OPTIMIZATION OF UNIT LOAD FORMATION TAKING INTO ACCOUNT THE MASS

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1 THE ARCHIVES OF TRANSPORT ISSN (print): Volume 32, Iue 4, 24 e-issn (online): DOI:.564/ OPTIMIZATION OF UNIT LOAD FORMATION TAKING INTO ACCOUNT THE MASS OF PACKAGING UNITS Kamil Popiela, Mariuz Waiak 2 Waraw Univerity of Technology, Faculty of Tranport, Diviion of Logitic and Tranport Sytem kamilpopiela@o2.pl, 2 mwa@wt.pw.edu.pl, Abtract: Thi article preent a mathematical formulation of the optimization problem of loading unit formation taking into account the ma of packaging unit. Propoed model can be applied to optimize the arrangement of non-uniform cubical loading unit in loading pace. The model enure poibility of defining variou dimenion, mae, reitance of particular packaging unit and their vertical axi rotation. Within the contraint of formulating optimization problem, taking into account mae and reitance enure that all packaging unit will ret on a pallet or on other packaging unit, and the urface of contact between loading unit guarantee tability of unit arranged in ubequent layer. The mathematical model wa verified. The paper provide an appropriate calculation example. Key word: optimalization, unit load, formation of loading unit. Introduction Forming loading unit i the one of the mot import ant element of logitical ytem. It ha an effect on both the performance and the cot of logitical procee []. An appropriate arrangement of packaging unit on auxiliary loading equipment (e.g. on a pallet) guarantee the bet poible ue of available loading pace. Additionally, the formed unit load hould be arranged o that packaging unit are afe from damage utained under the weight of other unit during tranport. Thi implie that the appropriate arrangement of packaging unit hould enure maximizing the ue of unit load dimenion, taking into account their mechanical reitance. The problem of loading unit formation occur primarily in manufacturing companie and logitic. With repect to manufacturing companie, unit loading formation i an iue which ha a one-time olution for each pecific type of manufactured product. Often, the individual unit of one product type are placed in protective packaging. Thi become the packaging unit. When we combine thee unit with auxiliary loading unit, they are formed into uniform loading unit. Taking into account the reitance and ma obtained by the expert or optimization method, a ingle olution to the packaging unit arrangement problem ha a long term application. In logitic companie, the problem of forming unit load i more complex. Here, unit load are formed from heterogeneou package and in term of reitance and ma. A one-time olution cannot be applied to forming many loading unit (becaue each of them i different) in thi ituation. A a reult, logitic companie are required to obtain real-time olution. Many optimization model of loading unit formation have been decribed in literature[2], [5], [6], [7], [2], [5], [6]. However, thee model have been formulated by taking into account variou implified aumption. Thi limit the application of exiting model to pecific cae of thi problem. Alo, mot of them do not take into conideration the ma and reitance of packaging unit [6], [7], [], [4]. On the other hand, in tak which include the ma and reitance of packaging unit, the contact requirement of bearing urface and the poibility of rotating packaging about the vertical axi unit i overlooked[], [2], [4], [8], [5], [6]. With thi in mind, the mathematical formulation of the optimization problem of forming loading unit wa developed by conidering the ma, reitance and the poibility of rotating the packaging unit about the vertical axi. 73

2 Kamil Popiela, Mariuz Waiak Optimization of unit load formation taking into account the ma of packaging unit 2. Review of Literature In literature, the problem of unit load formation i quite extenively decribed and include formulation of model in one, two and three dimenion. One of the implet optimization model of forming unit load i propoed by T. Tlili, S. Faiz and S. Krichen [5].In thi model, the number of container ued to tore object i minimized. Additionally, in the mathematical formulation of the problem it i aumed that each object i characterized by volume and ma, each container ha a predetermined maximum capacity, each object can be placed only in one container and mut fit in it, container capacity cannot be exceeded by the located cargo, and the container may be tored in one container lot, whoe loading area volume may not be exceeded. Conidering an object location in a container i baed on the criterion of capacity (not dimenion), it would be quetionable if an object could fit in a container under actual condition. Thi model allow only an initial etimate of the number of bin needed to arrange the cargo.applying thi tool in actual condition carrie an inevitable rik of acquiring an inufficient number of container ued to form unit load. A imilar problem wa formulated by M. Mongeau and Ch. Be [8]. In the cae of a two-dimenional loading unit formation problem, an intereting option i a mathematical model decribed by A. Savić, T. Sukilović and V. Filipović [4]. They modeled the formation of a loading unit a a arrangement problem of maller rectangle in one larger rectangle. In thi model, the poibility of rotating the rectangle wa not conidered. In the contraint of the model, it wa provided that mall rectangle are placed parallel to the wall and do not overlap or go beyond the outline of the large rectangle. Lacking the poibility of rotation and rectangle tacking, along with the relatively long optimization problem olving time for jut ix rectangle make thi an inefficient tool for the proce of forming loading unit. A. Lodi, S. Martello and M. Monaci [6] and A. Lodi, S. Martello and D. Vigo [7] propoe imilar mathematical model to the one previouly decribed. D. Piinger and M. Siguard [] alo formulated a related problem. However, their model had diverified type of loading container a well a an aigned cot with repect to each container. Examining the problematic apect of forming three-dimenional loading unit, it hould be mentioned that one of the firt formulation of an optimization problem wa uggeted by S.C. Chen, S.M. Lee and Q.Shen [3]. The author tackled the container loading problem, which i baed on determining the poition of packaging unit in a et number of container. In thi model, container and packaging unit were taken into account, which are characterized by variou ize, where the longet dimenion i the length of packaging unit, the middle dimenion i it width and the hortet dimenion i height. In formulating the optimization problem, the continuou and binary variable are taken into account. The coordinate location of each packaging unit in a given container wa decribed by dicrete variable yielding a real and poitive value. However, the location of individual packaging unit were mapped with the relative poition of packaging unit pair. To ummarize, in the optimization problem decribed in [3] it wa provided that there will be no overlap of packaging unit, a well a the poibility of rotating them on the three axi. Still, the lack of tacking requirement caue the obtained olution to not alway have bearing urface of packaging unit upported. M. Hifi, I. Kacem, S. Negre and L. Wu [5] propoed an analogou model. In thi model, the author developed cuboidal packaging unit which are placed in cubic container which have an identical hape, where coordinate of the lower lefthand corner provide a decription of it location in the pace of the container. A imilar mathematical model of forming loading unit wa preented by A. Ratkiewicz [3]. Each of the analyzed mathematical model of forming loading unit i characterized by certain implification, which are lack of rotating packaging unit, lack of contact requirement between bearing urface, and finally, omitting ma and reitance of packaging unit. Taking thi into conideration, the need for formulating an optimization problem which contain all of thee element wa identified. 74

3 AoT Vol. 32/Iue Propoal of mathematical model of threedimenional loading unit formation 3.. General Aumption of the Model The model propoed in thi article i baed on the model[2], which conider the requirement of contact between bearing urface a well a rotation of packaging unit about the vertical axi. Thi model alo ha additional contraint with regard to packaging unit reitance and loading of auxiliary loading equipment to ait with the formation of loading unit. The goal of formulating a three-dimenional optimization problem of loading unit conit of the following aumption: Packaging unit have a cuboidal hape, Each of the packaging unit i able to be rotated 9 o about a vertical axi (Fig. ), however, all unit mut be in a vertical poition at all time. The ma, reitance, and dimenion of each packaging unit may vary. The ma of each packaging unit i in a geometrical center. Fig.. Graphical illutration of packaging unit arrangement with the poibility of rotation about the vertical axi 3.2. Formulation of Optimization Problem For data including: et of packaging unit number JO = {,, i,, k,, n}, where: n i the amount of packaging unit, but i and k are number of packaging unit, length of packaging unit pi, i JO, width of packaging unit qi, i JO, height of packaging unit ri, i JO, ma of packaging unit mi, i JO, permiible preure on the bearing urface of packaging unit dmi, i JO, length of unit load L, width of unit load W, maximum height of unit H, height of auxiliary loading equipment to ait with the formation of loading unit rp, permiible loading of auxiliary loading equipment to ait with the formation of loading unit B, the value of the following variable hould be aigned: ti {, } for i JO, where the binary variable ti accept the value of when the i-th packaging unit i placed in a loading unit, otherwie ti =, xi R + for i JO i the coordinate of the left wall of the i-th packaging unit on the X axi, yi R + for i JO i the coordinate of the front wall of the i-th packaging unit on the Y axi, zi R + for i JO i the coordinate of the bottom wall of the i-th packaging unit on the Z axi, i {, } for i JO, where the binary variable i accept the value of, when the i- th packaging unit i placed in a loading unit lengthwie along the X axi, otherwie i =, lxi R + for i JO i the dimenion of the i- th packaging unit along the X axi, lyi R + for i JO i the dimenion of the i- th packaging unit along the Y axi, ttik {, } for i JO, where the binary variable ttik accept the value of, when the i-th or k-th packaging unit i placed in a loading unit, otherwie ttik =, a lp ik {,} for i, k JO: i k, where the lp binary variable accept the value of, where the i-th packaging unit i on the left ide of the k-th packaging unit, otherwie lp, a ik {,} for i, k JO: i k, where the binary variable a ik accept the value of, 75

4 Kamil Popiela, Mariuz Waiak Optimization of unit load formation taking into account the ma of packaging unit where i-th packaging unit i in front of the k- th packaging unit, otherwie ik {,} for i, k JO: i k, where the a binary variable accept the value of, where the i-th packaging unit i below the k- th packaging unit, otherwie aik {,} binary variable for i, k JO: i k, where the accept the value of, when the i-th packaging unit i underneath the k-th packaging unit, otherwie a,,, ik a tp k {,} for k JO, where the binary variable tp a k accept the value of, where k- th packaging unit i directly located on the tp palet, otherwie a k, ik {,} for i, k JO: i k, where the a t binary variable t accept the value of, where the i-th packaging unit i placed on the t ik k-th packaging unit, otherwie a, o that all the contraint are met: on the dimenion of packaging unit along the X axi: i JO lxi pi i qi ( i ) () on the dimenion of packaging unit along the Y axi: i JO lyi qi pi lxi (2) in the preence of a pair of packaging unit in a unit load: i, k JO : i k ttik ti tk (6) on not overlapping packaging unit in a loading unit: i, k JO: i k lp lp aik aki aik aki aik aki ttik (7) on the poibility of placing packaging unit relative to each other: i, k JO: i k lp aik aik aik aik 4ttik (8) on the poibility of contact between bearing urface for each pair of packaging unit: i, k JO : i k (9) t aik on the poibility of contact between the bearing urface of packaging unit and auxiliary loading equipment: tp k JO a k tk () on enuring contact between bearing urface for each pair of packaging unit: t i, k JO : i k aik ( zi ri zk ) () on enuring contact between bearing urface of the packaging unit and auxiliary loading equipment: on relative placement of packaging unit along the X axi: lp i, k JO : i k xk ( xi lxi ) aik (3) on relative placement of packaging unit along the Y axi: i, k JO : i k yk ( yi lyi ) aik (4) on relative placement of packaging unit along the Z axi: i, k JO : i k zk ( zi ri ) aik (5) tp k JO ak ( rp zk ) (2) on enuring placement of each packaging unit on top of other packaging unit or directly on the auxiliary loading equipment: tp k JO ak t aik tk (3) i JO: ik on ituating packaging unit above other along the X axi (coefficient of,5 tand for the geometrical center of the packaging unit edge): 76

5 AoT Vol. 32/Iue 4 24 i k : i k xi aik xk, 5lxk, JO (4) i, k JO: i k ( xk,5 lxk ) aik xi lxi (5) on ituating packaging unit above other along the Y axi (coefficient of,5 tand for the geometrical center of the packaging unit edge): i, k JO: i k yi aik yk,5 lyk i, k JO: i k ( yk,5 lyk ) aik yi lyi (6) (7) on ituating packaging unit above other along the Z axi: i k : i k aik ( zi ri ) zk, JO (8) on enuring the packaging unit will not go beyond the dimenion of the auxiliary loading equipment along the X axi: i JO xi lxi L (9) i JO xi (2) on enuring the packaging unit will not go beyond the dimenion of the auxiliary loading equipment along the Y axi: i JO yi lyi W (2) i JO yi (22) on not exceeding the maximum height of the loading unit: i JO zi ri H (23) on enuring the ituation of packaging unit above the urface of auxiliary loading equipment: i JO zi rp (24) on ecuring packaging unit from damage: i JO aik mk dmi (25) k JO: ki on ecuring againt damage from the auxiliary loading equipment: ti mi B ijo (26) and o that the criterion function with the interpretation of filling the loading unit: p i qi ri ti max (27) ijo accept a maximum value Sample Calculation The model of three-dimenional loading unit formation decribed in the previou ection wa implemented in LINGO. Next, the goal wa to ae it uefulne, verify correctne and olve many ample calculation. The reult obtained referred to olution of analytical problem. In the article, one of thee ample wa decribed. The problem of placing 8 packaging unit on a euro pallet with the dimenion of L = 2, W = 8, rp = 44 mm wa olved. The maximum height of the pallet loading unit H wa et at an even 344 mm and the permiible pallet loading capacity B wa et at kg. The parameter of packaging unit are lited in Tab.. The value of variable obtained a a reult of the olution of the decribed problem with the help of implemented optimization formulation of loading unit are how in Tab.2 (Some variable which reult were equal to zero were omitted). The criterion function for the obtained olution ha the following reult: ,746 mm In the examined ample calculation, 4 out of 8 packaging unit were placed on the pallet (Fig. 2) Additionally, the ma of the placed packaging unit did not exceed the permiible preure of the pallet and the packaging unit would not be damaged due to tacking. 77

6 Kamil Popiela, Mariuz Waiak Optimization of unit load formation taking into account the ma of packaging unit On the bai of the obtained olution, it i poible to reflect the viualization of an optimal pallet loading unit. It i proof that for olution obtained with the help of the developed tool, the pallet along with placed packaging unit create an integral whole. Solution to problem are poible thank to a logical phyical interpretation. Tab.. Characteritic of Packaging Unit Number of Packaging Unit (i) Length of Packaging Unit in mm (pi) Width of Packaging Unit in mm (qi) Height of Packaging Unit in mm (ri) Ma of Packaging Unit in kg (mi) Permiible Preure on the Bearing Surface of Packaging Unit in kg (dmi) Tab. 2. Reult of Solving the Optimization Problem of Palet Loading Unit Formation Variable Value Variable Value Variable Value x x 2 x 3 x 4 x 5 t 5 t 6 t 7 t 8 tt 53 a 2 a 3 a 52 lx 9 a 53 x 6 2 lx 2 9 a 23 x 7 lx 3 9 a 5 x 8 y 2 lx 4 5 t a 23 lx 5 9 t a 5 y 2 4 lx tp 6 a 2 y 3 4 lx 7 4 tp a 5 y 4 lx 8 45 a 23 y 5 ly 4 a 5 y 6 ly 2 4 tt 2 78

7 AoT Vol. 32/Iue 4 24 Variable Value Variable Value Variable Value y 7 y 8 ly 3 ly z 644 mm ly 5 4 tt 2 z 2 44 mm ly 6 8 z mm ly 7 4 tt 25 z 4 z 5 z 6 z 7 z mm ly t t 2 45 tt 3 tt 5 tt 23 tt 3 tt 32 tt 35 tt 5 tt 52 t 3 t Concluion The mathematical model of forming loading unit decribed in literature are characterized by certain implification. Thee implification include the inability to rotate packaging unit about a vertical axi, lack of contraint regarding contact between the bearing urface, lack of integrating ma a well a reitance of the packaging load. Conequently, applying thee kind of model to planning actual loading unit doe not alway allow u to obtain an optimal olution which ha the correct phyical interpretation. i=5 i= i=3 i=2 The optimization problem preented in thi article i free from flaw preent in the optimization model in the literature. The obtained olution to the optimization problem of forming loading unit allow for detailed projection of the optimal placement of packaging unit on a pallet or in a container. Mot importantly, reitance and ma of individual packaging unit are taken into account. The quality of the olution and tet performed for other ample calculation allow u to ae the developed formulation concerning the mathematical problem of forming unit load and it numerical implementation a valid. It hould, however, be noticed that the accepted aumption which implify the center of ma of packaging unit in their geometrical center in the cae of large packaging unit placed on top of many maller packaging unit may introduce a certain error. In order to eliminate thi decribed imperfection, the optimization problem hould be modified, accepting the aumption that the ma of each packaging unit i ditributed equally on the entire bae urface of the packaging unit. Fig. 2. Viualization of Sample Calculation Solution 79

8 Kamil Popiela, Mariuz Waiak Optimization of unit load formation taking into account the ma of packaging unit Reference [] Alian W., Baiong Ch., Zhang J., Liangfeng L., Three-dimenional Packing by Tabu Search Algorithm in Military Airlift Loading, 2 International Conference on Optoelectronic and Image Proceing, IEEE, 2. [2] Bomba I., Woźniak G., Kwiecień K.: Wykorzytanie programowania liniowego w projektowaniu paletowej jednotki ładunkowej, Czaopimo Logityka 3/22, [3] Chen C.S., Lee S.M., Shen Q.S., An analytical model for the container loading problem,theory and Methodology,European Journal of Operational Reearc, No. 8,pp.68-76, Elevier, 995. [4] Dahmani N., Krichen S., On olving the biobjective aircraft cargo loading,5th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), IEEE, 23. [5] Hifi M., Kacem I.,Negre S., Wu L.,A Linear Programming Approach for the Three- Dimenional Bin-Packing Problem,Electronic Note in Dicrete Mathematic, No. 36, pp. 993, 2. [6] Lodi A., Martello S., Monaci M., Twodimenional packing problem: A urvey, European Journal of Operational Reearch, No. 4,pp , Elevier Science Ltd., 22 [7] Lodi A., Martello S., Vigo D., Model and Bound for Two-dimenional Level Packing Problem, Journal of Combinatorial Optimization, No.8, pp , Kluwer Academic Publiher, 24 the Netherland. [8] Mongeau M., Be Ch., Optimization of Aircraft Container Loading, IEEE Tranaction on Aeropace and Electronic Sytem, Vol.39, No., IEEE, 23. [9] Parga R.P., Jain R., A Parallel Stochatic Optimization Algorithm for Solving 2D Bin Packing Problem, AIA,IEEE, 993 Orlando Florida United State. [] Piekarka M., Mrozek-Kantak J., Lewandowka J., Rozwiązania najważniejzych problemów wpółczenych łańcuchów dotaw FMCG w Polce, w: Najlee praktyki w logityce, Polki kongre Logityczny Intytut Logityki. Materiały konferencyjne, Poznań 26, [] Piinger D., Sigurd M., The two-dimenional bin packing problem with variable bin ize and cot, Dicrete Optimization, No. 2, pp.54 67, Elevier Science Ltd., 25. [2] Popiela K., Waiak M., Optymalizacja formowania jednotek ładunkowych, Czaopimo Logityka 4/24, [3] Ratkiewicz A., Rozprawa doktorka Optymalizacja proceu komijonowania w utalonej klaie łańcuchów tranportowomagazynowych, Wydział Tranportu Politechniki Warzawkiej, Warzawa 22. [4] Savić A.,Sukilović T., Filipović V., SOLVING THE TWO-DIMENSIONAL PACKING PROBLEM WITH m-m CALCULUS, Yugolav Journal of Operation Reearch 2, No., pp.93-2, 2. [5] Tlili T., Faiz S., Krichen S., A particle warm optimization for olving the one dimenional container loading problem, 5th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), IEEE, 23. [6] Yao Ch., Jia-Zhen H.,Hu L.,Optimal Model, Algorim and Deicion Support Sytem of Bulk Ship Loading Problem, IEEE, 28. 8