As technology and human knowledge have advanced, industrial and scientific investigators attempt to solve large-scale complex optimization problems that mostly fall in the category of NP-hard. The complexity comes not only from the high-dimensional solution domain that tests the computational capacity of hardware and software but also from cross-dimensional constraints. Since most traditional optimization methods are inefficient, if not infeasible, for these large-scale problems in today's real world, researchers search for new algorithms that can balance efficiency and accuracy. Metaheuristic algorithms sacrifice a part of accuracy to pursue extra efficiency and provide reasonable solutions to optimization problems using adequate resources. These algorithms typically sample a subset of the solution space that is too large to be completely enumerated. Metaheuristic algorithms can also handle multi-dimensional real-values problems without relying on the gradient of the objective functions, which enables them to search over solution spaces that are non-continuous, noisy, or changing over time.

As its primary class, nature-inspired metaheuristics can be further categorized into two main categories. Evolutionary algorithms, such as Genetic Algorithm (GA) (Goldberg, 2003), Genetic Programming (GP) (Cramer, 1985), Differential Evolution (DE) (Storn and Price, 1997), and many others, are inspired by Darwin's evolutional theory that allows only those with the fittest characteristics to survive in a competition among species members when individual variations randomly occur and are hereditary. On the other hand, swarm algorithms, such as Ant Colony Optimization (ACO) (Dorigo and Gambardella, 1997), Artificial Bee Colony (Dervis and Basturk, 2007), Particle Swarm Optimization (PSO) (Kennedy and Eberhart, 1995; Kennedy, 2010), Swarm Intelligence Based (SIB) method (Phoa et al., 2016; Phoa, 2017), and many others, mimic the collective behavior of self-organized and decentralized systems for characteristics improvements. Among all, PSO is one of the most representative swarm intelligence algorithms in engineering problems and some scientific research in the past decades. With well-defined physical meanings, it efficiently tackles high-dimensional optimization problems in continuous solution spaces, but it may not be the first choice for non-continuous solution spaces that commonly appear in mathematics, statistics, and many other fields, even via the remedy of a simple round-off (Kim et al., 2010). As a result, the SIB method (Phoa, 2017) was proposed for this manner with a wide range of applications, including the constructions of optimal experimental designs (Phoa et al., 2016), the uniform distribution of testing points (Phoa and Chang, 2016; Huang and Phoa, 2023), supercomputing scheduling (Lin and Phoa, 2019), hot spot determination (Hsu and Phoa, 2018), traveling salesman problem (Yen and Phoa, 2021), and many others.

First introduced in 1982, the concept of supply chain management (SCM) is to organize the flow of goods and services, including all processes from obtaining raw materials to delivering final products to customers. As the connection between suppliers and customers, SCM aims to minimize total costs within a supply chain and maximize a company's net profits. It increases a company's profit and gains its competitive advantage over the market. More about the basics of SCM are referred to Fredendall and Hill (2001) and Mentzer et al. (2001). Nonetheless, there is a growing gap between advanced optimization techniques and real applications in SCM. Even though advanced methods have been available in computer science and engineering for decades, many applications still use traditional optimization methods such as linear programming (Delloite, 2022). Without advanced techniques that can significantly reduce the computational cost, researchers in SCM confront difficulty in developing large-scale data analysis systems for optimizing multi-supplier selling schemes, a many-to-many network where products are delivered directly from multiple suppliers to multiple customers. Moreover, optimization in the SCM usually suffers from both the high-dimensional solution domain and cross-dimensional constraints. Without the latter constraint, the former complexity may simply be solved by decomposing the high-dimensional solution space into multiple low-dimensional ones and then optimizing each low-dimensional space once at a time. The existence of cross-dimensional constraints breaks the independency assumption among the decomposed low-dimensional domains; thus, the simple divide-and-conquer approach is no longer applicable for simplifying the problem.

This work introduces a metaheuristic optimization method via swarm intelligence that can solve high-dimensional problems and deal with cross-dimensional constraints simultaneously. Section 2 briefly reviews three nature-inspired metaheuristic optimization techniques related to our work. Section 3 introduces the implementation details of the SIB method for the optimization problem of multi-supplier selling schemes. Then, the proposed method is evaluated with several simulated supply chains on different scales and compared with the Genetic Algorithm (GA) in Section 4. Finally, some conclusions are in the last section.

As a popular selling strategy in E-commerce nowadays, multi-supplier selling is a process of selling products from many suppliers to many customers. This strategy is a kind of direct sales with no dealers or intermediaries in the selling scheme, and products are sold and delivered to the customers directly from the suppliers. A direct sale market structure not only increases suppliers' profits and decreases the prices of products for customers but also simplifies the complexity of the target optimization problem. However, even with a direct sale, the optimization task on multi-supplier selling schemes is still complicated due to high-dimensional solution space and cross-dimensional constraints. Thus, the SIB method is chosen to solve this optimization problem.

Denote a selling scheme with N customers, K product types, and M suppliers, and a tensor X with dimensions N×K×M to represent this selling scheme. Figure 1A illustrates the definition of a particle while the C-, S-, and P-axes correspond to customers, suppliers, and product types. Each entry x_nkm indicates the number of the kth product sold to the nth customer by the mth supplier, and each column (Figure 1B) with K entries indicates the selling scheme between a customer and a supplier.

The following assumptions are made for this work. First, the quantities of supply and demand are known in advance, and the supply is less than the demand to create a more challenging situation where prescriptive analytics is needed. Second, no further complications on resale, buy-back, or others exist. Third, customers are willing to purchase an identical product from multiple suppliers. Finally, each customer pays a constant price for each product, called the willingness to pay in economics. In real markets, a shortage in supply, which is the first assumption, is likely to make customers pay the maximum prices they are willing to pay.

In a SCM task, the objective function is generally the profit that a selling scheme can make, equal to the difference between sales and costs. The sales component is the products' prices multiplied by their quantities. Among all potential costs, this work only considers delivery and purchase costs. To calculate the delivery cost, participants of a supply chain are categorized into two geographical locations: North and South. The transportation cost per product between a specific combination of locations is a constant. The purchase cost per product from a specific supplier is also a constant. Mathematically, the objective function can be written as

A supply constraint is that a supplier must have a maximum production capacity, and a demand constraint is that a customer must have the desired quantity for each product. Both constraints are cross-dimensional in nature.

Table 1 is the pseudo-code of the proposed SIB algorithm for the SCM optimization. In the initialization part, a set of valid selling schemes (particles) are randomly generated and evaluated by the objective function. Then, the initial LBs are defined as the initial positions, and the GB is the best among all LBs. In the iteration part, the MIX operation generates new selling schemes by mixing current selling schemes with the best ones, and the MOVE operation picks the best among three candidate schemes for update if the newly generated schemes provide better objective function values, or a perturbed scheme via random jump otherwise. The iteration continues until the pre-defined stopping criteria are fulfilled, which can be the maximum number of iterations, achieving a pre-defined objective function value, or converging with pre-defined rules. The final GB is considered the optimal multi-supplier selling scheme suggested by the proposed SIB algorithm. The following subsections provide a more detailed description of every step of the SIB method.

The proposed SIB method has been applied to the small single-supplier and large multi-supplier supply chains (Phoa et al., 2021). This section evaluates the SIB method with supply chains in different scales and compares the performance with the GA algorithm. Moreover, the effect of the parallelization technique will be illustrated by showing the execution time.

To test the algorithm's general performance, supply chains' configurations in 11 different scales, including one-to-many scale, are used to set up several experiments; the detailed settings of the scales are listed in Table 2. For each scale, ten configurations are manually generated, containing different demand and supply constraints and different geographical locations for suppliers and customers. Specifically, there are some simulated supply chains in the egg market. In addition to the category and location details of suppliers and customers, the data consists of the supply amount of each supplier, the demand amount of each customer, the type of product that each supplier supplies, the cost of purchasing products from suppliers, the transport cost per mile for each egg, and product prices that are different among customers.

A specific metric is used to evaluate the optimization results and to compare the results among different configurations and scales. Singh et al. (2022) defined an improvement multiplier that can measure the progress of an algorithm from the initial random values to the end of the iterations. The following equation can calculate the improvement multiplier:

where obj_f is the objective function value of the optimized outcome, and obj₀ is the objective value of the best individual among all the initial particles.

In the first experiment, ten initial particle sets are generated based on each configuration and used as initial swarms in the GA and SIB algorithms for a fair comparison. Each algorithm is run for 300 iterations without stopping early, and the performance of the two algorithms is compared at different iterations. Table 3 shows the optimized results of two algorithms on each scale after 100, 200, and 300 iterations. According to the results, the SIB method outperforms the GA in all scales. The mean of the improvement given by the SIB is a multiple of that given by the GA, like over 1.7x, 2.5x, and 1.8x respectively in Configs. 1, 2, and 4. Furthermore, the SIB method improves faster than the GA in the early stage and converges. The means of the improvement multipliers given by the SIB in all the scales have already outperformed the GA's results after 100 steps and become stable in the rest of the steps. In contrast, the GA results start at a lower level and keep increasing until the 200th step or even the 300th step. The profit progress trend of one experiment on each scale is visualized in Figure 2, which shows our observations in an easy-understanding way.

The second experiment tests the effect of parallel computing for large data scales. Two versions of the SIB method are implemented, one with the parallelization techniques and one without. Three configurations, No. 1, No. 4, and No. 8, are chosen to test the effect's difference in data scales. The experiments are individually run on a server with 104 cores. Table 4 summarizes the computation time in seconds. With a significantly smaller mean, the parallelization helps reduce the computing time in all three configurations. While the computing time only reduces to one-third of the time without parallel on configuration No. 1, the parallelization reduces the computing time to almost one-fifth of the time without parallel on configuration No. 8. This may result from the complexity of the computation in terms of the particle size. Since parallelization costs additional time to copy data from one CPU to another CPU, it is worthier to use this technique in experiments with higher computation complexity.

This paper proposes a SIB method to handle the multi-supplier-multi-customer supply chain management problem. A modified MIX operation is designed to handle the high-dimensional solutions; the remaining quantity matrices that store additional information help to handle cross-dimensional constraints. Moreover, parallelization techniques accelerate the program to obtain the desired results within a reasonable time. The experiments show that the SIB method, compared to the GA method, offers better-optimized solutions in a shorter time. One must consider more practical factors if this method is applied. For example, prices and costs in the real world vary based on the quantities of demand and supply, and they require a predictive marketing model before optimization. Moreover, the proposed method is not limited only to the SCM optimization problems, but also to similar optimization problems with high-dimensional domains and cross-dimensional constraints.

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

H-PL: Formal analysis, Methodology, Software, Visualization, Writing – original draft. FP: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. Y-HC-B: Conceptualization, Data curation, Investigation, Project administration, Resources, Writing – review & editing. S-PL: Conceptualization, Investigation, Project administration, Resources, Writing – review & editing.