# Towards Productive and Ergonomic Order Picking: Multi-Objective Modeling Approach
## 4 Methodology for the Experiment  
As mentioned, the storage space assignments of boxes were made:  
* With a time-based objective model;
* With an energy expenditure-based model;
* With a health-risk based model;
* With a multi-objective model (all described in Section 4).  
Three single-objective models and one multi-objective model were developed. We choose three factors that significantly impact the decision-making process about storage assignment: picking time, energy expenditure and health risk.  
The first single-objective function is time-based, where the objective is to minimize the total time needed for order picking (\(T\)). The second single-objective function is an energy-based objective function (\(E\)), where the rack locations are determined based on the minimum energy expenditure for order picking. As a novelty, the third single-objective function is a health risk-based objective function (\(O\)), where the objective is to minimize the health risk for the order picker. The following Formulas (8)-(10) define single-objective models for the presented case (where we use the corresponding objective function for the selected optimization):  
\[\min(T)=\min\sum_{i=1}^{9}\sum_{j=1}^{45}t_{ij}\cdot x_{ij} \tag{8}\]  
\[\min(E)=\min\sum_{i=1}^{9}\sum_{j=1}^{45}e_{ij}\cdot x_{ij} \tag{9}\]  
\[\min(O)=\min\sum_{i=1}^{9}\sum_{j=1}^{45}o_{ij}\cdot x_{ij} \tag{10}\]  
where \(t_{ij}\) represents the time needed for the whole process of picking the \(i\)-th box from the \(j\)-th storage space, \(e_{ij}\) represents the energy usage for the entire process of picking the \(i\)-th box from the \(j\)-th storage space, and \(o_{ij}\) represents the value of health risk for the whole process of picking the \(i\)-th box from the \(j\)-th storage space.  
All three single-objective functions are subject to the same following constraints (11):  
\[\forall m,n\in\mathbb{N},n=1,2,\ldots,N,m=1,2,\ldots,M\]  
and  
\[\begin{array}{c}x_{m,j},x_{i,n}\in\{0,1\}\\ \text{we define :}\\ \sum\limits_{j=1}^{N}x_{m,j}=f_{m}\\ \sum\limits_{i=1}^{9}x_{i,n}\leq 1\end{array} \tag{11}\]  
\(f_{m}\) refers to the frequency of individual boxes which need to be picked in each scenario.  
The last is the multi-objective function where optimization of arrangements is based on all three objective factors (12):  
\[\min(T,E,O)=\min(\sum_{i=1}^{9}\sum_{j=1}^{45}w_{t}(t_{ij}\cdot x_{ij}),\sum_{i =1}^{9}\sum_{j=1}^{45}w_{e}(e_{ij}\cdot x_{ij}),\sum_{i=1}^{9}\sum_{j=1}^{45} w_{o}(o_{ij}\cdot x_{ij})) \tag{12}\]  
where \(w_{t}\) represents the weight of time function, \(w_{e}\) represents the weight of energy function, and \(w_{o}\) represents the weight of health risk function.  
Further, we analyze four differently weighted scenarios (Table 4). The advantage of the multi-objective function is that the decision-maker can determine the importance of specific criteria in a particular situation. The constraints are the same as (11).  
Before single- and multi-objective functions were used for storage assignment purposes, all data ware normalized. Sections 4.1-4.3 explain how we calculated total time, total energy expenditure, and overall health risk values in the minimization functions. All linear programing results were obtained using the MATLAB software tool.