# A Mathematical Model for Handling in a Warehouse Pareto-based multi-objective optimization for warehouse design. One of the key design variables is the number of aisles in the warehouse, and it can be any integer value between the minimum and maximum numbers of aisles. Given the maximum number of aisles and the number of storage classes, the dimensions of the particle for two-block warehouse is given in Eq. Suppose that the maximum number of aisle in a two-block warehouse equals 12 and the number of product class equals three. The solution to this warehouse design problem can be represented as a particle with 11 dimensions as shown in Figure 5 where Ri represents a randomly generated value.

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An example particle with 11 dimensions is shown in Figure 6. A particle with 11 dimensions. An example of a randomly generated particle. The first dimension represents the number of aisle by using Eq. The first dimension of the particle contains the value 0. In this case, the number of aisles quals 6. Once the aisle is known; the length of aisle can be determined by using the formulas The next dimensions are decoded to the partial aisle length for storage of items for each class. The rectangular warehouse with narrow aisle is planned to store multiple product group. The warehouse is operated with a manual order picking system and the class-based storage policy.

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The general assumptions for the warehouse are: - The warehouse size is designed for two-block warehouse that is 2, m 2. The minimum length of the storage rack is 10 m. The percentage of the total storage space used for each class equal 20, 25, 25, and 30, respectively. The parameters are set as follows. The number of particles is fixed at Moreover, the stopping criterion is set at iterations.

The inertia weight is linearly decreasing from 0. The acceleration constants for self-learning and social learning are 1 and 1, respectively. In addition, the upper limit of elite group is fixed at particles. Five replications are used in this experiment. The necessary data for warehouse design are input to the program. After that, the solutions are collected. The results of the two-block warehouse are shown in Table 1 and Figure 5. Table 1.

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Non-dominated solutions of two-block warehouse Figure. Non-dominated solutions of two-block warehouse. Seven alternatives for the aisle layout are shown in Table 1. From Table 1, if the minimum number of aisle is selected for warehouse design, the percentage usable storage space is a maximum.

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The length of aisle is the longest. However, it has an effect on the travel distance which means that the travel distance is increased. In Figure 7, if the maximum percentage usable storage space is selected, the increasing travel distance is Otherwise, if the minimum travel distance is selected, the usable storage space will reduced only 3.

This paper presents a multi-objective mathematical model for a two-block warehouse layout design that uses the class-based storage policy. There are three decision variables: the number of aisles, the length of aisle and the length of each pick aisle to allocate to each product class. Moreover, Pareto-based multi-objective PSO is applied to solve the mathematical model. The proposed algorithm is capable of providing the alternative solution.

## A Multi-Criteria Model for Warehouse Layout Design

The numerical example is given and the results from the example show that the proposed algorithm can provide design alternatives for the decision. There are many objectives when a warehouse is designed but this paper considers only two objectives. Therefore, it has room to improve the mathematical model. The author thanks the referees for their valuable comments. The author acknowledges the financial support from the National Science and Technology Development Agency.

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This research proposes a Pareto-based multi-objective optimization approach to class-based storage warehouse design,considering a two-block warehouse that operates under the class-based storage policy in a low-level, picker-to-partand narrow aisle warehousing system. A mathematical model is formulated to determine the number of aisles, thelength of aisle and the partial length of each pick aisle to allocate to each product class that minimizes the travel distanceand maximizes the usable storage space. A solution approach based on multiple objective particle swarm optimizationis proposed to find the Pareto front of the problems.

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Numerical examples are given to show how to apply theproposed algorithm. The results from the examples show that the proposed algorithm can provide design alternativesto conflicting warehouse design decisions. Therefore, 5. Caron, F. Garcia-Diaz, A. Gu, J. Hwang, H.

click Kachitvichyanukul, V. Lai, K. Larson, T. Le-Duc, T. Li, M. Nguyen, S. Onut, S. Park, Y. Petersen II, C. Petersen, C. Poulos, P.

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Sooksaksun, N. Tompkins, J. Van den Berg, J. The optimal total material handling cost by using this mathematical model equal to Rp It saves Rp6. Thus, all of the products can be allocated in accordance with the product flow type with minimal total material handling cost.

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