all three
mentioned axles are in the same compartment. A level algorithm may choose to pack this item
to the leftmost edge of the level. Subsequently, only lighter cargo items such as motorcycles may
be loaded on the level to the right of the tractor-trailer, or the remainder of the level could be
left empty. In either case the center of balance would deviate too far left of the preferred center
of balance, resulting in an unacceptable load. The solution is, when heavy items are selected, to
consider packing it centered on the X-axis, anticipating that there will exist room for other items
Figure 9. Lateral balancing on wide bodied aircraft.
Figure 8. C-5 Airframe loaded using Pyramid Method.

to be packed to the left and right. Generally, this approach is saved as a last resort after a left
to right packing fails. Referring to Figure 9, centering the tractor-trailer axle and the motorcycle
axles as a left to right level algorithm would do (a). The maximum tolerable deviation of a center
of balance from the X-axis would be exceeded, so that the aircraft would be out of balance. A
load planner would “centerline load” the tractor-trailer and pack the motorcycles on either side
(b), resulting in an aircraft with perfect lateral balance.
3.1.3. Axle Aggregation
All aircraft have limitations on the amount of weight a single axle can apply to the cargo hold
floor. Some also have special considerations when axles are loaded within a certain proximity
of each other. For example, on a C-5, if two axles are resting within 40 in. of each other (in some
places), their weights are to be aggregated so that their combined weight is treated as a single
application of weight to the floor [11]. Consequently, an item that, alone, causes no violations
may cause a violation when packed next to certain other items. Again, a simple level algorithm
can be insufficient. Two like items with heavy axles, placed side by side can create an aggregate
axle violation in this fashion (Figure 10).
Level algorithms provide no means of resolving this problem, other than creating another
level for the second cargo item, resulting in wasted space (a). When this happens, load planners
often overcome the violation by shifting one of the items forward until the axles are more than
40 in. apart (b).
4. COMPARISON OF OFF-LINE 2-DIMENSIONAL ALGORITHMS
A fair comparison of these algorithms is unrealistic. Though some clearly yield better results
than others, not all algorithms are conducive to some practical environments. For example,
consider an on-line environment involving material being loaded on a flatbed truck. The on-line
environment eliminates the possibility of all off-line algorithms. Since the cargo holding area is
accessible from the sides as well as the front, BFL would be a reasonable choice, since it would
yield better results than NFL or FFL. If the conveyance was a commercial van, however, BFL
and FFL could not be used, since without side access the only level accessible during the
packing process would be LMAX. In off-line environments, as processor speeds and overall
computing capabilities continue to improve, time goals will become easier to achieve.
4.1. Level/Shelf Algorithms
When the packing environment allows for off-line planning, BFLD tends to yield the best
results among traditional Level algorithms [7]. All 2-dimensional level algorithm packings can
Figure 10. Axle aggregation.
759

be observed as groupings of unidimensional packings where each level is one of the packings
within a group, and the items are packed into the level from the side using a Next, First, or Best
Fit algorithm based on width. This approach mandates that the left edge of all packed items be
adjacent to the left edge of the bin or only one other item, and that the base of each item be
adjacent to the left edge of the bin or only one other item, and that the base of each item be
adjacent to the lower barrier defining its respective level. These requirements result in an
inability to further consider the collective length components of items so that they can be stacked
within a level.
In a BFL packing of the items in Figure 11, the best result would be attained through a
preceding sort by decreasing height (BFLD), resulting in a packing as shown in (a). However,
if items could be stacked within the levels, a closer to optimal solution could be found (b).
Furthermore, if the levels could be “bent” where appropriate, an even more efficient packing
could be achieved (c).
4.2. CLS Algorithm
Item placement is not constrained by levels in this algorithm, and under certain distributions
of item size, CLS can yield extraordinarily good results. Its greatest drawbacks are that through
any cross section of the bin, there can be only two pieces of cargo, and if item widths are all or
mostly less that
1
2
bin width, there is substantial waste, with as few as only one item through any
cross section of the bin.
4.3. KLM Algorithm
In the event that items are not evenly distributed, KLM can be grossly inefficient. In the worst
of cases, the algorithm may have only one item per level and use less than 25% of the packing
area [8]. KLM shares the same level-bound constraint on item placement that the Level
algorithms have, and cannot contain more than four items per level.
Figure 11. Weakness of Level algorithm.

5. AN ALTERNATIVE APPROACH
Due to assorted item weight densities, special loading requirements, and prioritization, it is
futile to attempt to load cargo on conveyance systems using a sort based on a single attribute
such as length. Level/Shelf algorithms, even when using more realistic sorting methods, fail to
contend with rejected item placement. For balancing purposes alone, the CLS algorithm is
generally unacceptable. The KLM algorithm could easily create lateral balancing problems,
waste an intolerable amount of loadable area in many situations, and cannot accommodate items
longer than the bin width.
As discussed earlier, Level algorithms have some implied requirements or side-effects that,
if overcome, could produce better loads for cargo conveyance systems:
1. All items must be left attached to a bin wall or to one and only one other item.
2. All barriers defining levels must be straight lines dividing the bin.
3. All items must be attached to the base barrier of its level.
4. All items must be packed in the order they appear in the item list after it is sorted.
The following algorithm dismisses these requirements by providing two major modifications
to the Level/Shelf type algorithm. First, it uses a barrier that is “bendable” such that it can bend
at 90° angles at up to four places to define a (up to) five-segment barrier. Second, after selecting
an area to pack in the bin, it iterates through the item list trying multiple items until success is
met or all items have failed. The Appendix gives a general description of the algorithm. Further
details are provided by Heidelberg [8].
5.1. The Packing Barrier
In addition to the five Cartesian coordinates associated with each barrier, each barrier has a
type. The type can be determined from the barrier coordinates and may be one of any nine
values. The type is used in the algorithm to help decide which portion of the bin is to be packed
next. Figure 9 shows the nine barrier types.
The barriers are named with respect to the relative position of their “levels” or X values
(Figures 12 and 13). For example, in the L_C_R barrier, the left level is lowest, center is next,
and the right level is highest. The levels are listed in order of their magnitude to form the name.
Y values do not affect the naming scheme, and in the less complex barriers (C and L_R, for
example), the unused levels are simply left out.
5.2. Candidate Bin Sections
The first objective is to determine which section of the bin should be packed. This decision
is based on the barrier type, barrier coordinates, and sometimes bin dimensions. Like some
Figure 12. Barrier components.
761

Level algorithms, it is desirable to load the next item as close to the base of the bin as possible.
For example, assume the barrier type is C_L_R. Here the center level is the lowest point of the
barrier, so that it is on that level where we wish to pack the next item (see Figure 14).
The upper bound of the desired area to pack is dictated by the uppermost adjacent level to the
level being packed, in this case the right level. The edges of the bin or surrounding level edges
determine the width of the bin section.
5.3. Candidate Item Selection
Once a candidate bin section is selected, the item list is checked, in order, for the best item
to be packed in the section. “Best” is, of course, not clearly defined and may vary among
contexts, but generally it implies that the item fits dimensionally and causes no constraint
violations.
A subgoal when selecting an item for a candidate bin section is to return the barrier to as
normalized a state as possible. This is important, since most of the time three-leveled barriers
cannot accommodate large items. Barriers of type L_R, R_L, and especially C are most capable
of providing a candidate bin section capable of holding a relatively large item. This subgoal is
Figure 13. Barrier types.
Figure 14. Candidate bin section.

the reason that the right level of Figure 15(a) dictated the upper bound of the candidate bin
section. If the ceiling of the bin had been used (b), longer items could be packed in the bin
section, which is the fundamental goal in conveyance load planning, but the resulting barrier
would likely not be conducive to packing the largest subsequent items. This could result in
nonuse of the bin section and subsequently waste of its space. A compromise is to allow a
maximum tolerable extension t to the candidate bin section (c). This would allow the packing
of slightly oversized items, while at the same time helping to keep the barrier close to
normalization. Studies to date have not effectively resolved a means of selecting the best value
for t, though it is currently bound by minimum item length and aircraft loading constraints with
some success. Ignoring additional constraints, an extension of t 5 0 tends to yield results at least
as good as BFLD. Increasing the extension in small quantities can increase performance to a
degree [8].
Based on the current barrier type and dimensions, and the dimensions of the item to be
packed, the best position within the bin segment for the item is decided. This part of the
algorithm is fairly simple: Generally the item is packed on top of the lowest level on the barrier
and next to the tallest barrier edge to which the level is adjacent. In environments where
additional constraint checks must be made on the packing decisions, this operation would be
followed by the constraint checking [8].
5.4. Adjusting Barrier After Packing
Once an item has been packed, the appropriate barrier needs to be adjusted to reflect the
change in available space. This needs to be done to minimize waste of space.
Figure 15. Extension of candidate bin section.
Figure 16. Adjusting barrier for packed item.
763

Referring to Figure 16, in (a), a candidate item has been packed and it is necessary to adjust
the barrier. This can be done in one of two ways. The right level can be raised to rest on top of
the new item (b) or the right edge can be moved to the right of the new item (c). Using (b) would
result in less waste of area than (c). In some environments it may be convenient to consider the
applicability of the resulting barriers to the remaining items before committing to a new barrier.
After the barrier levels and edges are adjusted to accommodate the new item, it is sometimes
necessary to change the barrier type. In Figure 17, an L_R_C barrier is packed with a new item
(a). After the barrier levels and edges are adjusted, the barrier type changes (b).
5.5. Adjusting Failed Barrier
In the event that a bin section was selected for packing and no item would fit, the next step
is to adjust the barrier so that a larger bin section may be created with minimal wasted space.
Using the failed barrier type, the adjustment to a more general type is made so that the larger
bin section can subsequently be considered (Figure 18). The adjustment is made by raising the
lowermost level to the height of the second lowermost level. If all items fail at a barrier type C,
then no more items can be packed in the bin.
6. ANALYSIS OF THE ALGORITHM
Two types of analysis were performed on the algorithm. First, we evaluate the performance
versus other bin-packing algorithms on test data developed for this comparison. Second, we
summarize the operational experience with the heuristic.
Figure 17. Adjusting barrier type for packed item.
Figure 18. Adjusting failed barrier.

6.1. Test Cases
Several test cases were used to compare the behavior of the new algorithm versus BFLD and
CLS. KLM was omitted from the test since it cannot accommodate items with length greater
than the width of the bin. This analysis only compares the fundamental packing capability of the
algorithms and does not consider their applicability to cargo conveyance systems by considering
cargo constraints.
The four columns in each of Tables 1 and 2 reflect (1) a t under which the new algorithm has
minimum waste, followed by the percentages of waste for (2) the new algorithm, (3) BFLD, and
(4) CLS. For the test, bin size remained constant, but the quantity of items for each set fluctuated
at random. For each set of items, first minimum and maximum allowable lengths and widths
were determined: minimum length ranged from 20 to 55, maximum lengths ranged from 80 to
165, minimum widths ranged from 8 to 25, and maximum widths ranged between 60 and 95.
Bin width is 100. These ranges are not completely arbitrary. The goal in generating item sizes
was to create sets of items with dimensions that are comparable to relative dimensions of actual
cargo items loaded on actual aircraft. Table 1 shows the results of random item size of uniform
distribution within the dimensional bounds, while Table 2 shows the results of packing smaller
sets of item sizes which are duplicated between 5 and 10 times at random. This second test was
valuable since in actual air loading environments, duplicate items are often loaded. Also, BFLD
tends to yield better results when same sized items are being packed, and this approach provided
a stronger test of the new algorithm’s ability to exceed the performance of BFLD.
For all cases in Table 1, the new algorithm outperformed CLS and BFLD. The table makes
clear the variance that can occur in t across different data sets. Only in one case in Table 2 did
CLS outperform the new algorithm, and BFLD matched its performance twice.
6.2. Operational Performance
Compared to the other algorithms in this discussion, the greatest drawback of the proposed
algorithm is the computational expense involved. However, for cargo aircraft loading, this
Table 1. Waste for random item size.
t NEW BFLD CLS
67 13 16 31
77 12 14 20
4 13 17 37
12 12 16 24
9 9 17 24
0 12 16 32
0 13 18 40
70 18 23 18
44 8 13 23
58 17 21 21
46 11 17 38
73 13 17 24
63 11 17 20
0 9 14 25
54 10 14 25
58 8 12 19
24 12 17 35
1 10 15 22
0 6 12 16
5 11 14 26
765

expense in negligible, especially since the planning is done off-line and most normal situations
obtain acceptable results. On an Intel 486-based computer with 32 MB of memory to virtually
eliminate disk I/O during the packing process, 10 aircraft would normally be packed in about
one second. The algorithm, under the most complex scenario which involves approximately five
different sized aircraft and a division sized load list, can generate 700 acceptable loads in under
2 min, a feat that could take two expert loadmasters up to 2 months to complete.
The algorithm allows for easy shifting of cargo fore and aft during the loading process. This
aids in balancing the load longitudinally and resolves problems related to axle weight aggre-
gation. The algorithm, however, is not very conducive to lateral shifting of cargo items during
the loading process, since an item’s ability to move sideways is often directly dependent on
more than one other cargo item to which it is adjacent. Lateral shifting is used to resolve lateral
balancing problems and also height constraints, since the curvature of the fuselage on some
cargo aircraft tapers inward on the cargo hold.
The algorithm often matches the ability of an expert loadmaster to make the most efficient
loads. The algorithm is extremely effective in gross load planning, and makes recommendations
for load plans that can later be easily modified by an experienced load planner.
7. AREAS FOR FURTHER RESEARCH
The greatest weakness of this algorithm is its inability to shift cargo laterally to accommodate
constraints. Further research could greatly increase the performance of the algorithm should this
be overcome.
Finite barrier dimensions restrict any attempt to use underlying space after barrier adjustment.
Attempts to overcome this restriction could be approached through recursively packing wasted
spaces as separate bins, or through allowing a variable number of levels within a barrier, rather
Table 2. Waste for duplicated random item
size.
t NEW BFLD CLS
5 11 14 26
0 8 13 36
0 14 24 48
0 31 34 31
80 12 15 17
16 10 16 30
72 15 20 16
0 9 14 35
74 11 22 11
0 9 20 40
0 11 16 36
0 12 12 19
76 14 18 26
44 13 13 39
9 8 17 45
0 38 38 18
76 16 17 27
28 19 23 52
0 20 23 20
13 15 18 36

than only three. As exhibited in Figure 6, there are often two options for barrier adjustment.
Consideration of items remaining to be packed may affect the efficiency of one adjustment over
the other, and further study in this area may be worthwhile.
Figure 19.
767

The value of t often determines the comparative quality of the packing [8], and further
research into the assignment of that parameter is warranted.
8. CONCLUSION
Bin packing for cargo conveyance systems deals with numerous constraints far beyond
physical dimensions. As a result of their complexity, it is difficult to develop a concise
mathematical formulation to provide optimal solutions. Taking a heuristic approach can provide
reasonable results in a short amount of time. This goal is important in real-world scenarios
where load plans are required in an almost real-time fashion. Algorithms, such as the one
presented in this paper, successfully apply the heuristic approach, making computer planning of
cargo loads realistic. An implementation of the new heuristic has been used and reviewed by
professional loadmasters around the world since January 1994, and has generally been accepted
as quite satisfactory for gross load planning by the U.S. Air Force and Army. The performance
of the heuristic often matches the capability of expert loadmasters, usually requiring no
adjustment for better packing efficiency [8].
APPENDIX: GENERAL DESCRIPTION OF ALGORITHM
Figure 19 provides a general description of the algorithm. Anchor Point: point at which top
left corner of item is to be placed in the bin. Current Bin Full: no more items will fit in the bin.
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[5] Gehring, H., et al., “A Computer-Based Heuristic for Packing Pooled Shipment Containers,”
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Virginia Commonwealth University, 1995.
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[10] Sharpe, R., Loadmaster, Chief Master Seargent, USAF retired, personal communication, 1992.
[11] T.O. 1C-5A-9, Loading Instructions Manual, USAF Series C-5A and C-5B Airplanes, 23 January
1992, Secretary of the Air Force, Washington, DC.

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