MODIFICATION OF AN ANALYTICAL MODEL FOR CONTAINER LOADING PROBLEMS

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1 MODIFICATIO OF A AALYTICAL MODEL FOR COTAIER LOADIG PROBLEMS Reception date: DEC.99 otification to authors: 04 MAR Cevriye GECER Departent of Industrial Engineering, University of Gazi Maltepe, Ankara, Türkiye Abstract In this paper, the general container- loading proble where a certain nuber of cartons are loaded to a certain nuber of containers is studied. In the proble, diensions of cartons are not restricted to be standard. The objective of the proble is to iniize total unused volue. Chen, Lee and Shen forulated it as a zero-one ixed integer optiization odel for solution of the general container loading probles in But, in the forulation of their odel, solution procedure is cubersoe, not straightforward. By adding new constraints, it is ade easy to follow the solution process. Keywords: 3- diensional container loading; Matheatical odeling; 0-1 Integer optiization. 1. Introduction In the anufacturing and distribution industries loading of cartons into a container is an iportant activity. In this paper, a container is defined as a rectangular large box whereas cartons are defined sall stockable boxes to be packed into a container. All kinds of goods packed in cartons are loaded in a container for transportation and warehouse strorage. The general packing proble is known as a deterination of the optial nuber of containers to pack a given set of rectangular cartons of different diensions. The objective is to iniize the total unused space. 51

2 Container loading is classified as a 3-diensional rectangular packing proble. Several researchers study 3-diensional cargo loading probles. Soe of the are releated to practical applications of cargo loading probles. And soe of the are an attept to enlarge solution of two-diensional cargo loading proble to 3-diensional of that. The first paper on 3-diensional cargo-loading proble was introduced by George and Robinson 6. George and Robinson developed a heuristic ethod for packing sall boxes into a container where after locating a box, the reaining space of container is considered as a container with saller space and finally used space is axiised. Bischoff and Marriot 2 studied the sae subject by cobining 2-diensional cargo loading procedures by Bischoff and Dowsland 1 and George and Robinson's 3-diensional cargo loading work. Based on packing procedures of George and Robinson 6 and Haessler and Talbot 8, Gehring, Menscher and Meyer 5 proposed a heuristic axiising used space in 3-diensional cargo loading. Gilore and others 7, Dowsland 4 studied the sae subject fro several aspects. All the above entioned works gave heuristic solutions of proble and there has been no analytical odel for 3- diensional container loading proble until the work of Chen, Lee and Shen 3. They forulated the proble as 0-1 ixed integer optiization odel. In this work, the odel of Chen, Lee and Shen is exained and it is found out that their odel is cubersoe, not straightforward. A superior odel giving straightforward solution procedure is obtained. 2. Diensional Cargo Loading Proble First, packing proble of certain nuber of cartons into certain nuber of containers is considered. Diensions of containers and cartons need not to be standard. Assuptions of the odel are as below: 1. The diensions and nuber of each container are known, 2. The diensions and nuber of each carton are known, 3. The length of each container is placed at X-axis and the width of each carton is placed at Y-axis, 4. The front-left-botto (FLB) corner of container is fixed at origin, 5. The edges of a carton are either parallel to or perpendicular to the axes. Longest, iddle and shortest diensions of carton or container are respectively called length, width and height. 52

3 The paraeters and the variables used in the odel are defined as follows: : Total nuber of cartons to pack, : Total nuber of containers available, M : An arbitrarily large nuber, p i : The length of carton i, L j : The length of container j, q i : The width of carton i, W j : The width of container j, r i : The height of carton i, H j : The height of container j, S ij : A binary variable which is equal to 1 if carton nuber i is placed in container j; otherwise it is equal to 0, n j : A binary variable which is equal to 1 if the container j is used; otherwise it is equal to 0, (xi, yi, zi) : The coordinates of the front-left-botto corner of carton i respectively, (l xi, l yi, l zi ) : Binary variables indicating whether the length of carton i is parallel to X-, Y-, Z-axis. For exaple, the value of l xi is equal to 1 if the length of carton i is parallel to the X-axis; otherwise it is equal to 0, (w xi, w yi, w zi ) : Binary variables indicating whether the width of carton i is parallel to X-, Y-, Z-axis. For exaple, the value of w xi is equal to 1 if the width of carton i is parallel to the X-axis; otherwise it is equal to 0, (h xi, h yi, h zi ) : Binary variables indicating whether the height of carton i is parallel to X-, Y-, Z-axis. For exaple, the value of h xi is equal to 1 if the height of carton i is parallel to the X-axis; otherwise it is equal to 0. Binary variables a ik, b ik, c ik, d ik, e ik and f ik are defined to indicate the placeent of cartons relative to each other. The a ik is equal to 1 if carton i is on the left side of carton k. Siilarly, the variables b ik, c ik, d ik, e ik and f ik represent whether carton i is on the right of, behind, in front of, below, or above carton k, respectively. These variables are needed and defined only when i < k. Paraeter and variables are illustrated in Figure 1. 53

4 Y Z r i P i H j q i Carton i (x i,y i,z i ) r k qk W j p k Carton k (0,0,0) (x k,y k,z k ) L j Figure 1. Container j Cartons i and k are loaded in container j in Figure 1. Therefore, S Since carton i is located on the left-hand side of and behind carton k, a ij ik S 1. kj d 1. Since length and width of carton k are parallel to Z-axis and X-axis respectively, w h 1. zk xk yk Proposed by Chen, Lee and Shen (1995), 0-1 ixed integer optiization odel is forulated as: Min Z L j W j H j n j p i q i r i j 1 i 1 ik Subject to for all i, k, i k, x i p i xi q i w xi r i h xi x k M( 1 a ik ) (1) x k p k xk q k w xk r k h xk x i M( 1 b ik ) (2) y i q i w yi p i yi r i h yi y M( 1 c ) k ik (3) y k q k w yk p k yk r k h yk y i M( 1 d ik ) (4) z i r i h zi q i w zi p i zi z k M( 1 e ik ) (5) z k r k h zk q k w zk p k zk z i M( 1 f ik ) (6) a ik b ik c ik d ik e ik f ik S ij S kj 1 (7) S j ij 1 1 for all i (8) 54

5 S ij Mn j i 1 for all j (9) for all i, j, x i p i xi q i w xi r i h xi L j M( 1 S ij ) (10) y i q i w yi p i yi r i h yi W j M( 1 S ij ) (11) z i r i h zi q i w zi p i zi H j M( 1 S ij ) (12) xi, yi, zi, w xi, w yi, w zi, h xi, h yi, h zi, a ik, b ik, c ik, d ik, e ik, f ik, S ij, n j 0 1 (13) x i, y i, z i 0 (14) The solution to the odel provides an optial pattern for packing a given set of cartons in selected container(s). The objective of this odel is to iniize the total unused space of the container(s) selected. The constraint (1) - (6) in the odel prevents cartons fro overlapping each other. If several cartons are placed in the sae container, constraint (7) ensure the above- entioned easure. Constraint (8) ensures that all cartons are placed in containers. Usage of a container for packing is provided by constraint (9). Constrains (10) - (12) provide that cartons located in a container subject to capacity. 0-1 variables xi, yi, zi, w xi, w yi, w zi, h xi, h yi, h zi are dependent and obey the following relations: for all i, xi + y i + zi = 1, (1) w xi + w yi + w zi = 1, (2) h xi + h yi + h zi = 1, (3) xi + w xi + h xi = 1, (4) yi + w yi + h yi = 1, (5) zi + w zi + h zi = 1 (6) Related constraints (10) - (12) do not deterine easily which carton i to load to which container j. To reedy this difficulty, indices j ust be added to variables such as x ij, y ij, z ij and a ikj, b ikj, c ikj, d ikj, e ikj, f ikj (0-1 variables). Carton i to be loaded container j can be deterined easily and directly by adding the following constraint to the odel: x ij + y ij + z ij MS ij for all i, j; S ij 1, if carton i is loaded into container j 0, otherwise, then xij yij zij 0. 55

6 The odified odel is as follows Min Z L j W j H j n j p i q i r i j 1 i 1 Subject to for all i, k, i k, x ij p i xi q i w xi r i h xi x kj M( 1 a ikj ) (1) x kj p k xk q k w xk r k h xk x ij M( 1 b ikj ) (2) y ij q i w yi p i yi r i h yi y kj M( 1 c ikj ) (3) y kj q k w yk p k yk r k h yk y ij M( 1 d ikj ) (4) z ij r i h zi q i w zi p i zi z kj M( 1 e ikj ) (5) z kj r k h zk q k w zk p k zk z ij M( 1 f ikj ) (6) a ikj b ikj c ikj d ikj e ikj f ikj S ij S kj 1 (7) S ij 1 j 1 for all i (8) S ij Mn j for all j (9) i 1 for all i, j, x ij p i xi q i w xi r i h xi L j M( 1 S ij ) (10) y ij q i w yi p i yi r i h yi W j M( 1 S ij ) (11) z ij r i h zi q i w zi p i zi H j M( 1 S ij ) (12) xi + yi + zi = 1,,for all i (13) w xi + w yi + w zi = 1,,for all i (14) h xi + h yi + h zi = 1,,for all i (15) xi + w xi + h xi = 1,,for all i (16) yi + w yi + h yi = 1,,for all i (17) zi + w zi + h zi = 1,for all i (18) x ij + y ij + z ij MS ij,for all i, j (19) xi, yi, zi, w xi, w yi, w zi, h xi, h yi, h zi, aikj, bikj, cikj, d ikj, eikj, fikj, Sij, n j 0 1 (20) xij, yij, zij 0 (21) 56

7 3. Coparison of Models with Saple Proble The differences between the original and the odified odels were investigated by saple proble. Let place two cartons with (28x27x13) and (27x26x15) into two containers, which are the diensions of (30x30x30) and (50x30x15) respectively. The atheatical odels established according to the original and the odified odels were analysed by using Lindo progra. When the result of both odels copared, we obtained the sae result (as total unused space z = 9162), but the nuber of iterations are different. The nuber of iterations for original odel was obtained as a 254, although the nuber of restriction, which is ore than the original odel that the odified odel, was obtained as a 121. However, the different probles were investigated for the nuber of iterations; the nuber of iteration obtained for the odified odel is less. The solution of both two odels for the saple proble S 11 = S 21 = 0 S 12 = S 22 = 1are found. It eans that the second container (50x30x15) was used, but first container (30x30x30) was not used. The value of x i, y i, z i which give the coordinates of FLB corner of the cartons and used to deterined the place of i. carton in original odel were obtained as x 1, x 2, y 1, y 2, z 1, z 2 for the saple proble. But unfortunately it is not known that these values belong to which container. Adding the restriction of x ij + y ij + z ij MS ij for all i, j S ij 1, if carton i is loaded into container j 0, otherwise, then xij yij zij 0. to the original odel the solution of this odel is obtained as x 11 = x 21 = y 11 = y 21 = z 11 = z 21 = 0. It shows that the values of FLB coordinates has no value due to the first container was not used. This restriction provides to get the values of FLB coordinates for only used container. By adding the indices j to the a ik, b ik, c ik, d ik, e ik, f ik which are the 0-1 variables in the original odel for the deterination of the place of carton that are reletad each other, a ikj, b ikj, c ikj, d ikj, e ikj, f ikj is obtained. In solution of the original odel a 12, b 12, c 12, d 12, e 12, f 12 have a value but it is unknown that they are belongs to which carton. If this variables are defined as a ikj, b ikj, c ikj, d ikj, e ikj, f ikj, it can be deterined this value belong to which container. In saple proble s solution, a 121 = b 121 = c 121 = d 121 = e 121 = f 121 = 0 are found for the first container was not used. 57

8 4. Conclusion The original odel is odified by adding the indices and the constraint to the Chen, Lee and Shen s odel. As it is entioned in the saple proble, ıncreasing the nuber of constraints the nuber of iteration in the odified odel is less because of the values of FLB coordinates and 0-1 variables which is not calculated for not used containers. References 1. Bischoff, E.E. and Dowsland, W.B., An application of the icrocoputer to product design and distribution, Journal of the Operational Research Society, 33, (1982), Bischoff, E. E. and Marriot, M.D., A coparative evaluation of heuristic for container loading, European Journal of Operational Research 44, (1990), Chen, C.S., Lee, S.M. and Shen, Q.S., An analyticial odel for the container proble, European Journal of Operational Research, 80, (1995), Dowsland, K., Efficient autoated pallet loading, European Journal of Operational Research, 44, (1995), 1990, Gehring, H., Menscher, K. and Meyer, M., A coputer- based heuristic for packing pooled shipent containers, European Journal of Operational Research, 44, (1990), George J.A. and Robinson, D.F., A heuristic for packing, boxes into a container, Coputers and Operations Research 7, (1980) Gilore, J.F. Willias, E., du Fossat, E., Bohlander, R. and Elling, G., An expert syste approach to palletizing unequal- sized containers, SPIE 1095 (Applications of Artificial Entelligence VII),, (1989) Hoessler, R.W., and Talbot, F.B., Loading planning for shipents of low density products, European Journal of Operational Research, 44, (1990),