﻿ Mathematical Model for the Homogenization of Unit Load Formation Journal of Applied Mathematics and Physics, 2014, 2, 14-20 Published Online January 2014 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2014.21003 OPEN ACCESS 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: a ltilles@uni-miskolc.hu, v.bognar.gabriella@uni-miskolc.hu Received September 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 parameters 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 equip- ment for a given number of products under given constraints. KEYWORDS Unit Load; Unit Load Formation Device; Homogeneous Unit Load; Branch and Bound Method 1. Introduction In the development of transport, storage and distribution design within supply chain, great emphasis is put on the planning of low task consuming material flows. The fast and efficient way of handling and storing components, raw materials, semi-finished and finished products plays a significant role in this process. Therefore, one of the most important issues in storage and transport processes is the formation of unit loads. It is designed for simpli- fying the cargo and storage operations as well as for reducing their frequency. The most common areas applicable on any area of materials handling are the materials handling inside the plant, the materials handling between plants, the in-plant storage, the outside transportation, the commercial storage and the distribution systems [1-6,8]. According to the design of the unit load formation devices the most important devices are • standardized pallets, • columnar pallets, • box pallets, • platform pallets (wood, metal, wire mesh), • roller pallets, • shock absorbent pallets, • disposable pallets, • skids, • storage baskets, • tote pans, • containers The two main tasks during the unit load formation are  1. to choose the proper unit load formation equipment to the goods, 2. to determine the way of loading goods into the unit load formation equipment The following conditions must be satisfied during the selection of the equipment 1. the goods must fit into the unit load formation equipment, 2. the weight of the goods cannot exceed the carrying capacity of the unit load formation equipment, 3. the position of the goods must be fixed inside the equipment, 4. the unit load formation equipment must fit into the storage areas, loading devices and transporting vehicles B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 15 The unit load types can be classified as 1) Homogeneous (only one type of goods placed in the unit load formation equipment) 2) Non-homogeneous (or mixed when several kinds of goods placed in the equipment) We review the main factors which appear in the homogenization method of the unit load formation. An opti-mization process is introduced for the selection of the most appropriate equipments under different constraints. The method is exhibited through the solution to a given example. 2. Homogenization Process Definition: The method of choosing optimal number of unit load formation equipment types subject to given number of goods and maximized number of unit load formation equipments is called the homogenization of the unit load formation equipments. We review four different cases depending on the constraints. In our investigations the following indices will be applied: • product identifier: i = 1, ..., m • equipment model number: ν = 1, ..., r 2.1. No Constraint on the Type of Unit Load Formation Equipments Let us define the loading matrix A by A 11iraimνν=, (3 ) where the number of goods 0l is not greater than the type number of unit load formation equipments r, i.e., 0lr≥. In this case, we search for the maximal item of each row in the loading matrix and we choose the unit load formation equipment belonging to the column of goods with maximal item. 2.2. Constraint on the Unit Load Formation Equipment Types Let us suppose that 0lr<. 1) If 0ml≤, i.e., the product type number is not greater than the equipment type number, then the optimal unit load formation equipment type can be given by choosing a maximal per line item. 2) If 0ml>, i.e., the product type number is greater than the equipment type number, then 11rmiiiiKk a xmax!νννν= == =∑∑, where 1ixν=, if the ν-th equipment is chosen for the i-th product, otherwise 0ixν=, and ikν is the weight factor of the i-th product for the ν-th equipment. A further condition of the model is that for each product ex- actly one equipment has to be chosen; 11riixxνν== =∑. Let 1misgn xνννβ==∑ and 1νβ= if the ν-th equipment is chosen. Therefore the upper bound condition for the unit load formation equipment is 001rlννββ== ≤∑. B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 16 Summarizing the problem is formulated as: { }01ix,∈ 11riixxνν== =∑ 011rmisgn xlννν= =≤∑∑ 11rmiiiiKk a xmax!νννν= == =∑∑ This problem can be solved by Branch and Bound Method or dynamic programming. 2.3. Constraint on the Number of Unit Load Formation Equipments by Type If more type of equipments can be applied for a product then let us denote by iM the quantity of i-th product placed into the equipment. The following conditions has to be met 1) each product has to be placed such that no space is left on either equipment 1rii iax Mννν==∑, 2) the number of equipment is the bound: 1mixcννν=≤∑, 3) the product numbers are integers: ix Integerν= The objective function can be formulated as 11rmiiiiKk a xmax!νννν= == =∑∑ If there is no solution, the first strong condition can be substituted by a less rigorous one 1rii iax Mννν=≥∑. If only one type of equipment can be applied for a product then let us define matrix B B 11irbimνν=, which gives the equipment demand for the average amount of the product. Hence 05iiiMb Entier.zνν= +, where izν denotes the quantity of goods placed on the equipment. The conditions are the following 1) for each product only one equipment can be chosen, 2) 1ixν= if the ν-th equipment is chosen for the i-th product, otherwise 0ixν=, 11riixxνν== =∑, { }01ix,∈. B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 17 3) The equipment number is the bound 1miiyxb cννν νν== ≤∑. The objective function can be given by 11rmii iiKk bxmax!ννν= == =∑∑, where the weight factor ik can be chosen as an average quantity: iikM=. The model can be solved by linear programming. 2.4. Lower Bound on the Number of Equipment Purchases Only one equipment can be selected for a product: 1) 1ixν= if the ν-th equipment is chosen otherwise 0ixν=; 11riixxνν== =∑, { }01ix,∈ 2) the bound on the number of equipment 1miiyxb dννν νν== ≥∑ The objective function is 11rmiiiKkbxmin!νννν= == =∑∑ where kν denotes the cost of the equipment. The model is solvable by linear programming. 3. The Branch and Bound Method The selection of the unit load equipment is made by product . 1) If the number of product types m is less than the number of the unit load equipments p then maxlm=. 2) If mp> then maxlm=, 3) If the types of the allowed unit load equipments is 0l, then 0maxll=: a) when 0ll≤, there is no need to homogenization, b) when 0ll> then the asset diversity must be reduced. The weight factor applied for the selection of the device for the ν-th equipment and i-th product is ikν. In the weighting factor0iiiiuxtuxνν=, 0iuu denotes the relative frequency of the i-th product, where 01miiuu==∑ and 1iiuuνδδ==∑ is the relative fre- quency of the ν-th quantity. Let us form the matrices X 11ikkximν=, KO 11iirkbimννν=. B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 18 The objective function is 11pmii iiKMk bxmax!ννν= == =∑∑ and 1ixν= if the i-th product is selected for the ν-th equipment, otherwise 0ixν=, { }0, 1ix∈ 11riixxνν== =∑. For the selection of the initial value of 0l we form the sums of the matrix column KO 1miikbν νννβ==∑, then we form the equipments in descending order according to νβ. We introduce a column order investigation. Let mp>. It is advisable to take more objective functions into account. We form the efficiency matrix A 11ijjmain=, where { }ij ijamax fµµ= denotes the quantity of the i-th product applicable to the j-th equipment with the µ-th loading method. The optimal selection is achieved if we find the maximal element in the efficiency matrix{ }i ijjsmax a=. If we chose the most efficient equipment to each product then the upper bound for the efficiency is 01miiss==∑. In order to review the optimal variants we form the inefficiency matrix B 11ijjmbin=, with elements iji ijb sa= −. Let us reduce matrix B. We take the least element in each column { }i ijjdmin b=. If0jd>, then the column can be omitted (as these equipments are not optimal for none of the products). If 0jd= then the j-th column is kept (i.e., it is optimal for at least one product) B’ 11ijjkbin=, km≤, and k denotes the number of column left. Moreover, let p is the number of applicable equipments. If kp≤ then the obtained equipment number is allowed, and the applied number is k. If kp<, then the equipment numbers have to be reduced. Let us form the sum of the columns: 1nj ijicb==∑. We sort in order the obtained B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 19 values and take the pδ+ column, where ( )p'p'ccδε+−< for δ and ε is an appropriate small value; ( )()12 1'' p'p 'cc ccδ++≤ ≤≤≤≤. On the base of this order we form a possible p column combination from the pδ+-th column of A. Let the r-th combination of A is A’(r) 11ijj ppa'(r )in=, 1rw≤≤, 1, ,rw=. We form the maximal value of the rows: ( )( ){ }i ijjs'r maxa'r= and determine the resultant effi-ciency: ( )( )01miis'rs'r==∑. The optimum version of all the possible variants corresponds to the greatest effi-ciency: ()() () (){ }000001,2,,,,rs''maxs's's' rs' w=. 4. Example Let us select the most appropriate 4 unit load formation equipment from 5 different equipments for 5 different products. Then 5,5, 4nmp== =. The efficiency matrix and row maximums can be calculated as A = 123451 5921492 4243993 3585784 2662565 734117 and 5019986739iiss== =++++=∑. Let us form the inefficiency matrix by iji ijb sa= −: B = 123451 592142 424393 358574 266255 73411 00030 That gives the values of { }i ijjdmin b=: [ ]00030, i.e., the third column can be neglected as it is posi- tive. The reduced B matrix is B’ = 123451 40752 57503 53014 40015 0436. B. ILLÉS, G. BOGNÁR OPEN ACCESS JAMP 20 As kp=, we found the optimal solution. The optimal selection of the equipments for the given problem can be summarized as follows: Product Equipment 1 → 2 2 → 5 3 → 3 4 → 2-3 5 → 1 Acknowledgements The research work presented in this paper is based on the results achieved within the TÁMOP-4.2.1.B- 10/2/KONV-2010-0001 project and carried out as part of the TÁMOP-4.1.1.C-12/1/KON V -2012-0002 “Coop-eration between higher education, research institutes and automotive industry” project in the framework of the New Széchenyi Plan. The realization of this project is supported by the Hungarian Government, by the Euro-pean Union, and co-financed by the European Social Fund. REFERENCES  B. Illés and G. Bognár, “Mathematical Modeling of the Unit Load Formation,” Applied Mechanics and Materials, Vol. 309, 2013, pp. 358-365. http://dx.doi.org/10.4028/www.scientific.net/AMM.309.358  B. Illés and G. Bognár, “On the Multi-Level Unit Load Formation Model,” Key Engineering Materials, Vol. 581, 2014, pp. 519-526. http://dx.doi.org/10.4028/www.scientific.net/KEM.581.519  B. Illés and G. Bognár, “A Mathematical Model of the Unit Load Formation,” ASIMMOD Asian Simulation and Modeling Conference Proceedings, 19-21 January 2013, pp. 3-10.  J. Cselényi and B. Illés, “Design and Management of Material Flow Systems I,” Miskolci Egyetemi Kiadó, Miskolc, 2006, pp. 1-384.  B. Illés, E. Glistau and N. I. C. Machado, “Logistik und Qualitätsmanagement,” Budai Nyomda, Miskolc, 2007, pp. 1-195.  B. Illés, E. Glistau and N. I. C. Machado, “Logistics and Quality Management,” Budai Nyomda, Miskolc, 2007, pp. 1-197.  J. A. Tompkins, J. A. White, Y. Bozer, E. Frazelle, J. Tanchoco and J. Trevino, “Facilities Planning,” John Wiley & Sons, New York, 1996.  D. R. Delgado Sobrino, O. Moravčík, D. Cagáňová and P. Košťál, “Hybrid Iterative Local Search Heuristic with a Multiple Cri- teria Approach for the Vehicle Routing Problem,” ICMST 2010: 2010 IEEE International Conference on Manufacturing Sci- ence and Technology, Malaysia, Kuala Lumpur, 26-28 November 2010, pp. 1-5.