Paper Title (Use Style: Paper Title) s50

Journal of Applied Mathematics and Physics, 2013, *, ** doi:10.4236/jamp.2013.***** Published Online ** 2013 (http://www.scirp.org/journal/jamp) Mathematical Model for the Homogenization of Unit Load Formation

Béla Illés, Gabriella Bognár Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc, Hungary Email: [email protected], [email protected]

Received **** 2013

Abstract One of the most important issues in storage and transport processes is the formation of unit loads. Our main goal is to investigate the homogenization of unit load formation cases. We provide a model involving the major factors and pa- rameters for the optimal selection of the unit load formations. Objective functions and constraints related to the basic tasks are formulated. We give a method for the selection of the optimal unit load formation equipment for given number of products under given constraints.

Keywords: Unit Load, Unit Load Formation Device, Homogeneous Unit Load, Branch and Bound Method

1. Introduction 1. to choose the proper unit load formation equipment to the goods, In the development of transport, storage and distribu- 2. to determine the way of loading goods into the unit tion design within supply chain, great emphasis is put on the planning of low task consuming material flows. The load formation equipment. fast and efficient way of handling and storing compo- nents, raw materials, semi-finished and finished products The following conditions must be satisfied during play a significant role in this process. Therefore, one of the selection of the equipment the most important issues in storage and transport pro- 1. the goods must fit into the unit load formation equip- cesses is the formation of unit loads. It is designed for ment, simplifying the cargo and storage operations as well as 2. the weight of the goods can not exceed the carrying for reducing their frequency. capacity of the unit load formation equipment, The most common areas applicable on any area of 3. the position of the goods must be fixed inside the materials handling are the materials handling inside the equipment, plant, the materials handling between plants, the in-plant 4. the unit load formation equipment must fit into the storage, the outside transportation, the commercial stor- storage areas, loading devices and transporting vehicles age and the distribution systems [1-6, 8]. According to the design of the unit load formation The unit load types can be classified as devices the most important devices are 1. Homogeneous (only one type of goods placed in the - standardized pallets unit load formation equipment) - columnar pallets, 2. Non-homogeneous (or mixed when several kinds of - box pallets, goods placed in the equipment) - platform pallets (wood, metal, wire mesh) - roller pallets, We review the main factors which appear in the homoge- - shock absorbent pallets, nization method of the unit load formation. An optimiza- - disposable pallets, tion process is introduced for the selection of the most - skids, appropriate equipments under different constraints. The - storage baskets, method is exhibited through the solution to a given ex- - tote pans, ample. - containers

The two main tasks during the unit load formation are 2. Homogenization Process

Copyright © 2013 SciRes. JAMP 2 A. N******* ET AL. (an abbreviation of the first author’s name) Definition: The method of choosing optimal number of weight factor of the i-th product for the  -th equip- unit load formation equipment types subject to given ment. A further condition of the model is that for number of goods and maximized number of unit load each product exactly one equipment has to be cho- formation equipments is called the homogenization of r sen; xi   xi  1 . Let the unit load formation equipments. We review four dif-  1 ferent cases depending on the constraints.  m    sgn  xi  In our investigations the following indices will be ap-  1  plied: and   1 if the  -th equipment is chosen. There- • product identifier: i = 1, ..., m  fore the upper bound condition for the unit load for- • equipment model number: ν =1, ..., r mation equipment is r 0     l0 .  1 2.1. No constraint on the type of unit load forma- Summarizing the problem is formulated as: tion equipments xi 0,1 Let us define the loading matrix A by r xi   xi  1 1 ⋯  ⋯ r  1 1   r  m  ⋮   sgn  xi   l0 A    , (3)  1  1  i  ai  r m ⋮   K    ki ai xi  max!    1i1 m   This problem can be solved by Branch and Bound where the number of goods l0 is not greater than the Method or dynamic programming. type number of unit load formation equipments r , i.e., 2.3. Constraint on the number of unit load for- l0  r . mation equipments by type If more type of equipments can be applied for a product In this case, we search for the maximal item of each row then let us denote by M the quantity of i -th product in the loading matrix and we choose the unit load forma- i tion equipment belonging to the column of goods with placed into the equipment. The following conditions has maximal item. to be met (i) each product has to be placed such that no 2.2. Constraint on the unit load formation equip- space is left on either equipment ment types r  ai xi  Mi , Let us suppose that  1 (ii) the number of equipment is the bound: l  r 0 . m  xi  c ,  1 (i) If m  l0 , i.e., the product type number is (iii) the product numbers are integers: not greater than the equipment type num- x  Integer ber, then the optimal unit load formation i equipment type can be given by choosing a maximal per line item. The objective function can be formulated as r m (ii) If m  l0 , i.e., the product type number is K    ki ai xi  max! greater than the equipment type number,  1i1 then If there is no solution, the first strong condition can be r m substituted by a less rigorous one K    ki ai xi  max! ,  1i1 r where xi  1 , if the  -th equipment is chosen for  ai xi  M i .  1 the i-th product, otherwise xi  0 , and ki is the

If only one type of equipment can be applied for a prod- where k denotes the cost of the equipment. The model uct then let us define matrix B is solvable by linear programming. 1 ⋯  ⋯ r

1   ⋮   B=    , 3. The Branch and Bound Method i  bi  ⋮   The selection of the unit load equipment is made by m   product [3].   (i) If the number of product types m is less which gives the equipment demand for the average than the number of the unit load equip- amount of the product. Hence ments p then lmax  m . (ii) If m  p then l  m ,  M  max  i  bi  Entier  0.5 , (iii) If the types of the allowed unit load equip-  zi  ments is l0 , then lmax  l0 : l  l where zi denotes the quantity of goods placed on the a. when 0 , there is no need to ho- equipment. The conditions are the following mogenization, (i) for each product only one equipment can be b. when l  l0 then the asset diversity chosen, must be reduced. The weight factor ap- (ii) xi  1 if the  -th equipment is chosen plied for the selection of the device for the  -th equipment and i-th product is for the i -th product, otherwise xi  0 , r ki . x  x  1 x  0,1 i  i , i  . ui xi  1 In the weighting factor ti  , u x (iii) The equipment number is the bound 0 i m ui y   xi bi  c . denotes the relative frequency of the i-th product,  1 u0 m  The objective function can be given by where u0  ui and ui  ui is the relative frequen- i1  1 r m cy of the  -th quantity. Let us form the matrices K    kibi xi  max!,  1i1 1 ⋯ k ⋯  where the weight factor ki can be chosen as an average 1   ⋮   quantity:   X=  i , ki  M i .  xik  ⋮   The model can be solved by linear programming.   m   2.4. Lower bound on the number of equipment purchases 1 ⋯  ⋯ r 1   ⋮   Only one equipment can be selected for a product,:   KO=  i . (i) xi  1 if the  -th equipment is chosen other-  ki bi  r ⋮     wise xi  0 ; xi   xi  1 , xi 0,1 m  1   (ii) the bound on the number of equipment The objective function is m p m y   xi bi  d KM  k b x  max!  1   i i i  1i1 The objective function is x  1  r m and i if the i-th product is selected for the -th K    k bi xi  min! equipment, otherwise xi  0 , xi 0,1  1i1

Copyright © 2013 SciRes. JAMP 4 A. N******* ET AL. (an abbreviation of the first author’s name) r 1 ⋯ j ⋯ k xi   xi  1 .  1 1   ⋮     For the selection of the initial value of l0 we form the B’  , k  m , i  bij  sums of the matrix column KO ⋮     m n      ki bi ,  1 and k denotes the number of column left. Moreover, let then we form the equipments in descending order ac- p is the number of applicable equipments. If k  p then cording to  . the obtained equipment number is allowed, and the applied number is k . If k  p , then the equipment num- We introduce a column order investigation. Let m  p . bers have to be reduced. Let us form the sum of the col- It is advisable to take more objective functions into ac- n count. We form the efficiency matrix umns: c j  bij . We sort in order the obtained values i1 1 ⋯ j ⋯ m and take the p   column, where c( p )'  cp'   for   1  and  is an appropriate small value; ⋮     c1'  c2'    c( p1)'    c( p )' . On the base of this A=  i ,  aij  p ⋮   order we form a possible column combination from   the p   -th column of A. Let the r -th combination of n   A is 1 ⋯ j ⋯ pp where aij  maxfij  denotes the quantity of the i-th  1   product applicable to the j-th equipment with the  -th ⋮     loading method. The optimal selection is achieved if we A’(r)  , 1  r  w , i  a'(r)ij  find the maximal element in the efficiency matrix ⋮     si  maxaij . If we chose the most efficient equipment n j   to each product then the upper bound for the efficiency is m r  1,, w . We form the maximal value of the rows: s  s 0  i . In order to review the optimal variants we si '(r)  maxaij '(r) i1 j and determine the resultant effi- form the inefficiency matrix m 1 ⋯ j ⋯ m ciency: s0 '(r)   si '(r) . The optimum version of all the i1   1 possible variants corresponds to the greatest efficiency: ⋮     s0 '' maxs0 '(1), s0 '(2),, s0 '(r),, s0 '(w) B  , r . i  bij  ⋮     n 4. Example   Let us select the most appropriate 4 unit load formation with elements bij  si  aij . Let us reduce matrix B. We equipment from 5 different equipments for 5 different products. di  minbij  take the least element in each column j . If n  5, m  5, p  4 d j  0 , then the column can be omitted (as these equip- Then . The efficiency matrix and row ments are not optimal for none of the products). If maximums can be calculated as d j  0 then the j-th column is kept (i.e., it is optimal for 1 2 3 4 5 at least one product) 1 5 9 2 1 4 9 2 4 2 4 3 9 9 A=   3 3 5 8 5 7 8 4 2 6 6 2 5 6 5   7 3 4 1 1 7

Copyright © 2013 SciRes. JAMP A. N******* ET AL. 5 5 2  5 and s0   si  9  9  8  6  7  39 . Let us form the in- i1 3  3 b  s  a efficiency matrix by ij i ij : 4  2-3 1 2 3 4 5 5  1 1 5 9 2 1 4 2 4 2 4 3 9 B=   3 3 5 8 5 7 5. Acknowledgements 4 2 6 6 2 5 5   This research was carried out in the framework of the 7 3 4 1 1 Center of Excellence of Mechatronics and Logistics at the University of Miskolc. 0 0 0 3 0 REFERENCES di  minbij  That gives the values of j : 0 0 0 3 0 , i.e., the third column can be neglected as it is positive. [1] Illés B., G. Bognár, Mathematical modeling of the unit The reduced B matrix is load formation, Applied Mechanics and Materials, 309 (2013), 358-365. 1 2 3 4 5 [2] Illés B., G. Bognár, On the multi-level unit load 1 4 0 7 5 formation model, Key Engineering Materials, 581 (2014), 2 5 7 5 0 519-526. B’=   . [3] B. Illés, G. Bognár, A mathematical model of the unit 3 5 3 0 1 load formation, ASIMMOD Asian Simulation and 4 4 0 0 1 Modeling 2013, Conference Proceedings, January 19-21, 5   0 4 3 6 2013. 3-10. [4] J. Cselényi, B. Illés, Design and Management of Material Flow Systems I. Miskolci Egyetemi Kiadó, Miskolc, As k  p , we found the optimal solution. The optimal (2006) 1-384 (in Hungarian) ISBN 963 661 6728 selection of the equipments for the given problem can be [5] B. Illés, E. Glistau, N.I.C. Machado, Logistik und summarized as follows: Qualitätsmanagement. Budai Nyomda, Miskolc, (2007) 1-195 (in German) ISBN 978 877 3814 [6] B. Illés, E. Glistau, N.I.C. Machado, Logistics and Quality Management. Budai Nyomda, Miskolc., (2007) 1-197 (in Hungarian) ISBN 978 963 877 3807 [7] J. A. Tompkins, J. A. White, Y. Bozer, E. Frazelle, J. Tanchoco, J. Trevino, Facilities Planning, John Wiley & Sons, New York, 1996. [8] D. R. Delgado Sobrino, O. Moravčík, D. Cagáňová, P. Košťál. Hybrid Iterative Local Search Heuristic with a Multiple Criteria Approach for the Vehicle Routing Problem. In: ICMST 2010 : 2010 International Conference on Manufacturing Science and Technology. Malaysia, Kuala Lumpur, 26-28, November, 2010. - : Product Equipment IEEE, 2010. - ISBN 978-1-4244-8758-5. - S. 1-5 1  2