O PTIMIZATION OF U NIT L OAD FORMATION TAKING INTO ACCOUNT THE MASS OF PACKAGING UNITS

This article presents a mathematical formulation of the optimization problem of loading unit formation taking into account the mass of packaging units. Proposed model can be applied to optimize the arrangement of non-uniform cubical loading units in loading spaces. The model ensures possibility of defining various dimensions, masses, resistances of particular packaging units and their vertical axis rotation. Within the constraints of formulating optimization problem, taking into account masses and resistances ensures that all packaging units will rest on a pallet or on other packaging units, and the surface of contact between loading units guarantees stability of units arranged in subsequent layers. The mathematical model was verified. The paper provides an appropriate calculation example.


Introduction
Forming loading units is the one of the most import ant elements of logistical systems. It has an effect on both the performance and the cost of logistical processes [10]. An appropriate arrangement of packaging units on auxiliary loading equipment (e.g. on a pallet) guarantees the best possible use of available loading space. Additionally, the formed unit loads should be arranged so that packaging units are safe from damage sustained under the weight of other units during transport. This implies that the appropriate arrangement of packaging units should ensure maximizing the use of unit load dimensions, taking into account their mechanical resistance. The problem of loading unit formation occurs primarily in manufacturing companies and logistics. With respect to manufacturing companies, unit loading formation is an issue which has a one-time solution for each specific type of manufactured product. Often, the individual units of one product type are placed in protective packaging. This becomes the packaging unit. When we combine these units with auxiliary loading units, they are formed into uniform loading units. Taking into account the resistance and mass obtained by the expert or optimization methods, a single solution to the packaging unit arrangement problem has a long term application. In logistics companies, the problem of forming unit loads is more complex. Here, unit loads are formed from heterogeneous packages and in terms of resistance and mass. A one-time solution cannot be applied to forming many loading units (because each of them is different) in this situation. As a result, logistics companies are required to obtain real-time solutions. Many optimization models of loading

Review of Literature
In literature, the problem of unit load formation is quite extensively described and include formulation of models in one, two and three dimensions. One of the simplest optimization models of forming unit loads is proposed by T. Tlili, S. Faiz and S. Krichen [15].In this model, the number of containers used to store objects is minimized. Additionally, in the mathematical formulation of the problem it is assumed that each object is characterized by volume and mass, each container has a predetermined maximum capacity, each object can be placed only in one container and must fit in it, container capacity cannot be exceeded by the located cargo, and the container may be stored in one container slot, whose loading area volume may not be exceeded.

Proposal of mathematical model of threedimensional loading unit formation 3.1. General Assumption of the Model
The model proposed in this article is based on the model [12], which considers the requirement of contact between bearing surfaces as well as rotation of packaging units about the vertical axis. This model also has additional constraints with regard to packaging unit resistance and loading of auxiliary loading equipment to assist with the formation of loading units. The goal of formulating a three-dimensional optimization problem of loading units consists of the following assumptions:  Packaging units have a cuboidal shape,  Each of the packaging units is able to be rotated 90 o about a vertical axis ( Fig. 1), however, all units must be in a vertical position at all times.  The mass, resistance, and dimension of each packaging unit may vary.
 The mass of each packaging unit is in a geometrical center.  , so that all the constraints are met:  on the dimension of packaging units along the X axis:  on the dimension of packaging units along the Y axis:  on relative placement of packaging units along the X axis:

JO
(3)  on relative placement of packaging units along the Y axis:

JO
(4)  on relative placement of packaging units along the Z axis:  on situating packaging units above others along the Z axis:  on ensuring the packaging unit will not go beyond the dimensions of the auxiliary loading equipment along the X axis:  on ensuring the packaging unit will not go beyond the dimensions of the auxiliary loading equipment along the Y axis:  on not exceeding the maximum height of the loading unit:  on ensuring the situation of packaging units above the surface of auxiliary loading equipment:  on securing packaging units from damage: : JO JO (25)  on securing against damage from the auxiliary loading equipment: and so that the criterion function with the interpretation of filling the loading unit: accepts a maximum value.

Sample Calculation
The model of three-dimensional loading unit formation described in the previous section was implemented in LINGO. Next, the goal was to assess its usefulness, verify correctness and solve many sample calculations. The results obtained referred to solutions of analytical problem. In the article, one of these samples was described.
In the examined sample calculation, 4 out of 8 packaging units were placed on the pallet (Fig. 2) Additionally, the mass of the placed packaging units did not exceed the permissible pressure of the pallet and the packaging units would not be damaged due to stacking.

Conclusion
The mathematical models of forming loading units described in literature are characterized by certain simplifications. These simplifications include the inability to rotate packaging units about a vertical axis, lack of constraints regarding contact between the bearing surfaces, lack of integrating mass as well as resistance of the packaging loads. Consequently, applying these kinds of models to planning actual loading units does not always allow us to obtain an optimal solution which has the correct physical interpretation.

Fig. 2. Visualization of Sample Calculation Solution
The optimization problem presented in this article is free from flaws present in the optimization models in the literature. The obtained solutions to the optimization problem of forming loading units allows for detailed projection of the optimal placement of packaging units on a pallet or in a container. Most importantly, resistance and mass of individual packaging units are taken into account. The quality of the solution and tests performed for other sample calculations allow us to assess the developed formulation concerning the mathematical problem of forming unit loads and its numerical implementation as valid. It should, however, be noticed that the accepted assumption which simplify the center of mass of packaging units in their geometrical center in the case of large packaging units placed on top of many smaller packaging units may introduce a certain error. In order to eliminate this described imperfection, the optimization problem should be modified, accepting the assumption that the mass of each packaging unit is distributed equally on the entire base surface of the packaging unit.