Download Shelf Space Optimization using a Genetic

Shelf Space Optimization using a Genetic Algorithm
Siripong Malasri1and Pili Segui2
Abstract: Space optimization is a classic packaging problem. Shelf space optimization is the most basic problem,
where only the widths of packages to be placed on the shelf are considered. More complex problems include
warehouse space optimization, in which widths and depths of packages are considered, and container space
optimization, in which widths, depths, and heights of packages are considered. In this work, a genetic algorithm was
used to optimize the space used by three box sizes on a five-level shelf. The paper describes the algorithm used as
well as an implemented computer program.
Keywords: Packaging, optimization, genetic algorithm
Introduction
Space optimization is a common problem in packaging, including shelf optimization (only package widths are
considered, a one dimensional problem), warehouse space optimization (widths and depths or package footprints are
considered, a two dimensional problem), and container space optimization (package widths, depths, and heights are
all considered, a three dimensional problem).
Genetic algorithms, developed in the 1960’s by John Holland at the University of Michigan, are computer algorithms
based on Darwin’s rules of evolution. The natural selection process is a powerful tool for solving optimization
problems [1]. A typical algorithm is shown in Fig. 1, while many variations exist.
Figure 1. A typical genetic algorithm
1 Christian Brothers University, 650 East Parkway South, Memphis, TN 38104,
[email protected]
2
Carrer Sa Vinyeta, 3, Ciutadella de Menorca, I.B, 07760, Spain, [email protected]
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Typically, a fixed population size and discrete intervals (i.e., all members of a generation start and end at the same
time) are used to simplify programming efforts. Each potential solution to the problem is represented by a
chromosome, which consists of genes representing different parts of a solution as shown in Figure 2. New possible
solutions are created through the processes of mutation and cross-over. A mutation is a process in which certain
genes in a chromosome change their values. This is a method for nature to insert diversity into the population.
Cross-over is a process in which a part of a chromosome is exchanged with the same part of another chromosome.
This is a method for nature to pass on traits from parents to offspring. Figures 3 and 4 illustrate mutation and crossover, respectively.
Figure 2. Chromosome and genes
Figure 3. Mutation illustration
Figure 4. Cross-over illustration
To apply a genetic algorithm, there must be a way to measure the fitness of each chromosome. How fitness is
measured depends on the problem. A fitness value is computed based on the gene values in that chromosome.
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Shelf Space Optimization
A prototype genetic algorithm was developed for a five-level shelf for three sizes of boxes. The objective is to
minimize the unused space of the shelf as shown in Fig. 5.
Figure 5. Shelf space optimization problem
A chromosome, which is a possible solution, consists of five sections, representing the five levels of the shelf (Fig.
6). Each section consists of three genes, representing the quantities of the three types of boxes to be placed on that
level.
Figure 6. Chromosome and genes used in shelf space optimization problem
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The total used space can be calculated based on gene values and input values of box lengths (Fig. 7). If the total
required length of a level exceeds the given shelf length, a penalty is applied. Knowing the total used shelf space,
the total unused shelf space can be found, as shown in Fig. 8. The total unused shelf space is the objective function
of the problem, which is to be minimized with respect to two constraints, i.e., shelf length and box availability (Fig.
9). When a constraint is violated, a penalty is applied to decrease the fitness value of the chromosome. A less fit
chromosome has a smaller chance to be selected into the next generation using a spinner concept (Fig. 10). A slice
on the spinner represents a chromosome, and its size represents the chromosome’s fitness value. After being spun,
the spinner has a larger chance to stop at a larger slice. This is how a chromosome is selected into the next
generation. Commonly used mutation and cross over methods, as shown in Figs. 3 and 4, are used. A computer
program written in C was developed with a sample run shown in Figs. 9 – 13.
Figure 7. Used space of the five levels
Figure 8. Total unused shelf space
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Figure 9. Problem constraints
Figure 10. An analogy of using a spinner for selecting a chromosome into the next generation
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Figure 9. Input of a sample run
Figure 10. Randomly created first generation of a sample run
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Figure 11. Fitness and penalty values
Figure 12. Best solution from the first generation
Figure 13. Overall best solution from all generations
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Conclusion
This prototype demonstrates the potential of applying a genetic algorithm to the space optimization problem, which
is of interest to the packaging industry. The prototype can be expanded to be more general and applicable to various
types of shelves and boxes. The concept can be expanded to include warehouse space optimization as well as
container space optimization.
References
1.
S. Malasri, “Genetic Algorithms,” Magill’s Survey of Science: Applied Science, Supplement, Salem Press,
Pasadena, CA, pp. 3050-3057, May 1998.
Authors:
Siripong Malasri – Professor of Civil Engineering and Packaging Activities/Healthcare Packaging Consortium
Coordinator at Christian Brothers University.
Pili Segui – Graduated in Computer Engineering from La Salle University in Barcelona, Spain, and currently is a
Business Intelligence Analyst for SDG Consulting in Barcelona. She implemented this shelf space optimization
genetic algorithm while she was an exchange student at CBU.
MAESC 2011 Conference, May 3, 2011, Christian Brothers University, Memphis, Tennessee