1. Introduction

2. Literature Review

3. Problem Settings

4. Methodology

4.2. Mathematical Programming Model

There are various ways to convey a problem statement and its solution when presenting an optimization model. The common approach is to express the problem using simplified math equations to provide a concise representation. However, in this section, we start with presenting the code in Lingo’s mathematical modeling language. This approach offers several advantages in terms of readability, usability, and the understanding of the potential impact of the problem formulation on the solver’s performance. This allows us to include annotations, which can provide additional explanations as presented in “Model E: Lingo Optimization Code (Block A)”. An MINLP model is created for each single-block to maintaining the current class-based storage blocks for fast-moving and slow-moving items while implement the proposed CSAS. The optimization aims to maximize the total zone correlation score while penalizing deviations from weighted averages, referred to as centroids in this model. By defining centroids from achieved total correlation score and total picking frequency based on ratios, we can conduct scenario analysis to examine their influence on travel distance optimization. To better measure deviation from centroids and refine the objective function, we have introduced the utilization of PD and ND as a more effective alternative penalty terms for achieving targets. These measures aim to address the positive or negative sources of deviation, particularly to prohibit it from increasing beyond or below a target. The purpose of this approach is to quantify the feasible solution to allow for better comparison against other states while approaching the target from the opposite side of deviation. By incorporating deviation penalty terms, we can expedite convergence towards maximizing the objective function that ultimately lead to higher total correlation score in addition to achieving specific targets.

In mathematical programming models, the positive or negative deviation is commonly defined through logical conditions, or employing functions that determine the maximum or minimum value between the deviation result and the zero value since, to the best of our knowledge, there is no standard formula available to measure this deviation. However, when dealing with optimization problems, the solvers translate the formulation into a mathematical representation. Therefore, to facilitate the calculation of these deviations and promote better convergence, we have employed the following simple formulas that enable the quantification of positive centroid deviation (PCD) and negative centroid deviation (NCD) for achieved score (x) and a targeted centroid (C):

$$PCD\left(x,C\right)=\frac{\left|x-C\right|+\left(x-C\right)}{2}$$

(1)

$$NCD\left(x,C\right)=\frac{\left|x-C\right|-\left(x-C\right)}{2}$$

(2)

To further refine our model, we incorporate a derivative-based deviation penalty term (P) that penalizes the deviations for each zone. The penalty parameter (t) adjusts the weight of the penalty term, striking a balance between maximizing the total correlation score and enforcing the desired weighted centroids per zone. We calculate the penalty (P) as a function of D, where (D) represents the PCD or NCD and (t) denotes the penalty parameter as presented in equation [3]:

$$P\left(D\right)=D\cdot \left(\frac{D}{t\cdot D+1}-1\right)$$

(3)

By setting (t) to 10, the penalty approaches ($-\frac{9}{10} \mathrm{D})$for larger deviations, resulting in higher accuracy for targeted ratios compared to setting (t) to 2, and the model converges in less solving time compared to using the absolute deviation measure. On the other hand, if we set (t) to 2, the solving time decreases and the objective function achieves higher correlation scores as the penalty term approaches ($-\frac{1}{2} \mathrm{D})$ due to reduced penalization. Although it may seem that the term $\left(\frac{\mathrm{D}}{\mathrm{t}\cdot \mathrm{D}+1}-1\right)$ can be replaced by a fixed percentage, it actually plays a crucial role in expediting convergence. For instance, when solving one of the proposed scenarios in Model B with (t) set to 2, the solving time is 10 minutes and 53 seconds. However, replacing the term with ($-\frac{1}{2}$ ) increases the solving time to 27 minutes and 5 seconds, which is 2.5 times slower. It’s important to note that both solutions are identical, but the difference in solving time demonstrates the formulation impact of the penalty term on the solver’s performance. Overall, incorporating the penalty term with an appropriate penalty parameter allows us to fine-tune the model’s performance, achieving targets while optimizing solving time.

In this section, we introduce Model E, which has been developed to assess the effectiveness of implementing CSAS in optimizing travel distance within an order picking system that utilizes a zoning strategy. The model incorporates weighting coefficients to guide convergence towards targeted correlation scores and picking frequencies. By adjusting the ratios of targeted centroids, the model enables customization of item correlation distribution across zones, effectively balancing workload according to different desired customizations. The primary objective of the model is to achieve various local optimal solutions based on these defined coefficients, maximizing item correlation per zone and consequently increasing the average order size. To further increase the average order size in distant zones, we propose reducing the total picking frequency of items in these zones. This can be achieved by prioritizing the placement of items that collectively exhibit higher correlation and lower picking frequency in distant zones. Such approach effectively reduces travel distance by minimizing the need for frequent travel to the depot, thereby optimizing the picker’s travel distance. In the result section, we present several scenarios to analyze the model’s performance by testing different parameter configurations. For instance, in a system with three zones where the first zone is the furthest from the depot, setting the correlation score ratio for the coefficients to 3:2:1 and the picking frequency ratio to 1:2:3, as in scenario fourteen, enables the model to find a local optimal solution that satisfies these criteria. This leads to larger average order sizes in distant zones, ultimately minimizing travel distance. To quantify deviations from the targeted centroids, the presented models employ penalization using either the PCD or the NCD. In Model E, NCD is applied for all correlation centroids, while PCD is applied for all picking frequency centroids. This arrangement allows correlation scores to surpass the defined centroids, but restricts picking frequency centroids from exceeding their targets. However, it’s important to note that setting extreme ratios and unattainable centroids may render this configuration unachievable. In such cases, when extreme targets are set, the model penalizes multiple targets and leans towards maximizing the total correlation score.

Here is the mathematical representation of Model B, D, and E:

$$\mathrm{MAXIMIZE} \mathrm{Z}={\displaystyle \sum }_{k=1}^n{C}_k+X_k+Y_k$$

(4)

Subject to:

$$S_{ak}\in \left\{0,1\right\},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall 1 \le \mathrm{a}\le \mathrm{m} , 1\le \mathrm{k} \le \mathrm{n}$$

(5)

$${\displaystyle \sum }_{k=1}^n{S}_{ak}=1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall 1 \le \mathrm{a}\le \mathrm{m}$$

(6)

$${\displaystyle \sum }_{a=1}^m{S}_{ak}=L_k,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall 1 \le \mathrm{k}\le \mathrm{n}$$

(7)

$L_k defined as the total available slots in zone k.$

where:

$$C_k={\displaystyle \sum }_{a=1}^m{\displaystyle \sum }_{a<b\le m}^m\left(T_{ab}\cdot S_{ak}\cdot S_{bk}\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall 1 \le \mathrm{k}\le \mathrm{n}$$

(8)

$T_{ab} is defined as the correlation frequency for item a and b.$

$$S_{ak}=\left\{\begin{array}{cc}1,\hfill & if item a is selected in zone k\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

(9)

$$X_k=\left(R_k\cdot \left(\frac{R_k}{t\cdot R_k+1}-1\right)\right), \forall 1 \le k\le n$$

(10)

$$Y_k=\left(F_k\cdot \left(\frac{F_k}{t\cdot F_k+1}-1\right)\cdot \left(\frac{{\sum }_{k=1}^n\left(C_k\right)}{{\sum }_{k=1}^n\left(P_k\right)}\right)\right), \forall 1 \le k\le n$$

(11)

$t is the penalty parameter set to a value of 10 in Model \left(E\right).$

${\mathrm{R}}_k is defined as the deviation from correlation centroid \left(C{C}_k\right) given acheived correlation$ $score \left(C_k\right) and defined type of deviation i_k that is calculated by equation \left[12\right] for zone k:$

$$R_k=\frac{\left|C_k-C{C}_k\right|+\left( i_k\cdot \left(C_k-C{C}_k\right)\right)}{2}$$

(12)

$$i_{k } =\left\{\begin{array}{c}-1, if NCD is implemented for C{C}_k in zone k \\ +1, if PCD is implemented for C{C}_k in zone k\end{array}\right.$$

(13)

$$C{C}_k=\left({\displaystyle \sum }_{k=1}^n{C}_k\right)\cdot e_k$$

(14)

${\mathrm{e}}_k is defined as the weighting coefficient for correlation centroid \left(C{C}_k\right) for zone k.$

$F_k is defined as the deviation from picking centroid \left(P{C}_k\right)given acheived picking frequency \left(P_k\right) and$

$defined type of deviation j_k that is calculated by equation \left[15\right] for zone k:$

$$F_k = \frac{\left|P_k-P{C}_k\right|+\left( j_k\cdot \left(P_k-P{C}_k\right)\right)}{2}$$

(15)

$$P_k ={\displaystyle \sum }_{a=1}^m\left(T_{aa}\cdot S_{ak}\right) ,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall 1 \le \mathrm{k}\le \mathrm{n}$$

(16)

$T_{aa} is defined as the picking frequency for item a$

$$j_{k } =\left\{\begin{array}{c}-1, if NCD is implemented for P{C}_k in zone k \\ +1, if PCD is implemented for P{C}_k in zone k\end{array},\right.$$

(17)

$$P{C}_{k }=\left({\displaystyle \sum }_{k=1}^n{P}_k\right)\cdot h_k$$

(18)

${\mathrm{h}}_k is defined as the weighting coefficient for picking centroid \left(P{C}_k\right)for zone k$

The objective function in equation [4] aims to maximize the sum of item correlation score $\left(C_k\right)$ for each zone $\left(k\right)$, while considering deviation penalty terms $\left(X_k\right)$ and $\left(Y_k\right)$, calculated using equations [10] and [11] respectively. For every zone $\left(\mathrm{k}\right)$, the deviations $\left(R_k\right)$ and $\left(F_k\right)$ in the penalty terms are calculated by equations [12] and [15] for the achieved item correlation score $\left(C_k\right)$and the picking frequency $\left(P_k\right)$. These penalty terms account for the zone’s correlation centroid $\left(C{C}_k\right)$ and picking centroid $\left(P{C}_k\right)$, based on the deviation types $\left(i_k\right)$ and $\left(j_k\right)$ that are defined for each zone. These terms are formulated using acceleration convergence technique, with a penalty parameter $\left(t\right)$ set to 10 in Model E, resulting in approximately 90% penalization for each deviation. The decision variables $\left(S_{ak}\right)$ in equation [5] are binary constraints representing the assignment of each item $\left(a\right)$ to a zone $\left(k\right)$ as in equation [9]. We ensure that the sum of $\left(S_{ak}\right)$ for all zones $\left(k\right)$ is equal to 1 for every item$\left(a\right)$, restricting each item to be located in only one zone. This constraint is expressed in equation [6]. Furthermore, the sum of $\left(S_{ak}\right)$ for all items is constrained by the value of $L_k$ for each zone $\left(k\right)$, as indicated in equation [7]. It should be noted that in the "E" model, the exported result for "Aisle_Item_Rank" is a combination of the item’s correlation score and its picking frequency. This approach, referred to as method A, which was used initially for item ranking within zones. However, the presented results in Table 1 for scenarios under Model E demonstrate the incorporation of method C. This method takes into account the average order size that is calculated from the order history data. Furthermore, an acceleration technique, which was previously used by Kuo et al. (2016) in their model, is incorporated in this model. This technique involves calculating the item correlation only once for two correlated items, resulting in a more efficient computation process. As a result, the total zone correlation score achieved in the model represents half of the actual zone correlation score, as shown in Table 1. In this paper, however, the item correlation used in formulas for item ranking is defined as the sum of correlation frequencies for all items located in that particular zone.

5. Optimization Result Analysis

▪

Ranking Method A is the sum of item correlation $\left(C_{ka}\right)$and item picking frequency $\left(P_{ka}\right)$ for item $\left(a\right)$in a particular zone $\left(k\right)$:

$$R_{ka}= C_{ka}+P_{ka}$$

(19)

▪

Ranking Method B involves calculating equation [20] for ranking an item $\left(\mathrm{a}\right)$ in a particular zone $\left(\mathrm{k}\right)$:

$$R_{ka}=\frac{ {\left(P_{ka}\right)}^2}{C_{ka}+P_{ka}}$$

(20)

▪

Ranking Method C for an item $\left(a\right)$involves calculating equation [21] for all orders $\left(w\right)$in the order history binary matrix $\left( O_{wb}\right)$ to determine the order size in the newly generated suborders with the optimization selection $\left(S_{bk}\right)$of all items denoted by $\left(b\right)$ in a particular zone $\left(k\right)$:

$$R_{ka}={\displaystyle \sum }_{w=1}^i\left(\frac{1}{{\sum }_{b=1}^m\left( O_{wa}\cdot O_{wb}\cdot S_{ak}\cdot S_{bk}\right)}\right) , O_{wa}, O_{wb},S_{ak},S_{bk}\in \left\{0,1\right\}$$

(21)

6. Conclusion

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