The threshold accepting is one of the existing optimization methods often used in search of nearly uniform designs because of its implementing simplicity and effectiveness. It was refined from simulated annealing technique. The difference between two methods lies on the process of acceptance or rejection of the candidate solution. However, like many other optimization methods, threshold accepting algorithm allows to have an approximate value of the minima of the target function; but it does not necessarily guarantee obtaining the global minimum of the target function (discrepancy) because, generally, the design space is too large so that it is impossible to compare all designs of this space. The principle of implementing of the threshold accepting algorithm involved following three fundamental concepts:

• Target function: The target function in this context corresponds to the selected discrepancy function.

• Neighborhood: Let X c be an initial design randomly chosen for given values of n , s , and q . Any design constructed by exchanging two entries in one or more columns of the initial design X c is considered a member of its neighborhood. The entire set of neighboring designs is denoted by N ( X c ) .

• Threshold sequences: The threshold sequence is related to the two previous concepts (target function and neighborhood). First, choose a large number of feasible solutions, n s , and a number of thresholds, n T , which is smaller than the former. Then, compute the absolute differences of the target function values between each pair of feasible solutions. The decreasing quantiles of these differences define the threshold sequences T r ( r = 1 , ⋯ , n T ). The pseudo-code (Equation Algorithm 1) for computing the threshold sequences is as follows:

Once threshold sequences are computed, the TA algorithm always accepts a solution that improves the target function until given threshold sequence is exhausted. The pseudo code (Equation Algorithm 2) of TA algorithm can be written as follows:

It is an iterative process which starts from an initial solution moves to another solution called local optimum improving the criterion of selection or target function. This moving from a solution to another one is made according to a neighborhood structure. A neighborhood structure is a change that can be made in the structure of initial solution to provide a new one. There are many neighborhood structures among which the swap neighborhood or neighborhood by interchange which consists in interchanging two entries of a column.

Let X i be an initial solution randomly chosen with n size and q dimensions (column). A Design X i n e w obtained by interchanging two (2) entries of a column (both column and entries are randomly chosen) of the initial design X i is called neighbor of X i and N ( X i ) is a set of such designs.

Once neighbor is generated, discrepancy of this new design is computed and compared with discrepancy value of the current design. The new design, X i n e w is accepted as current solution if only if its discrepancy value is smaller than that of X i meaning,

Otherwise, the current design X i is maintained and another neighbor of X i is taken among the set of neighbor solutions N ( X i ) . The process stops when all the N solutions are evaluated and the best one that improved the target function is taken as final solution, otherwise the current design is delivered as final solution. This is known as firstimprovement local search and its pseudo code (Equation Algorithm 3) is given as follows:

VNS is an optimization like TA based on local search algorithm [13]. The main idea is to use several neighborhood structures in the local search process in order to get an approximation of the real optimum known as global optimum. The strategy of using several neighborhood structures in VNS is motivated by three (3) observations:

1) A local minimum with respect to one of them is not necessarily a local minimum for another;

2) With respect to all possible neighborhood structures, a global minimum is a local minimum;

3) For many problems, local minimum with respect to one or more neighborhoods are relatively close to each other.

In this work, two types of structures are used: the first consists of interchanging two entries of a column randomly chosen and the second is to interchange two successive entries of column as shown below.

Like previous algorithms, VNS is simple to implement and consists of two main steps: Initialization and iteration process.

Initialization is to choose the number (M) and types neighborhood structures ( N k , k = 1 , ⋯ , M ) that will be used in the iteration process. Then, select also randomly a design, X i with size n and dimension q and define the stopping rule that is in our case, the number of time algorithm will be repeated (number of iterations I ).

In the iteration process, at each iteration ( i ) shaking, local search and accepting rule are applied for each neighborhood structure until stopping rule is reached. For each neighborhood structure, the current design ( X i ) is shacked with respect to the given neighborhood structure ( k m ) and local search is applied with this shacked design X ' (neighbor of initial design, X i ) to provide a new design X i → i + 1 . The values of discrepancy is computed for both designs ( X i → i + 1 and X i ) and if new design X i → i + 1 improves the uniform criterion or discrepancy, it is taken as current solution and the search continues with the same neighborhood structure; Otherwise, the search continues with another neighborhood structure. There are different variants of VNS among which basic variable neighborhood search (BVNS) particularly applied in this work. Its pseudo code (Equation Algorithm 4) can be written as follows.

VNS is a local search optimization algorithm that used different neighborhood structures in the process for a better solution. Despite using several neighborhood structures, local optimization applied at each iteration led to a local optimum. Employing an algorithm easy to implement allowing to get global optimum in VNS can improve considerably its effectiveness because not only different neighborhood structures used allow diversifying design space exploration but also the TA used to find an approximation of real optimum. The new algorithm (TA-VNS) is obtained by replacing local search process applied in VNS described previously by threshold accepting algorithm (TA) presented above. It is easy to implement because it doesn’t require any adjustment of parameters unlike others optimization algorithms.

TA-LSP is a hybrid optimization based on TA algorithm enhanced with LS algorithm. TA is an optimization method which explores in depth the design space in order to get an approximate of the true optimum said global optimum. LSP is a simple optimization method that allows finding a local optimum. The application of LSP at each iteration of TA algorithm aims to improve the later to provide a better solution. TA-LSP is also easy to implement as previous method.

The objective in this section is to see whether combined methods namely TA-VNS and TA-LSP provide better results than threshold accepting method and variable neighborhood search taken individually by comparing the values of wrap around discrepancy and computational times for constructed designs. The value of the discrepancy is strongly related to construct designs in particular the size, levels and dimension. The computational time depends on design but also the fixed number of solution to be assessed and that of iteration in the algorithm. An acceptable comparison requires constructing several designs of different sizes and dimensions. It is for this reason that we have envisaged constructing several designs with different sizes in low and high dimensions.

The numbers of design points (n), factor’s levels (q) and dimensions (s) have been voluntarily fixed in the order to assess the effectiveness of methods according size and dimension of designs. Designs used to compare the three (3) methods are symmetric designs whose factor’s levels vary from 3 to 5 distributed in three (3) categories according to the size and dimension. The first category consists of designs whose sizes are less or equal to 100 in 3 and 4 dimensions. The second and last categories consist of designs having large sizes in low dimensions (4, 5) and higher dimensions (10 - 30) respectively as shown in Table 1.

Table 1. List of designs constructed with their parameters.

3.4.1. Parameters of the Algorithms

In our study, the key parameters of the algorithms were set based on computational resource limits (memory size) to ensure convergence while remaining computationally feasible.

a. Tabu Algorithm (TA)

• Number of solutions (ns): the number of candidate solutions generated per iteration.

• Number of thresholds (nT): corresponds to the number of iterations.

The specific values used in our experiments are summarized in Table 2.

Table 2. Tabu algorithm parameters and total number of designs evaluated.

b. Variable Neighborhood Search (VNS)

• Number of neighborhood structures (M): 2 types of neighborhoods were used:

1. Interchanging two random entries in a column.

2. Interchanging two successive entries in a column.

• Number of iterations (I): set according to computational resources.

• Number ofcandidatedesigns per structure: generated according to the neighborhood type; the total number of designs evaluated across all iterations is denoted as Ntotal.

Selection of Parameters

The choice of these values was based on computational feasibility rather than exhaustive search. Despite these constraints, the algorithms converged quickly to local optima. At each iteration, a sequence of candidate solutions is generated and the optimal one is retained for the next iteration.

This reporting ensures that all algorithmic parameters are now fully documented, making the results reproducible and enabling a fair comparison between TA and VNS.

3.4.2. Computing Target Function

The iterative process of an algorithm is based on the evaluation of the target function of the current solution and the generated neighbors. Target functions being less easy to compute, it is essential to define a way to compute this function in the various possible cases. As a target function, wrap around discrepancy is used in this work because of its desirable feature as shown in previous chapter.

a. Computing Wrap around Discrepancy for the Current Design

Let P c be the current design and X c = ( x i j ) , its induced matrix. The wrap around discrepancy of X c is given by,

Let α = ( α i k ) be a n × n matrix.

Then, the wrap around discrepancy for the current design can be written as follow,

b. Computing WD for the Neighbor of Current Design

In the new design neighbor of the current design, computing discrepancy concerns only the value of the current design’s discrepancy and some elements of the matrix of the new design precisely lines of the two entries ( x i j , x k j ) exchanged. First, mathematical expressions of α ′ and β ′ are computed for lines comprising the entries exchanged and then they are subtracted from values of α and β of the same lines but for the current design. Finally, the obtained difference is added to the current design’s discrepancy.

For 1 ≤ t ≤ n , α ′ t i can be expressed as follows:

Then,

Figure 1 provides a comparative analysis of the performance of different optimization methods (TA, TA-LSP, and TA-VNS) using two types of metrics: solution quality, assessed through wrap-around discrepancy differences (Figures 1(a)-(c)), and computational cost, evaluated in terms of execution time (Figure 1(d), Figure 1(e)), as a function of the problem size n . Figure 1(a), Figure 1(b) indicate that the discrepancies between TA and its variants (TA-LSP and TA-VNS) remain generally small, with a few localized peaks at specific values of n , suggesting occasional but not systematic differences in performance. In contrast, Figure 1(c) reveals a higher variability between VNS and TA-VNS, highlighting that the hybrid TA-VNS approach can significantly influence solution quality depending on the problem size. Regarding computational time (Figure 1(d), Figure 1(e)), all methods exhibit an increasing trend with n , reflecting the growing computational complexity. However, TA remains the fastest method overall, followed by TA-LSP, while TA-VNS and VNS incur higher computational costs, particularly for larger problem sizes. This behavior illustrates the classical trade-off between solution accuracy and computational efficiency, and emphasizes the relevance of hybrid approaches such as TA-VNS, which can yield improved solution quality at the expense of increased computational time.

Figure 1. Comparison of wrap-around discrepancy differences and computational times across TA, TA-LSP, VNS, and TA-VNS methods for increasing problem sizes.

Figure 2 presents a detailed comparative assessment of the performance of TA, TA-LSP, VNS, and TA-VNS methods for larger problem sizes, combining both solution quality and computational efficiency. Figure 2(a), Figure 2(c) illustrate the evolution of wrap-around discrepancy differences as a function of the design size n . Overall, the discrepancies remain relatively low across methods, indicating comparable solution quality, although a pronounced peak around intermediate sizes (notably near n ≈ 400 ) suggests a temporary degradation in performance for all comparisons. Beyond this region, the discrepancies tend to decrease and stabilize, especially for larger values of n , highlighting improved robustness of the algorithms at scale. Figure 2(c) further shows that the variability between VNS and TA-VNS is slightly more marked, confirming that hybridization can alter solution behavior depending on the problem size. In contrast, Figure 2(d), Figure 2(e) reveal a sharp increase in computational time with increasing n , reflecting the growing algorithmic complexity. TA remains the most computationally efficient approach, followed by TA-LSP, whereas TA-VNS and VNS exhibit significantly higher execution times, particularly for large-scale instances (up to n = 3000 ). This trend clearly emphasizes the trade-off between computational cost and potential gains in solution refinement offered by hybrid methods, with TA-VNS achieving competitive accuracy at the expense of substantially increased runtime.

Figure 2. Comparative performance of TA, TA-LSP, VNS, and TA-VNS methods for large problem sizes. Discrepancy differences remain generally low with a noticeable peak around n ≈ 400 , while computational time increases sharply with n . TA is the fastest method, whereas hybrid approaches, particularly TA-VNS, provide competitive accuracy at the expense of higher computational cost.

Figure 3 provides a comprehensive comparison of the behavior of TA, TA-LSP, VNS, and TA-VNS methods across moderate problem sizes, considering both solution quality and computational efficiency. Figures 3(a)-(c) display the differences in wrap-around discrepancy, revealing generally low values for small and intermediate sizes, which indicates that all methods produce solutions of comparable quality in these regimes. However, noticeable peaks emerge at specific sizes (particularly around n ≈ 90   -   120 ), where discrepancies increase sharply, suggesting that certain configurations are more challenging and may induce instability or sensitivity in the optimization process. Beyond these points, the discrepancies tend to decrease again and stabilize, highlighting improved consistency of the methods. Figure 3(c) further emphasizes a higher variability between VNS and TA-VNS, indicating that the hybridization effect can significantly modify solution quality depending on the problem structure. Figure 3(d) and Figure 3(e) show that computational time increases steadily with n for all methods, with a more pronounced acceleration for larger sizes. TA remains the most efficient method in terms of runtime, followed by TA-LSP, while TA-VNS and VNS require substantially more computational effort, especially as n approaches 800 - 1000. Overall, the figure illustrates a clear trade-off between computational cost and potential improvements in solution refinement, with hybrid methods such as TA-VNS offering enhanced performance at the expense of increased execution time.

Figure 3. Comparative performance of TA, TA-LSP, VNS, and TA-VNS for moderate problem sizes. Discrepancy differences remain low overall, with peaks around n ≈ 90   -   120 , while computational time increases with n . TA is the most efficient method, whereas hybrid approaches improve solution quality at a higher computational cost.

Table 3 summarizes the wrap-around discrepancies ( W D 2 ) and computation times obtained using TA, VNS, TA-LSP, and TA-VNS. In general, hybrid methods yield smaller discrepancies and slightly higher computation times, indicating improved uniformity.

Table 3. Wrap-around discrepancies for selected designs.