Elsevier

Computers & Operations Research

Volume 34, Issue 9, September 2007, Pages 2576-2588
Computers & Operations Research

A heuristic for the label printing problem

https://doi.org/10.1016/j.cor.2005.09.021Get rights and content

Abstract

Label printing finds many applications in industry. However, this task is still labor intensive in many printing factories. Since each template can only accommodate a fixed number of labels, an important task is to work out the compositions of templates by allocating suitable labels to each template in order to fulfill the order requirements effectively. The template design could be rather arbitrary, which usually ends up with a lot of excessive printed labels. Enhancing the template design will significantly improve the efficiency of the printing process, and, at the same time, reduce the waste of resources. This motivates the study of more automatic design methods. In this paper, the problem is first formulated as a nonlinear integer programming problem. The main variables in the formulation are the compositions and the printing frequencies of templates. For practical purpose, each type of label is confined to one template only which allows automated packing and handling. The structure of the problems is carefully analyzed and a new algorithm is proposed. Numerical results show that the proposed method is a simple but effective way of generating good template designs.

Introduction

Label printing has many applications in industry. In the manufacturing sector, different types and quantities of labels are required. These include labels on clothes, labels on a variety of consumer products, and so on. For a particular product, the requirements of a label are usually unique and excessive printed labels are often not re-usable. In a printing factory, templates are made before the printing process commences. The number of labels that can be allocated to each template is fixed because of the size of the template. One has to determine the number of templates needed and the corresponding composition of each template. A simple method that many factories often adopt is to make as few templates as possible. Without analyzing the problem carefully and systematically, the composition of each template is often determined based on experience. Different combinations of labels are assigned arbitrarily to the templates until all the required labels have been assigned. The printing frequency of templates is then determined in order to fulfill the customer order. Clearly, this process requires a lot of human effort and might lead to a lot of wastage due to poor template designs. As labels are getting fancier and more resourceful and the templates for each customer order are usually discarded after use, better methods are definitely required to minimize the number of templates and the label wastage.

The label printing problem considered in this paper is formulated as a nonlinear integer programming problem. In order to avoid manual separation and aggregation before packing, the allocation of each type of label is confined to one template only. The resulting formulation is a multidimensional nonlinear integer programming problem which have a large number of possible solutions. Some nonlinear branch-and-bound methods have been proposed [1], [2] and more recently by [3], [4]. However, they are like exhaustive searching techniques which rely on tree enumeration, and are computationally expensive to implement. Hence, alternative numerical algorithms are sought here. By exploiting the bi-level structure, the original problem can be converted into a bi-level optimization problem. For each template, the outer level optimizes on the combination of labels while the inner level optimizes on the allocation of the assigned labels. The inner level optimization problem is analyzed and a heuristic is proposed. For the outer level problem, the technique of simulated annealing [5], [6] is applied. A large number of customer orders are first generated randomly to test the overall proposed algorithm. Then two real customer orders from a printing factory are presented and the results are compared.

Section snippets

Mathematical model

When a customer order arrives at a printing factory, the requirement in terms of sizes, styles and quantities for the labels is specified. A set of templates, denoted by n, are then designed for the printing of the required labels. Fig. 1 shows an example of a typical layout of labels in a template. Depending on the size of the labels, each template can only accommodate a fixed number of labels, denoted by M. In a typical order, a total number of different labels, denoted by m, are requested.

System analysis and algorithm

Comparing with formulation (F1), since each label appears in one template only, the number of possible ways to allocate the labels to the templates in formulation (F2) is reduced to the order O((nM)m) which is a lot lower than that of (F1). However, the complexity of the problem is still increasing exponentially with the number of labels.

Since the set of Ti defined in (24) is determined fully by I, this formulation has a very nice bi-level structure. The outer level optimization is to minimize

Numerical results

In the following, the parameters in the simulated annealing algorithm are chosen to be Nc=30, α=0.9, and initial T=0.9. The number of random perturbations N is set to 50 000. In the formulation, δ=0.9 is applied. All the calculations were carried out on a Pentium IV 2.4G PC.

The algorithm is first tested against randomly generated cases. The number of labels, the number of templates and the template capacities are varied to give different problem settings. For each set of parameters, order

Conclusions

In this paper, the label printing problem frequently occurs in printing factories has been studied. The problem has been formulated as a nonlinear integer programming problem. A heuristic has been proposed and its performance has been tested thoroughly by a large number of cases. Numerical examples from a printing factory have also been applied and compared. In general, the method is efficient and the calculated results are consistent for most of the problem settings. The solutions obtained can

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