three dimensional-pallet loading problem by abdulrhman al-otaibi
DESCRIPTION
Project Objective Minimizing the unused pallet volume subject to many constraints. Developing a three-dimensional pallet-packing algorithm that can be solved using LINDO software.TRANSCRIPT
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THREE DIMENSIONAL-PALLET LOADING PROBLEM
BY
ABDULRHMAN AL-OTAIBI
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Project Importance
• Everyday many items are shipped from one place to another.
• These items are put in containers or pallets.
• To ship more items while spending less energy, time and money, the items should be packed optimally, or at least near optimally.
• This problem becomes even more imprtant when we start to talk about air shipping.
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Project Objective
• Minimizing the unused pallet volume subject to many constraints.
• Developing a three-dimensional pallet-packing algorithm that can be solved using LINDO software.
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Review of Relevant Literature
1. A wall-based algorithm ( Liu and Chen, 1981).
2. A branch-and-bond method (Martello, Pisinger and Vigo ,2000).
3. A mathematical formulation (Ballew,2000) similar to the analytical method of Liu and Chen.
4. A model producing a high degree of stability( Bischoff, Janetz and Ratcliff ,1995)
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Solution Methodology
Chen, Lee, and Shen (1995) presented a zero- one mixed integer linear programming model .
The model considers the issues of : -carton orientations -multiple container sizes,
-multiple carton sizes -space utilization .
-avoidance of carton overlapping,
This model implement Chen, Lee, and Shen model with additional weight restriction constraint.
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The constraints
Maximize Z =
n
k 1
v k P k (10)
The objective function is:
Subject to:1 -no two boxes in the pallet overlap.
2 -each box is contained entirely with the pallet, with its sides parallel to the sides of the pallet.
3 -the proportion of the number of boxes of a given size to the total number of boxes of a full pallet load must
closely approximate the user’s specification.
4 -the total of boxes' weights must be less than the weight allowed to be in the pallet.
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Initial NotationS = a collection of n boxes to be considered = {b1, b2, … , b n )
(l I ,w I ,h i) = the dimensions of box i, bi, in set S. = length, width, and height,
respectively.
(L, W, H) = the dimensions of a pallet cube = length, width, and height, respectively.
(Xo, Yo, Zo) = pallet location in Cartesian coordinate space along the x-, the y-, and the
z-axis, respectively.
(x i , y i , z i ) = decision variables = the x, y, z coordinates of placement location of the
front bottom left corner of box i.
P k = a binary decision variable associated with the k-th box in set S.
Box k is loaded onto the pallet if P k = 1
Box k is discarded from set S if P k = 0
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V = the volume of the pallet = L.W.H.
Vk = the volume of box k = l i . w i . h i.
R g = the desired box proportion of type K.
G k = the wight of box k.
G = total boxes' weight allowed.
Cg = a subset of S; consists of all boxes of size g regardless of box
orientation:
= { b k ( l k . w k . h k) = (l g . w g. h g) or (w g . l g. h g), l k n} ;
M = an extremely large number.
r = total number of box types, r n.
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Preventing Box Overlaps
Figure 1 , two overlapped boxes
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Suppose the location of box A is fixed, and that box B is free to move arbitrarily in Cartesian coordinate space :
To avoid overlap of these two boxes, the following conditions must be satisfied:
x B – x A l A (1)
or
x A – x B l B (2)
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y B – y A w A (3)
or
y A – y B w B (4)
Figure 3. illustrates overlap condition in X and Z
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Figure 4. illustrates overlap condition in Y and Z
z B – z A h A (5)
or
z A – z B h B (6)
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where :
l A ,l B = lengths of boxes A and B, respectively.
w A , w B = widths of boxes A and B.
h A ,h B = heights of boxes A and B.
(x A , y A , z A) = front bottom left comer coordinate of box A.
(x B , y B, z B) = front bottom left comer coordinate of box B.
At least one of these six constraints must hold to prevent overlap of the two boxes.
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Determination of proportion of assigned number of boxes in a pallet:
•The number of boxes of each type to be considered in set S can be determined using the following two equations:
where n g = the number of boxes of type g to be considered in set S.
•By solving Equations (7) and (8), the number of boxes for type g can be obtained as:
n g = (9)
r
iiivR
RgV
1
n g / n i = R g (7)
n i . v i = V (8)
r
i 1
r
i 1
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Formulation of Three-Dimensional Model
The three-dimensional pallet loading problem can now be formulated as a mixed 0-1 integer programming model :
Maximize Z =
n
k 1
v k P k (10)
subject to:
Objective Function:
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1 (Avoid overlap of boxes:
x j – x i – l j ij (11)
or
x i – x j – l i ij (12)
or
y j – y i – w j ij (13)
or
y i – y j – w i ij (14)
or
z j – z i – hj ij (15)
or
z i – z j – h i ij (16)
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2 (Confine placement boundary:
x k Xo P k k (17)
y k Yo P a k (18)
z k Zo P k k (19)
x k (Xo + L) – l k k (20)
y k (Yo + W) – wk k (21)
z k (Zo + H) – h k k (22)
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3 (Weight limitation:
n
k 1
Gk . P k G (23)
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4 (Proportion of boxes assigned:
Pm R g .
(24)
P k {0,1}
X k ,y k ,x k 0
i = 1, 2, … , n – 1
j = i + 1, i +2, … , n
k = 1, 2, … , n
g = 1, 2, … , r
gCm
1m
mP
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Converting multiple-choice constraints
• The multiple choice (either/or) in equations (11-16) must be converted to standard “AND” constraints :
The six possible combinations of different binary values are:
u1 u2 u3
____________________________
1 0 0
0 1 0
0 0 1
1 1 0
0 1 1
1 0 1
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The multiple choice constraints of Equations (11)-(16) in he model are equivalent to:
x j – x i - l j + M(u2 + u3) (25)
x i – x j – l i + M(u1 + u3) (26)
y j – y i – w j + M(2 – (u1 + u2)} (27)
y i – y j – w i + M[2 – (u1 + u2)} (28)
z j – z i – h j + M[2 – (u2 + u3)] (29)
z i – z j – hi + M[2 – (u1 + u3)] (30)
where: 1 u1 + u2 + u3 2
u1, u2, u3, {0, 1}.
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Table 1. Binary Variable and Associated RHS Values
Binary variables RHS values of equations
U1 U2 U3 (25) (26) (27) (28) (29) (30)
Applicable
Constraint
Equation
1
0
0
1
0
1
0
1
0
1
1
0
0
0
1
0
1
1
-l j
M
M
M
2M
M
M
-l j
M
M
M
2M
M
M
-w j
2M
M
M
M
M
2M
-wj
M
M
2M
M
M
M
-hj
M
M
2M
M
M
M
-hj
(25)
(26)
(27)
(28)
(29)
(30)
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b1= 12 x 24 x16 b1= 24 x 12 x16
b3= 24x24x8b4= 24x24x8
A numerical Example
A pallet of 36x 24 Satacking height is 16
(Xo, Yo, Zo) = (100,100,100)Two boxes are required to be shipped:
Two orientations for each box:
24248241216
S = {b1, b2, b3, b4}.
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x 2 – x 1 - 24 + 500(u12 + u13)
x 1 – x 2 – 12 + 500(u11 + u13)
y 2 – y 1 – 12 + 500(u11 + u12)
y 1 – y 2 – 24 + 500[2 – (u11 + u12)]
z 2 – z 1 – 16 + 500[2 – (u12 + u13)]
z 1 – z 2 – 16+ 500[2 – (u11 + u13)]
1 u11 + u12 + u13 2
x 1 100 P1
y 1 100 P1
z 1 100 P1
x 1 (100 + 36) – 12
y 1 (100 + 24) – 24
z 1 (100 + 16) – 16
(P 1+ P 2) (1/3)( P 1+ P 2+ P 3+ P 4)
(P 3+ P 4) (2/3)( P 1+ P 2+ P 3+ P 4)
20 P 1 + 20 P 2 + 40 P 3 + 40 P 4 <= 100
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b1
b3
b4
b2
1224
16
8
8
24 24
12
(P 1, P 2, P 3, P 4) = (1,0,1,1)
(x 1 , y 1 , z 1 ) = (124,100,100)
(x 2, y 2 , z 2 ) = (100,100,84)
(x 3, y 3 , z 3 ) = (100,100,108)
(x 4 ,y 4 , z 4 ) = (100,100,100)
One of the possible solution is:
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• another four problems were randomly selected from OR library and run in LINDO software.
• It was noticed with increasing number of boxes included in the pallet, the execution time is significantly increased.
• In some problems which have a large number of orientations, the execution time exceeds 23 hours and the computer stop running showed out a sign of “out of memory” .
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• A nother six problems were chosen in which 2 problems with two different box sizes.
• A nother two problems with three different box sizes
• Finally two problems with four different box sizes. • All have different orientations forming 12 positions shapes
• Unfortunately, non of them succeed to present a solution except when one of the constraint is removed such eliminating proportion constraints.
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Conclusion• This project has presented an exact mixed 0-1
integer programming model wherever the set• s = {b1, b2..,bn}.
• The drawback of the presented model is only applicable for small problems .
• The computational time requirements of the presented model prevent its use in real-time palletizing applications